--- /srv/rebuilderd/tmp/rebuilderd9mwklS/inputs/macaulay2-common_1.24.11+ds-5_all.deb +++ /srv/rebuilderd/tmp/rebuilderd9mwklS/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 504876 2025-02-09 22:54:37.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 29427704 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) 6498 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) 4651 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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22:54:37.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5370 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5385 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4475 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4211 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2939 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./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/_make__B_sq__Slice.html │ │ │ @@ -3789,19 +3789,19 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2151 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1447 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Chain__Complex__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 578 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Exact.out │ │ │ -rw-r--r-- 0 root (0) root (0) 721 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Exact_lp..._cm__Length__Limit_eq_gt..._rp.out │ │ │ -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) 1960 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 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│ │ -rw-r--r-- 0 root (0) root (0) 7229 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_is__Quasi__Isomorphism.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9284 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_is__Quasi__Isomorphism_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5254 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7099 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9459 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9457 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6219 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/ │ │ │ --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) 4376 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) 1502 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) 2013 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) 4377 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 759 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 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 624 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 623 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1049 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Multi__Proj__Coord__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Output.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3297 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Segre.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Toric__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_is__Multi__Homogeneous.out │ │ │ -rw-r--r-- 0 root (0) root (0) 759 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_probabilistic_spalgorithm.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 783 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 22815 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5958 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22816 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5957 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10706 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Toric__Variety__Valid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17471 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chern.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9130 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6126 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6952 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Toric__Chow__Ring.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9704 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17878 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5477 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9703 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17877 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5476 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5575 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6056 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6055 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8040 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Multi__Proj__Coord__Ring.html │ │ │ -rw-r--r-- 0 root (0) root 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22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_cartesian__Code.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 12347 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 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_choose__Strat.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_codewords.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3134 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_cyclic__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 382 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_cyclic__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1326 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_dim_lp__Linear__Code_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 700 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_dual__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/example-output/_enumerate__Vectors.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 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./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) 24584 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 │ │ │ @@ -4158,23 +4158,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 37688 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27376 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21171 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 13414 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 13261 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13260 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1045 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg_lp__Normal__Toric__Variety_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 97 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3605 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/___Silent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9802 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8109 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24175 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24174 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5805 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3702 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 107581 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1103 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/___Coincident__Root__Locus_sp_st_sp__Coincident__Root__Locus.out │ │ │ @@ -4289,16 +4289,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1851 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 856 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1286 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10481 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_new__Ext.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1273 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_odd__Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_regularity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1285 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_splittings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 351 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ --rw-r--r-- 0 root (0) root (0) 419 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 350 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 420 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5846 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___A__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4714 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Augmentation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5949 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__G__G__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8657 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5781 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Check.html │ │ │ @@ -4351,18 +4351,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10523 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_module__As__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22129 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_new__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8318 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_odd__Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5737 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_psi__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6734 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_regularity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7009 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_splittings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5046 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_stable__Hom.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5790 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5789 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5406 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_tensor__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4691 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_to__Array.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6130 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6131 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 57456 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 36450 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17259 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Complexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Complexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 717340 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Complexes/example-output/ │ │ │ @@ -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 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2310 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Codim__Bs__Inv.out │ │ │ --rw-r--r-- 0 root (0) root (0) 19796 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out │ │ │ --rw-r--r-- 0 root (0) root (0) 524 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1793 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 19793 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 527 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1794 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2581 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6044 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out │ │ │ --rw-r--r-- 0 root (0) root (0) 42548 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7622 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6047 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 42547 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1470 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_coefficients_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 33238 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_describe_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 515 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_exceptional__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1047 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_flatten_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_force__Image.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6560 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6561 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_graph_lp__Ring__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4858 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_ideal_lp__Rational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 11551 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4856 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_ideal_lp__Rational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 11550 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 303 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 43387 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out │ │ │ 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./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6113 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1086 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_map_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 18541 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_parametrize_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1461 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4714 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4713 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2351 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_quadro__Quadric__Cremona__Transformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3884 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2826 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2825 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5782 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Polynomial__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6396 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ring_cm__Tally_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2775 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./usr/share/doc/Macaulay2/Divisor/example-output/_dualize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1174 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_embed__As__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 500 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_find__Element__Of__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_gbs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 831 2025-02-09 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) 4355 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/example-output/_reflexify.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1094 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) 11562 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) 19079 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Divisor/html/_reflexify.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9038 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 │ │ │ @@ -5976,22 +5976,22 @@ │ │ │ -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) 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) 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) 890 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) 7548 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.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) 7549 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/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/dump/ │ │ │ @@ -6181,35 +6181,35 @@ │ │ │ -rw-r--r-- 0 root (0) 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./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2168 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) 10965 2025-02-09 22:54:37.000000 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(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) 1276 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 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_insert_lp__Priority__Queue_cm__Thing_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 248 2025-02-09 22:54:37.000000 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./usr/share/doc/Macaulay2/FastMinors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./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 │ │ │ @@ -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 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Frobenius__Thresholds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 793 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Guess__Strategy.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_compare__F__P__T.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4033 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2458 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4034 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2459 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 760 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__Jumping__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 552 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__P__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 828 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__Simple__Normal__Crossing.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 12 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5150 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Bounds.html │ │ │ @@ -6728,16 +6728,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4657 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Global__Frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10043 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Guess__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4336 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Return__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4440 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Search.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4429 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Standard__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5625 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Use__Special__Algorithms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14109 2025-02-09 22:54:37.000000 ./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) 25102 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23558 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 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 339 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 732 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 170 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_normal__Toric__Variety_lp__G__K__M__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8072 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8068 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pullback_lp__Equivariant__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 615 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pushforward.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_set__Indicator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 437 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_trivial__K__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 401 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_underlying__Graph_lp__Moment__Graph_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/ │ │ │ @@ -6840,15 +6840,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7594 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8106 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8970 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8163 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 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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 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Constraints.out │ │ │ -rw-r--r-- 0 root (0) root (0) 598 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Vector.out │ │ │ -rw-r--r-- 0 root (0) root (0) 30368 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2709 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Stratum.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_linear__Part.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13718 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) 15160 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) 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) 15154 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) 23345 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 │ │ │ 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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 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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 │ │ │ @@ -8200,18 +8200,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 466 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Frac__P_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 707 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Fractions.out │ │ │ -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) 28090 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 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./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/ │ │ │ @@ -8235,18 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22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_is__Invariant.out │ │ │ @@ -8327,22 +8327,22 @@ │ │ │ -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-- 0 root (0) root (0) 6513 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_group.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6196 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_group__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9791 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hilbert__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6631 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hilbert__Series_lp__Ring__Of__Invariants_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11357 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hironaka__Decomposition.html │ │ │ --rw-r--r-- 0 root (0) root (0) 22456 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22457 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8582 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariant__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11283 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8577 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8578 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14134 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5840 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Subring__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7933 2025-02-09 22:54:37.000000 ./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) 8675 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 │ │ │ -rw-r--r-- 0 root (0) root (0) 10083 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Linearly__Reductive__Action_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10871 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Linearly__Reductive__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6908 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_is__Abelian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11810 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_is__Invariant.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 │ │ │ @@ -8515,22 +8515,22 @@ │ │ │ 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) 2260 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) 3424 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) 1003 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) 267 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) 6874 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 │ │ │ @@ -8543,22 +8543,22 @@ │ │ │ -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) 9471 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) 10204 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) 6618 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6589 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) 14514 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 │ │ │ @@ -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) 595 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) 7427 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 │ │ │ @@ -8923,30 +8923,30 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 60434 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/___Linear__Truncations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 231 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_comp__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_diagonal__Multidegrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Mins.out │ │ │ --rw-r--r-- 0 root (0) root (0) 701 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 700 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ -rw-r--r-- 0 root (0) root (0) 754 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1046 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Linear__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 904 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Quasi__Linear.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1150 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 976 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_multigraded__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_partial__Regularities.out │ │ │ -rw-r--r-- 0 root (0) root (0) 577 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_regularity__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_support__Of__Tor.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 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│ │ │ -rw-r--r-- 0 root (0) root (0) 6873 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7719 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_is__Linear__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8763 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_is__Quasi__Linear.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7953 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8029 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10218 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_multigraded__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8571 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_partial__Regularities.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 │ │ │ @@ -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 │ │ │ @@ -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 │ │ │ @@ -9496,15 +9496,15 @@ │ │ │ -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 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compact__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compare__Exchange.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_complete_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compose.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compositions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compress.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 │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cot.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_coth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 297 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cover__Map_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 521 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cover_lp__Module_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 317 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 314 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 243 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_create__Task.out │ │ │ -rw-r--r-- 0 root (0) 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) 21212 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 │ │ │ @@ -9783,15 +9783,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 565 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Mutable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Open.out │ │ │ -rw-r--r-- 0 root (0) root (0) 290 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Output__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Prime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Primitive.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1881 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1880 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ready_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 115 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Real.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 460 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Skew__Commutative.out │ │ │ @@ -9888,32 +9888,32 @@ │ │ │ -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 │ │ │ -rw-r--r-- 0 root (0) root (0) 310 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 632 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min_lp__Graded__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 925 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Groebner__Basis_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2181 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 737 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingle.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1805 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1804 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 643 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimize__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 504 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minors_lp__Z__Z_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out │ │ │ @@ -9988,15 +9988,15 @@ │ │ │ -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) 1736 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) 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 │ │ │ @@ -10035,15 +10035,15 @@ │ │ │ -rw-r--r-- 0 root (0) 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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 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_push__Forward_lp__Ring__Map_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 796 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_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 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_saving_sppolynomials_spand_spmatrices_spin_spfiles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 369 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_scan.out │ │ │ -rw-r--r-- 0 root (0) root (0) 295 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_scan__Keys.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_scan__Pairs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 301 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_scan__Values.out │ │ │ --rw-r--r-- 0 root (0) root (0) 351 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_schedule.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 331 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_schedule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1372 2025-02-09 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 │ │ │ @@ -10226,15 +10226,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 595 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_tests.out │ │ │ -rw-r--r-- 0 root (0) root 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 186 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 141 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__Absolute__Path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 296 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__C__C.out │ │ │ -rw-r--r-- 0 root (0) root (0) 428 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__External__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 593 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__Field_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 94 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__Lower.out │ │ │ @@ -10276,15 +10276,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_value_lp__Symbol_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 201 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_values.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_variables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 770 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__Monoid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 759 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 842 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_what_spa_spclass_spis.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_what_spis_spa_sp__Groebner_spbasis_qu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 754 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_while.out │ │ │ -rw-r--r-- 0 root (0) root (0) 133 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_width_lp__Net_rp.out │ │ │ @@ -10346,15 +10346,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3058 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___B__L__A__S.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4070 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Bag.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3937 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Bareiss.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4358 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Base__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3683 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Base__Row.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18195 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Basic__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8418 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Basic__List_sp_sh_sp__Z__Z.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10186 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) 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 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896 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_mixed__Multiplicity.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2839 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2838 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal_lp..._cm__Variable__Base__Name_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 229 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_sec__Milnor__Numbers.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 762 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7458 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_hom__Ideal__Polytope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6808 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_hom__Ideal__Polytope_lp..._cm__Coefficient__Ring_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7988 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_m__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9676 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_mixed__Multiplicity.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12028 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12027 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9050 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal_lp..._cm__Variable__Base__Name_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7453 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_sec__Milnor__Numbers.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12575 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8290 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4949 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/dump/ │ │ │ @@ -12944,15 +12944,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver__Options.out │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 243 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 389 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_compute__Mixed__Volume.out │ │ │ -rw-r--r-- 0 root (0) root (0) 378 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_create__Seed__Pair.out │ │ │ --rw-r--r-- 0 root (0) root (0) 938 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 944 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9539 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out │ │ │ -rw-r--r-- 0 root (0) root (0) 398 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_cm__Abstract__Point_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1366 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 956 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__E.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__Lower__Bound.out │ │ │ @@ -12967,15 +12967,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5593 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/___Point__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4288 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4340 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4476 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4298 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4941 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_compute__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7542 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_create__Seed__Pair.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7425 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7431 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4610 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4271 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4697 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6107 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_homotopy__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4409 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_indices_lp__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4667 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_is__Member_lp__Abstract__Point_cm__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4565 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_length_lp__Point__Array_rp.html │ │ │ @@ -13326,15 +13326,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 507 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_eq_eq_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 390 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 905 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1341 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1277 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1273 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1200 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_lt_lt_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 693 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1848 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 772 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 803 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 763 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 744 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multirational__Map.out │ │ │ @@ -13353,42 +13353,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_coefficient__Ring_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3165 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cone__Of__Lines.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_conormal__Variety_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cycle__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 671 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_decompose_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 158 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1006 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1007 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 428 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degrees_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 722 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2263 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2265 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 155 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dim_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 482 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dual_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 809 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_entries_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 239 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_euler_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2030 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_factor_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1459 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_fiber__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 673 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_force__Image_lp__Multirational__Map_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1858 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_hilbert__Polynomial_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 384 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_ideal_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1018 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1017 2025-02-09 22:54:37.000000 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(0) 386 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Member_lp__Multirational__Map_cm__R__A__T_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 754 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 752 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Subset_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Well__Defined_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2025-02-09 22:54:37.000000 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root (0) 665 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2393 2025-02-09 22:54:37.000000 ./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 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./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10297 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Nauty/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6820 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Nauty/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13765 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Nauty/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7669 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Nauty/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7978 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Nauty/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13768 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14129 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Graphs.html │ │ │ @@ -13974,15 +13974,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9031 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Nauty/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 131317 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 493 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1406 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1405 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 826 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1498 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 124 2025-02-09 22:54:37.000000 ./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 │ │ │ @@ -13996,15 +13996,15 @@ │ │ │ -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) 8036 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 │ │ │ @@ -14220,18 +14220,18 @@ │ │ │ -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) 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) 1002 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) 1965 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 │ │ │ -rw-r--r-- 0 root (0) root (0) 1640 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1605 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1337 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_polytope_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2219 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_projective_spspace.out │ │ │ @@ -14335,18 +14335,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9495 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_making_spnormal_sptoric_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11122 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12037 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10687 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_matrix_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8626 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_max_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9957 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10393 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_nef__Generators_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10124 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10125 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18104 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13133 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Matrix_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12903 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12905 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9355 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11308 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10177 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11400 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10031 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10082 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_polytope_lp__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10169 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_projective_spspace.html │ │ │ @@ -14654,46 +14654,46 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 40015 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24975 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6785 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 146171 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1170 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1178 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 578 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Numerical__Interpolation__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1600 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Pseudo__Witness__Set.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1626 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1627 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_is__On__Image.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1348 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1333 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Degree.out │ │ │ --rw-r--r-- 0 root (0) root (0) 702 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 703 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 712 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Sample.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Nullity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1569 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Source__Sample.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1298 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_pseudo__Witness__Set.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1180 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_real__Point.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1181 2025-02-09 22:54:37.000000 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│ -rw-r--r-- 0 root (0) root (0) 5143 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Free__O__I__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5379 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Module__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5462 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5110 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Polynomial__O__I__Algebra_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5467 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8059 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html │ │ │ @@ -15628,15 +15628,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11755 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Parsing/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9312 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Parsing/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8801 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Parsing/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 138371 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4679 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4680 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1184 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_centers.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11148 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_ci__Module__To__Clifford__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1537 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_ci__Module__To__Matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 766 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_clifford__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1561 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_clifford__Module__To__C__I__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 860 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_clifford__Module__To__Matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 864 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_clifford__Operators.out │ │ │ @@ -15665,15 +15665,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1735 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_tensor__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 584 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_translate__Isotropic__Subspace.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_vector__Bundle__On__E.out │ │ │ -rw-r--r-- 0 root (0) root (0) 484 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_y__Action.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 47 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6946 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Clifford__Module.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13008 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13009 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4702 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Random__Nice__Pencil.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7140 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Vector__Bundle__On__E.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5067 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_base__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9720 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_centers.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19227 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_ci__Module__To__Clifford__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10434 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_ci__Module__To__Matrix__Factorization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8515 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_clifford__Module.html │ │ │ @@ -15962,15 +15962,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 14441 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12451 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5274 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 47379 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1729 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1730 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 619 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Make__Ring__Maps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1456 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1010 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 463 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points__Mat.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_expected__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 234 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Points/example-output/_min__Max__Resolution.out │ │ │ @@ -15982,15 +15982,15 @@ │ │ │ -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) 592 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) 7920 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) 21987 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) 45477 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) 1084 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) 7701 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/ │ │ │ @@ -17169,21 +17169,21 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4077 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2789 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50294 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 237 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_is__Smooth__Curve.out │ │ │ --rw-r--r-- 0 root (0) root (0) 400 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 399 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5792 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/___Details.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5845 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/___Printing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5191 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_is__Smooth__Curve.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8664 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8663 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7558 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve__Genus14.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7304 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve__Genus15.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7395 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve__Via__Plane__Model.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7294 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve__Via__Space__Model.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10904 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10388 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5194 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/toc.html │ │ │ @@ -17207,33 +17207,33 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6337 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4337 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 85886 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Finding_sp__Extreme_sp__Examples.out │ │ │ --rw-r--r-- 0 root (0) root (0) 410 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 429 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__Chain__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__From__Shelling.out │ │ │ -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) 251 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) 8771 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 │ │ │ @@ -17242,28 +17242,28 @@ │ │ │ -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) 5987 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) 17522 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) 25818 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 │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 151319 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/___C__M__Stats.out │ │ │ @@ -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) 4962 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 ./usr/share/doc/Macaulay2/RationalMaps/html/___Rational__Mapping_sp_st_sp__Rational__Mapping.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5003 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/___Rees__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4351 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/___Saturate__Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4924 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/___Saturation__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5154 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/___Simis__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8821 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_base__Locus__Of__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8617 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_ideal__Of__Image__Of__Map.html │ │ │ --rw-r--r-- 0 root (0) root (0) 19021 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) 19023 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 ./usr/share/doc/Macaulay2/RationalMaps/html/_jacobian__Dual__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7810 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_map__Onto__Image.html │ │ │ @@ -17517,26 +17517,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1462 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/___Rational__Points2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1247 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_base__Change.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_charpoly.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1084 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_ext__Field.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_global__Height.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_hermite__Normal__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 361 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_integers.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4102 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4103 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1244 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_zeros.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5366 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/___Projective__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9909 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_base__Change.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6299 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_charpoly.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11810 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_ext__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5532 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_global__Height.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5560 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_hermite__Normal__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5896 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_integers.html │ │ │ --rw-r--r-- 0 root (0) root (0) 20408 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20409 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7426 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_zeros.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20061 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16030 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5070 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ReactionNetworks/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ReactionNetworks/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 93186 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ReactionNetworks/dump/rawdocumentation.dump │ │ │ @@ -17689,15 +17689,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_is__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3424 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_jacobian__Dual.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1022 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_minimal__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_minimal__Reduction_lp..._cm__Tries_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 255 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 943 2025-02-09 22:54:37.000000 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./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) 16084 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 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./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) 7311 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/ │ │ │ @@ -18290,30 +18290,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 66470 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 47495 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/master.html │ │ │ -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) 2825 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) 11393 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) 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) 1859 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) 25841 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/ │ │ │ @@ -18394,15 +18394,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1688 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Generation_spof_spformulas.out │ │ │ -rw-r--r-- 0 root (0) root (0) 902 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Grassmannian_spof_splines_spin_sp__P3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 639 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Hilbert_sppolynomial_spand_sp__Todd_spclass_spof_spprojective_sp3-space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 803 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Lines_spon_spa_spquintic_spthreefold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 511 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Riemann-__Roch_spformulas.out │ │ │ -rw-r--r-- 0 root (0) root (0) 569 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__The_spnumber_spof_spelliptic_spcubics_spon_spa_spsextic_sp4-fold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 247 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1587 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1585 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Abstract__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 913 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Quotient__Bundles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spcurve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1741 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spsurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7722 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spwithout_spdenominators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Ring_sp_us_sp__Chern__Class__Variable.out │ │ │ @@ -18514,15 +18514,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5409 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Examples_spfrom_sp__Schubert_cm_sptranslated.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8828 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Flag__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5881 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5438 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Ideas_spfor_spfuture_spdevelopment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6052 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Incidence__Correspondence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3872 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Intersection__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3899 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Isotropic.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7598 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7596 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4926 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Abstract__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5798 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3997 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Pull__Back.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6241 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Quotient__Bundles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4677 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spcurve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7216 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spsurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11353 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spwithout_spdenominators.html │ │ │ @@ -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 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./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 │ │ │ @@ -19152,15 +19152,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 597 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│ │ │ --rw-r--r-- 0 root (0) root (0) 623 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_rehomogenize__Polynomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 619 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_rehomogenize__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1011 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_set__Ones__Forest.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6891 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_slack__From__Gale__Circuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 852 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_slack__From__Gale__Plucker.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_slack__From__Plucker.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1105 2025-02-09 22:54:37.000000 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│ │ │ -rw-r--r-- 0 root (0) root (0) 14641 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Gale__Circuits.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5147 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Gale__Circuits_lp..._cm__Tolerance_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7943 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Gale__Plucker.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8534 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Plucker.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7226 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Plucker_lp..._cm__Object_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11162 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__Ideal.html │ │ │ @@ -19345,18 +19345,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1122 2025-02-09 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22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1360 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 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_reverse__Shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_ring_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 457 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2104 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sort__Shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 12400 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Discriminant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 55253 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 55247 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ -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 ./usr/share/doc/Macaulay2/SparseResultants/html/_char_lp__Sparse__Discriminant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5035 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_char_lp__Sparse__Resultant_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6090 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6089 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8038 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8081 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8502 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ +-rw-r--r-- 0 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./usr/share/doc/Macaulay2/SparseResultants/html/_exponents_lp__Sparse__Resultant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7474 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_flattening.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7539 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_generic__Laurent__Polynomials.html │ │ │ @@ -19413,15 +19413,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6390 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_random__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8042 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_rank_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7378 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_reverse__Shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5354 2025-02-09 22:54:37.000000 ./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) 65060 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 │ │ │ @@ -19464,15 +19464,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 209 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_num__Columns_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 209 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_num__Rows_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 445 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_permutation__Matrix_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 389 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_permutation__Sign.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_permute__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 297 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_power__Sum__Symmetric__Polynomials_lp__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_reading__Word_lp__Young__Tableau_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1936 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1935 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 410 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_row__Permutation__Tableaux_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 335 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_row__Stabilizer_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 563 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_schur__Polynomial_lp__List_cm__Partition_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 45653 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1508 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_semistandard__Tableaux_lp__Partition_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 206 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_size_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 603 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_sort__Columns__Tableau_lp__Specht__Module__Element_rp.out │ │ │ @@ -19528,15 +19528,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4816 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_num__Columns_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4817 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_num__Rows_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5156 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_permutation__Matrix_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 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./usr/share/doc/Macaulay2/SpechtModule/html/_row__Permutation__Tableaux_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5539 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_row__Stabilizer_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7260 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_schur__Polynomial_lp__List_cm__Partition_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 53336 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7035 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_semistandard__Tableaux_lp__Partition_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 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 │ │ │ @@ -19559,15 +19559,15 @@ │ │ │ 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 │ │ │ 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22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 38469 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34410 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16670 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SpectralSequences/ │ │ │ @@ -20168,15 +20168,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 212 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_no__Packed__All__Subs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_no__Packed__Sub.out │ │ │ -rw-r--r-- 0 root (0) root (0) 174 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_squarefree__Gens.out │ │ │ -rw-r--r-- 0 root (0) root (0) 169 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_squarefree__In__Codim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 651 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symb__Power__Prime__Pos__Char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Defect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Polyhedron.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1211 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1209 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power__Join.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_waldschmidt.out │ │ │ -rw-r--r-- 0 root (0) root (0) 200 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_waldschmidt_lp..._cm__Sample__Size_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 15 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4318 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/___A_spquick_spintroduction_spto_spthis_sppackage.html │ │ │ @@ -20209,15 +20209,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5721 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_no__Packed__All__Subs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5455 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_no__Packed__Sub.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5447 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_squarefree__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5595 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_squarefree__In__Codim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6225 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symb__Power__Prime__Pos__Char.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6326 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Defect.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5767 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Polyhedron.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9294 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9292 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5595 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power__Join.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7627 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_waldschmidt.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6742 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_waldschmidt_lp..._cm__Sample__Size_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21847 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20336 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12449 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/SymmetricPolynomials/ │ │ │ @@ -20505,23 +20505,23 @@ │ │ │ -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) 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) 487 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) 1823 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) 1491 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 │ │ │ @@ -20556,23 +20556,23 @@ │ │ │ -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) 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) 7827 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) 14965 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) 15662 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 │ │ │ @@ -20765,31 +20765,31 @@ │ │ │ -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) 7070 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) 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) 934 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2069 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) 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) 6605 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12349 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) 21003 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 │ │ │ @@ -22293,23 +22293,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 38831 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/WeylGroups/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 44599 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/WeylGroups/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/ │ │ │ drwxr-xr-x 0 root (0) root 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22:54:37.000000 ./usr/share/info/SpechtModule.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 31902 2025-02-09 22:54:37.000000 ./usr/share/info/SpecialFanoFourfolds.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 290127 2025-02-09 22:54:37.000000 ./usr/share/info/SpectralSequences.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 12259 2025-02-09 22:54:37.000000 ./usr/share/info/StatGraphs.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 12265 2025-02-09 22:54:37.000000 ./usr/share/info/StatGraphs.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2421 2025-02-09 22:54:37.000000 ./usr/share/info/StatePolytope.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7566 2025-02-09 22:54:37.000000 ./usr/share/info/StronglyStableIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2025-02-09 22:54:37.000000 ./usr/share/info/Style.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 28990 2025-02-09 22:54:37.000000 ./usr/share/info/SubalgebraBases.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 15675 2025-02-09 22:54:37.000000 ./usr/share/info/SumsOfSquares.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5073 2025-02-09 22:54:37.000000 ./usr/share/info/SuperLinearAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2471 2025-02-09 22:54:37.000000 ./usr/share/info/SwitchingFields.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 13586 2025-02-09 22:54:37.000000 ./usr/share/info/SymbolicPowers.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 13585 2025-02-09 22:54:37.000000 ./usr/share/info/SymbolicPowers.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1898 2025-02-09 22:54:37.000000 ./usr/share/info/SymmetricPolynomials.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13753 2025-02-09 22:54:37.000000 ./usr/share/info/TSpreadIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1313 2025-02-09 22:54:37.000000 ./usr/share/info/TangentCone.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 46335 2025-02-09 22:54:37.000000 ./usr/share/info/TateOnProducts.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 21989 2025-02-09 22:54:37.000000 ./usr/share/info/TensorComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2065 2025-02-09 22:54:37.000000 ./usr/share/info/TerraciniLoci.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 30208 2025-02-09 22:54:37.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 30211 2025-02-09 22:54:37.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14210 2025-02-09 22:54:37.000000 ./usr/share/info/Text.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 21596 2025-02-09 22:54:37.000000 ./usr/share/info/ThinSincereQuivers.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8198 2025-02-09 22:54:37.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 7958 2025-02-09 22:54:37.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13497 2025-02-09 22:54:37.000000 ./usr/share/info/Topcom.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7287 2025-02-09 22:54:37.000000 ./usr/share/info/TorAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3783 2025-02-09 22:54:37.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3067 2025-02-09 22:54:37.000000 ./usr/share/info/ToricTopology.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 30590 2025-02-09 22:54:37.000000 ./usr/share/info/ToricVectorBundles.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 9253 2025-02-09 22:54:37.000000 ./usr/share/info/TriangularSets.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 13213 2025-02-09 22:54:37.000000 ./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) 17995 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) 31781 2025-02-09 22:54:37.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 5860 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 │ │ │ +-rw-r--r-- 0 root (0) root (0) 48525 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 │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/Macaulay2/Style/katex/contrib/render-a11y-string.min.js -> ../../../../javascript/katex/contrib/render-a11y-string.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-02-09 22:54:37.000000 ./usr/share/Macaulay2/Style/katex/fonts/KaTeX_AMS-Regular.ttf -> ../../../../fonts/truetype/katex/KaTeX_AMS-Regular.ttf │ │ ├── ./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 │ │ │ + -- 1.45904s 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 │ │ │ + -- 1.71631s 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
│ │ │ + -- 1.45904s 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
│ │ │ + -- 1.71631s 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 │ │ │ │ + -- 1.45904s 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 │ │ │ │ + -- 1.71631s 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 => {{1}, {2}} │ │ │ + 1 => {{1, 2}} │ │ │ │ │ │ o2 : AbstractSimplicialComplex │ │ │ │ │ │ i3 : prune HH simplicialChainComplex K │ │ │ │ │ │ 1 │ │ │ o3 = ZZ │ │ │ @@ -20,16 +20,16 @@ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : setRandomSeed(currentTime()); │ │ │ │ │ │ i5 : L = randomAbstractSimplicialComplex(6,3) │ │ │ │ │ │ o5 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ - 0 => {{3}, {6}} │ │ │ - 1 => {{3, 6}} │ │ │ + 0 => {{1}, {6}} │ │ │ + 1 => {{1, 6}} │ │ │ │ │ │ o5 : AbstractSimplicialComplex │ │ │ │ │ │ i6 : prune HH simplicialChainComplex L │ │ │ │ │ │ 1 │ │ │ o6 = ZZ │ │ │ @@ -38,44 +38,44 @@ │ │ │ │ │ │ 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, 6}, {2, 4}, {2, 5}, {3, 6}, {4, 5}, {4, 6}} │ │ │ + 2 => {{1, 3, 6}, {1, 4, 6}, {2, 4, 5}} │ │ │ │ │ │ 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 => {{1}, {2}} │ │ │ + 1 => {{1, 2}} │ │ │ │ │ │ o11 : AbstractSimplicialComplex │ │ │ │ │ │ i12 : J = randomSubSimplicialComplex(K) │ │ │ │ │ │ o12 = AbstractSimplicialComplex{-1 => {{}}} │ │ │ - 0 => {{2}} │ │ │ + 0 => {{1}} │ │ │ │ │ │ o12 : AbstractSimplicialComplex │ │ │ │ │ │ i13 : inducedSimplicialChainComplexMap(K,J) │ │ │ │ │ │ 2 1 │ │ │ o13 = 0 : ZZ <--------- ZZ : 0 │ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/example-output/_random__Abstract__Simplicial__Complex.out │ │ │ @@ -1,36 +1,33 @@ │ │ │ -- -*- 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 => {{4}} │ │ │ │ │ │ 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 => {{4}, {6}} │ │ │ + 1 => {{4, 6}} │ │ │ │ │ │ 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}} │ │ │ + 1 => {{1, 3}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 4}, {3, 5}, {3, 6}} │ │ │ + 2 => {{1, 3, 5}, {1, 3, 6}, {2, 3, 4}} │ │ │ │ │ │ o6 : AbstractSimplicialComplex │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/example-output/_random__Sub__Simplicial__Complex.out │ │ │ @@ -1,18 +1,20 @@ │ │ │ -- -*- M2-comint -*- hash: 13473104809235542297 │ │ │ │ │ │ i1 : setRandomSeed(currentTime()); │ │ │ │ │ │ i2 : K = randomAbstractSimplicialComplex(4) │ │ │ │ │ │ -o2 = AbstractSimplicialComplex{-1 => {{}}} │ │ │ - 0 => {{1}} │ │ │ +o2 = AbstractSimplicialComplex{-1 => {{}} } │ │ │ + 0 => {{2}, {3}} │ │ │ + 1 => {{2, 3}} │ │ │ │ │ │ o2 : AbstractSimplicialComplex │ │ │ │ │ │ i3 : J = randomSubSimplicialComplex(K) │ │ │ │ │ │ o3 = AbstractSimplicialComplex{-1 => {{}}} │ │ │ + 0 => {{2}} │ │ │ │ │ │ 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 => {{1}, {2}}
│ │ │ +                               1 => {{1, 2}}
│ │ │  
│ │ │  o2 : AbstractSimplicialComplex
│ │ │ │ │ │ │ │ │ 
i3 : prune HH simplicialChainComplex K
│ │ │  
│ │ │         1
│ │ │ @@ -80,16 +80,16 @@
│ │ │            
│ │ │                
i4 : setRandomSeed(currentTime());

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

│ │ │            
│ │ │            
│ │ │                
i6 : prune HH simplicialChainComplex L
│ │ │  
│ │ │         1
│ │ │ @@ -106,28 +106,28 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
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, 6}, {2, 4}, {2, 5}, {3, 6}, {4, 5}, {4, 6}}
│ │ │ +                               2 => {{1, 3, 6}, {1, 4, 6}, {2, 4, 5}}
│ │ │  
│ │ │  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.

│ │ │          

│ │ │ @@ -135,24 +135,24 @@
│ │ │            
│ │ │                
i10 : setRandomSeed(currentTime());

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

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

│ │ │            
│ │ │            
│ │ │                
i13 : inducedSimplicialChainComplexMap(K,J)
│ │ │  
│ │ │              2              1
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,16 +10,16 @@
│ │ │ │  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 => {{1}, {2}}
│ │ │ │ +                               1 => {{1, 2}}
│ │ │ │  
│ │ │ │  o2 : AbstractSimplicialComplex
│ │ │ │  i3 : prune HH simplicialChainComplex K
│ │ │ │  
│ │ │ │         1
│ │ │ │  o3 = ZZ
│ │ │ │  
│ │ │ │ @@ -27,16 +27,16 @@
│ │ │ │  
│ │ │ │  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}}
│ │ │ │ +                               0 => {{1}, {6}}
│ │ │ │ +                               1 => {{1, 6}}
│ │ │ │  
│ │ │ │  o5 : AbstractSimplicialComplex
│ │ │ │  i6 : prune HH simplicialChainComplex L
│ │ │ │  
│ │ │ │         1
│ │ │ │  o6 = ZZ
│ │ │ │  
│ │ │ │ @@ -48,40 +48,40 @@
│ │ │ │  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, 6}, {2, 4}, {2, 5},
│ │ │ │ +{3, 6}, {4, 5}, {4, 6}}
│ │ │ │ +                               2 => {{1, 3, 6}, {1, 4, 6}, {2, 4, 5}}
│ │ │ │  
│ │ │ │  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 => {{1}, {2}}
│ │ │ │ +                                1 => {{1, 2}}
│ │ │ │  
│ │ │ │  o11 : AbstractSimplicialComplex
│ │ │ │  i12 : J = randomSubSimplicialComplex(K)
│ │ │ │  
│ │ │ │  o12 = AbstractSimplicialComplex{-1 => {{}}}
│ │ │ │ -                                0 => {{2}}
│ │ │ │ +                                0 => {{1}}
│ │ │ │  
│ │ │ │  o12 : AbstractSimplicialComplex
│ │ │ │  i13 : inducedSimplicialChainComplexMap(K,J)
│ │ │ │  
│ │ │ │              2              1
│ │ │ │  o13 = 0 : ZZ  <--------- ZZ  : 0
│ │ │ │                   | 1 |
│ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/html/_random__Abstract__Simplicial__Complex.html
│ │ │ @@ -50,36 +50,33 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
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 => {{4}}
│ │ │  
│ │ │  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 => {{4}, {6}}
│ │ │ +                               1 => {{4, 6}}
│ │ │  
│ │ │  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.

│ │ │          

│ │ │ @@ -88,16 +85,16 @@
│ │ │                
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}}
│ │ │ +                               1 => {{1, 3}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 4}, {3, 5}, {3, 6}}
│ │ │ +                               2 => {{1, 3, 5}, {1, 3, 6}, {2, 3, 4}}
│ │ │  
│ │ │  o6 : AbstractSimplicialComplex

│ │ │            
│ │ │          
│ │ │        
│ │ │        

│ │ │          
See also

│ │ │ ├── html2text {}
│ │ │ │ @@ -6,42 +6,39 @@
│ │ │ │  ************ 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 => {{4}}
│ │ │ │  
│ │ │ │  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 => {{4}, {6}}
│ │ │ │ +                               1 => {{4, 6}}
│ │ │ │  
│ │ │ │  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}}
│ │ │ │ +                               1 => {{1, 3}, {1, 5}, {1, 6}, {2, 3}, {2, 4},
│ │ │ │ +{3, 4}, {3, 5}, {3, 6}}
│ │ │ │ +                               2 => {{1, 3, 5}, {1, 3, 6}, {2, 3, 4}}
│ │ │ │  
│ │ │ │  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 +50,25 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
i1 : setRandomSeed(currentTime());

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

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

│ │ │  		          

│ │ │        

│ │ │        

│ │ │          
Ways to use randomSubSimplicialComplex:

│ │ │ ├── html2text {}
│ │ │ │ @@ -6,20 +6,22 @@
│ │ │ │  ************ 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}}
│ │ │ │ +                               1 => {{2, 3}}
│ │ │ │  
│ │ │ │  o2 : AbstractSimplicialComplex
│ │ │ │  i3 : J = randomSubSimplicialComplex(K)
│ │ │ │  
│ │ │ │  o3 = AbstractSimplicialComplex{-1 => {{}}}
│ │ │ │ +                               0 => {{2}}
│ │ │ │  
│ │ │ │  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
│ │ │ + -- .0263787s 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
│ │ │ + -- .525438s 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
│ │ │ + -- 5.20627s 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
│ │ │ + -- .0144479s 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
│ │ │ + -- .0644871s 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
│ │ │ + -- 1.5835s 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
│ │ │ + -- .0263787s 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
│ │ │ │ + -- .0263787s 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
│ │ │ + -- .525438s 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

│ │ │ + -- 5.20627s 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
│ │ │ │ + -- .525438s 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
│ │ │ │ + -- 5.20627s 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
│ │ │ + -- .0144479s 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
│ │ │ + -- .0644871s 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

│ │ │ + -- 1.5835s 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
│ │ │ │ + -- .0144479s 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
│ │ │ │ + -- .0644871s 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
│ │ │ │ + -- 1.5835s 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 0.567372s (cpu); 0.3696s (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 0.60833s (cpu); 0.400106s (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 0.567372s (cpu); 0.3696s (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
│ │ │ │ + -- .729899s 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
│ │ │ │ + -- 29.2631s 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
│ │ │ │ + -- 13.3288s 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 0.191955s (cpu); 0.192218s (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
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
              
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.796378s (cpu); 0.57534s (thread); 0s (gc)
│ │ │ + -- used 0.60833s (cpu); 0.400106s (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 0.567372s (cpu); 0.3696s (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 0.60833s (cpu); 0.400106s (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
│ │ │ +# GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:38 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,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
│ │ │  dGlvbiB0byAgTWFjYXVsYXkyIiwgRGVzY3JpcHRpb24gPT4gKERJVntQQVJBe1RFWHsiV2UgYXNz
│ │ ├── ./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
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3RhbmRhcmQgTWFjYXVsYXkyIGJlbmNo
│ │ │  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 Sun Mar  1 17:14:21 UTC 2026
│ │ │ +-- Linux sbuild 6.12.73+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.73-1 (2026-02-17) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- 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: .187964 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 Sun Mar  1 17:14:21 UTC 2026
│ │ │ +-- Linux sbuild 6.12.73+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.73-1 (2026-02-17) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- 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: .187964 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 Sun Mar  1 17:14:21 UTC 2026
│ │ │ │ +-- Linux sbuild 6.12.73+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.73-1 (2026-02-17) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998
│ │ │ │  -- 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: .187964 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 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,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
│ │ │  RGxvY2FsaXplQWxsKElkZWFsLFJpbmdFbGVtZW50KQ==
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhEbG9jYWxpemVBbGwsSWRlYWwsUmluZ0VsZW1lbnQp
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,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
│ │ │  U3RlcHNGb3JJbmNyZWFzZQ==
│ │ │  #:len=268
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzM3Mywgc3ltYm9sIERvY3VtZW50VGFn
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│ │ │  #:len=29
│ │ │  aW1wb3J0SW5jaWRlbmNlTWF0cml4KFN0cmluZyk=
│ │ │  #:len=281
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzIyMiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaW1wb3J0SW5jaWRlbmNlTWF0cml4LFN0cmluZyks
│ │ │  ImltcG9ydEluY2lkZW5jZU1hdHJpeChTdHJpbmcpIiwiQmVydGluaSJ9LCBQcmltYXJ5VGFnID0+
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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/M2-123499-0/0", 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/M2-123499-0/0", 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/M2-123499-0/0", 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/M2-123499-0/0", 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/M2-123499-0/0", 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/M2-123499-0/0", 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
│ │ │ + -- .410611s 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
│ │ │ + -- .469718s 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
│ │ │ + -- .729899s 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
│ │ │ + -- 29.2631s 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
│ │ │ + -- 13.3288s 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
│ │ │ + -- .410611s 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
│ │ │ │ + -- .410611s 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
│ │ │ + -- .469718s 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
│ │ │ │ + -- .469718s 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
│ │ │ + -- .729899s 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
│ │ │ + -- 29.2631s 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
│ │ │ + -- 13.3288s 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 0.191955s (cpu); 0.192218s (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 0.191955s (cpu); 0.192218s (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__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__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 0.32831s (cpu); 0.25809s (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 0.111752s (cpu); 0.109925s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.202116s (cpu); 0.149978s (thread); 0s (gc)
│ │ │ + -- used 0.227794s (cpu); 0.157808s (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 0.32831s (cpu); 0.25809s (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 1.76596s (cpu); 0.493084s (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 0.275997s (cpu); 0.0955222s (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 0.32831s (cpu); 0.25809s (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 0.111752s (cpu); 0.109925s (thread); 0s (gc)
│ │ │      		          
                  
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.202116s (cpu); 0.149978s (thread); 0s (gc)
    
│ │ │ + -- used 0.227794s (cpu); 0.157808s (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 0.111752s (cpu); 0.109925s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.202116s (cpu); 0.149978s (thread); 0s (gc)
│ │ │ │ + -- used 0.227794s (cpu); 0.157808s (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
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyZXZlcnNlRmFjdG9ycyxDaGFpbkNvbXBsZXgsQ2hh
│ │ ├── ./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 1.76596s (cpu); 0.493084s (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 0.275997s (cpu); 0.0955222s (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 0.298737s (cpu); 0.13246s (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 0.611477s (cpu); 0.294227s (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 1.42866s (cpu); 0.439195s (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 2.50916s (cpu); 2.11648s (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 0.434213s (cpu); 0.226125s (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 0.278297s (cpu); 0.13134s (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 0.192649s (cpu); 0.0623477s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.316118s (cpu); 0.167256s (thread); 0s (gc)
│ │ │ + -- used 0.366454s (cpu); 0.172117s (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 0.362718s (cpu); 0.116966s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.203774s (cpu); 0.0900468s (thread); 0s (gc)
│ │ │ + -- used 0.256705s (cpu); 0.105768s (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 6.45424s (cpu); 1.51177s (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 6.5305s (cpu); 1.53296s (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 1.52913s (cpu); 0.578525s (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 0.157861s (cpu); 0.0577471s (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 0.0643643s (cpu); 0.0387317s (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 4.22872s (cpu); 1.34651s (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 0.917352s (cpu); 0.255874s (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 1.76596s (cpu); 0.493084s (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 0.275997s (cpu); 0.0955222s (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 0.298737s (cpu); 0.13246s (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 0.611477s (cpu); 0.294227s (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 0.298737s (cpu); 0.13246s (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 0.611477s (cpu); 0.294227s (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 1.42866s (cpu); 0.439195s (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 2.50916s (cpu); 2.11648s (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 0.434213s (cpu); 0.226125s (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 0.278297s (cpu); 0.13134s (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 1.42866s (cpu); 0.439195s (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 2.50916s (cpu); 2.11648s (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 0.434213s (cpu); 0.226125s (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 0.278297s (cpu); 0.13134s (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 0.192649s (cpu); 0.0623477s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    
│ │ │      		          
                  
    i5 : time Euler I
│ │ │ - -- used 0.316118s (cpu); 0.167256s (thread); 0s (gc)
│ │ │ + -- used 0.366454s (cpu); 0.172117s (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 0.362718s (cpu); 0.116966s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    
│ │ │      		          
                  
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.203774s (cpu); 0.0900468s (thread); 0s (gc)
│ │ │ + -- used 0.256705s (cpu); 0.105768s (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 0.192649s (cpu); 0.0623477s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : time Euler I
│ │ │ │ - -- used 0.316118s (cpu); 0.167256s (thread); 0s (gc)
│ │ │ │ + -- used 0.366454s (cpu); 0.172117s (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 0.362718s (cpu); 0.116966s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 2
│ │ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ │ - -- used 0.203774s (cpu); 0.0900468s (thread); 0s (gc)
│ │ │ │ + -- used 0.256705s (cpu); 0.105768s (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 6.45424s (cpu); 1.51177s (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 6.5305s (cpu); 1.53296s (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 6.45424s (cpu); 1.51177s (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 6.5305s (cpu); 1.53296s (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 1.52913s (cpu); 0.578525s (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 0.157861s (cpu); 0.0577471s (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 0.0643643s (cpu); 0.0387317s (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 1.52913s (cpu); 0.578525s (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 0.157861s (cpu); 0.0577471s (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 0.0643643s (cpu); 0.0387317s (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 4.22872s (cpu); 1.34651s (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 0.917352s (cpu); 0.255874s (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 4.22872s (cpu); 1.34651s (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 0.917352s (cpu); 0.255874s (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
│ │ │  YWx1YXRpb24gY29kZSIsICJsaW5lbnVtIiA9PiA1MTU4LCBJbnB1dHMgPT4ge1NQQU57VFR7IkMi
│ │ ├── ./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
│ │ │ @@ -45,73 +45,73 @@
│ │ │  
│ │ │  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   a+1 |
│ │ │                                                             | 1   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 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   1 0 0 1 |
│ │ │ +                                                                | 0   0   a a a+1 0 0 0 1 |
│ │ │ +                                             Generators => {{a + 1, a + 1, a, a, 1, 1, 0, 0, 1}, {0, 0, a, a, a + 1, 0, 0, 0, 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 a+1 0 0 0   |
│ │ │ +                                                                  | 0 1 0 0 0 a+1 0 0 0   |
│ │ │ +                                                                  | 0 0 1 0 0 0   0 0 a   |
│ │ │ +                                                                  | 0 0 0 1 0 0   0 0 a   |
│ │ │ +                                                                  | 0 0 0 0 1 a   0 0 a+1 |
│ │ │ +                                                                  | 0 0 0 0 0 0   1 0 0   |
│ │ │ +                                                                  | 0 0 0 0 0 0   0 1 0   |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, 0, 0}, {0, 1, 0, 0, 0, a + 1, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, a, 0, 0, a + 1}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ +                    Points => {{0, a}, {a, 0}, {a, 1}, {1, a}, {a, a}, {0, 0}, {1, 0}, {0, 1}, {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   a+1 |
│ │ │                                | 1   0   |
│ │ │                                | 0   0   |
│ │ │                                | 0   0   |
│ │ │                                | 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   1 0 0 1 |
│ │ │ +                                   | 0   0   a a a+1 0 0 0 1 |
│ │ │ +                Generators => {{a + 1, a + 1, a, a, 1, 1, 0, 0, 1}, {0, 0, a, a, a + 1, 0, 0, 0, 1}}
│ │ │ +                ParityCheckMatrix => | 1 0 0 0 0 a+1 0 0 0   |
│ │ │ +                                     | 0 1 0 0 0 a+1 0 0 0   |
│ │ │ +                                     | 0 0 1 0 0 0   0 0 a   |
│ │ │ +                                     | 0 0 0 1 0 0   0 0 a   |
│ │ │ +                                     | 0 0 0 0 1 a   0 0 a+1 |
│ │ │ +                                     | 0 0 0 0 0 0   1 0 0   |
│ │ │ +                                     | 0 0 0 0 0 0   0 1 0   |
│ │ │ +                ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, 0, 0}, {0, 1, 0, 0, 0, a + 1, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, a, 0, 0, a + 1}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │  
│ │ │  o5 : LinearCode
│ │ │  
│ │ │  i6 : F=GF(4);
│ │ │  
│ │ │  i7 : R=F[x,y];
│ │ │  
│ │ │ @@ -119,34 +119,34 @@
│ │ │  
│ │ │  o8 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │                      LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │                                               Code => image | 0   0   |
│ │ │ -                                                           | 0   0   |
│ │ │ -                                                           | a+1 1   |
│ │ │ -                                                           | a   a+1 |
│ │ │                                                             | 1   a+1 |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │ +                                                           | a   a+1 |
│ │ │ +                                                           | 0   0   |
│ │ │ +                                                           | a+1 1   |
│ │ │                                                             | 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}}
│ │ │ +                                             GeneratorMatrix => | 0 1   0 0 0 a   0 a+1 1 |
│ │ │ +                                                                | 0 a+1 0 0 0 a+1 0 1   1 |
│ │ │ +                                             Generators => {{0, 1, 0, 0, 0, a, 0, a + 1, 1}, {0, a + 1, 0, 0, 0, a + 1, 0, 1, 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 0 0   0 |
│ │ │ +                                                                  | 0 1 0 0 0 0 0 1   a |
│ │ │ +                                                                  | 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 1 0 a+1 0 |
│ │ │ +                                                                  | 0 0 0 0 0 0 1 0   0 |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 1, a}, {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, 1, 0, a + 1, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ +                    Points => {{0, a}, {a, a}, {a, 0}, {0, 0}, {0, 1}, {a, 1}, {1, 0}, {1, a}, {1, 1}}
│ │ │                                           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/_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/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   a+1 |
│ │ │                                                             | 1   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 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   1 0 0 1 |
│ │ │ +                                                                | 0   0   a a a+1 0 0 0 1 |
│ │ │ +                                             Generators => {{a + 1, a + 1, a, a, 1, 1, 0, 0, 1}, {0, 0, a, a, a + 1, 0, 0, 0, 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 a+1 0 0 0   |
│ │ │ +                                                                  | 0 1 0 0 0 a+1 0 0 0   |
│ │ │ +                                                                  | 0 0 1 0 0 0   0 0 a   |
│ │ │ +                                                                  | 0 0 0 1 0 0   0 0 a   |
│ │ │ +                                                                  | 0 0 0 0 1 a   0 0 a+1 |
│ │ │ +                                                                  | 0 0 0 0 0 0   1 0 0   |
│ │ │ +                                                                  | 0 0 0 0 0 0   0 1 0   |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, 0, 0}, {0, 1, 0, 0, 0, a + 1, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, a, 0, 0, a + 1}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ +                    Points => {{0, a}, {a, 0}, {a, 1}, {1, a}, {a, a}, {0, 0}, {1, 0}, {0, 1}, {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   a+1 |
│ │ │                                | 1   0   |
│ │ │                                | 0   0   |
│ │ │                                | 0   0   |
│ │ │                                | 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   1 0 0 1 |
│ │ │ +                                   | 0   0   a a a+1 0 0 0 1 |
│ │ │ +                Generators => {{a + 1, a + 1, a, a, 1, 1, 0, 0, 1}, {0, 0, a, a, a + 1, 0, 0, 0, 1}}
│ │ │ +                ParityCheckMatrix => | 1 0 0 0 0 a+1 0 0 0   |
│ │ │ +                                     | 0 1 0 0 0 a+1 0 0 0   |
│ │ │ +                                     | 0 0 1 0 0 0   0 0 a   |
│ │ │ +                                     | 0 0 0 1 0 0   0 0 a   |
│ │ │ +                                     | 0 0 0 0 1 a   0 0 a+1 |
│ │ │ +                                     | 0 0 0 0 0 0   1 0 0   |
│ │ │ +                                     | 0 0 0 0 0 0   0 1 0   |
│ │ │ +                ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, 0, 0}, {0, 1, 0, 0, 0, a + 1, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, a, 0, 0, a + 1}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │  
│ │ │  o5 : LinearCode
    
│ │ │      		            
    
│ │ │          
│ │ │          
    
│ │ │            
    a ring, a list and a Matrix are given
    
│ │ │ @@ -301,34 +301,34 @@
│ │ │  
│ │ │  o8 = EvaluationCode{cache => CacheTable{}                                                                                                                                                                                                                                       }
│ │ │                                                                 9
│ │ │                      LinearCode => LinearCode{AmbientModule => F                                                                                                                                                                                                                }
│ │ │                                               BaseField => F
│ │ │                                               cache => CacheTable{}
│ │ │                                               Code => image | 0   0   |
│ │ │ -                                                           | 0   0   |
│ │ │ -                                                           | a+1 1   |
│ │ │ -                                                           | a   a+1 |
│ │ │                                                             | 1   a+1 |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │                                                             | 0   0   |
│ │ │ +                                                           | a   a+1 |
│ │ │ +                                                           | 0   0   |
│ │ │ +                                                           | a+1 1   |
│ │ │                                                             | 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}}
│ │ │ +                                             GeneratorMatrix => | 0 1   0 0 0 a   0 a+1 1 |
│ │ │ +                                                                | 0 a+1 0 0 0 a+1 0 1   1 |
│ │ │ +                                             Generators => {{0, 1, 0, 0, 0, a, 0, a + 1, 1}, {0, a + 1, 0, 0, 0, a + 1, 0, 1, 1}}
│ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0 0 0   0 |
│ │ │ +                                                                  | 0 1 0 0 0 0 0 1   a |
│ │ │ +                                                                  | 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 1 0 a+1 0 |
│ │ │ +                                                                  | 0 0 0 0 0 0 1 0   0 |
│ │ │ +                                             ParityCheckRows => {{1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 1, a}, {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, 1, 0, a + 1, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ +                    Points => {{0, a}, {a, a}, {a, 0}, {0, 0}, {0, 1}, {a, 1}, {1, 0}, {1, a}, {1, 1}}
│ │ │                                           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   a+1 |
│ │ │ │                                                             | 1   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 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}}
│ │ │ │ +                                             GeneratorMatrix => | a+1 a+1 a a 1
│ │ │ │ +1 0 0 1 |
│ │ │ │ +                                                                | 0   0   a a
│ │ │ │ +a+1 0 0 0 1 |
│ │ │ │ +                                             Generators => {{a + 1, a + 1, a,
│ │ │ │ +a, 1, 1, 0, 0, 1}, {0, 0, a, a, a + 1, 0, 0, 0, 1}}
│ │ │ │                                               ParityCheckMatrix => | 1 0 0 0 0
│ │ │ │ -a+1 0 a   0 |
│ │ │ │ -                                                                  | 0 1 0 0 0 a
│ │ │ │ -0 0   0 |
│ │ │ │ +a+1 0 0 0   |
│ │ │ │ +                                                                  | 0 1 0 0 0
│ │ │ │ +a+1 0 0 0   |
│ │ │ │                                                                    | 0 0 1 0 0 0
│ │ │ │ -0 0   0 |
│ │ │ │ +0 0 a   |
│ │ │ │                                                                    | 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 a   |
│ │ │ │ +                                                                  | 0 0 0 0 1 a
│ │ │ │ +0 0 a+1 |
│ │ │ │ +                                                                  | 0 0 0 0 0 0
│ │ │ │ +1 0 0   |
│ │ │ │                                                                    | 0 0 0 0 0 0
│ │ │ │ -0 1   1 |
│ │ │ │ +0 1 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, a + 1, 0, 0, 0}, {0, 1, 0, 0, 0, a + 1, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0,
│ │ │ │ +a}, {0, 0, 0, 1, 0, 0, 0, 0, a}, {0, 0, 0, 0, 1, a, 0, 0, a + 1}, {0, 0, 0, 0,
│ │ │ │ +0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}}
│ │ │ │ +                    Points => {{0, a}, {a, 0}, {a, 1}, {1, a}, {a, a}, {0, 0},
│ │ │ │ +{1, 0}, {0, 1}, {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   a+1 |
│ │ │ │                                | 1   0   |
│ │ │ │                                | 0   0   |
│ │ │ │                                | 0   0   |
│ │ │ │                                | 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   1 0 0 1 |
│ │ │ │ +                                   | 0   0   a a a+1 0 0 0 1 |
│ │ │ │ +                Generators => {{a + 1, a + 1, a, a, 1, 1, 0, 0, 1}, {0, 0, a,
│ │ │ │ +a, a + 1, 0, 0, 0, 1}}
│ │ │ │ +                ParityCheckMatrix => | 1 0 0 0 0 a+1 0 0 0   |
│ │ │ │ +                                     | 0 1 0 0 0 a+1 0 0 0   |
│ │ │ │ +                                     | 0 0 1 0 0 0   0 0 a   |
│ │ │ │ +                                     | 0 0 0 1 0 0   0 0 a   |
│ │ │ │ +                                     | 0 0 0 0 1 a   0 0 a+1 |
│ │ │ │ +                                     | 0 0 0 0 0 0   1 0 0   |
│ │ │ │ +                                     | 0 0 0 0 0 0   0 1 0   |
│ │ │ │ +                ParityCheckRows => {{1, 0, 0, 0, 0, a + 1, 0, 0, 0}, {0, 1, 0,
│ │ │ │ +0, 0, a + 1, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, a}, {0, 0, 0, 1, 0, 0, 0, 0,
│ │ │ │ +a}, {0, 0, 0, 0, 1, a, 0, 0, a + 1}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0,
│ │ │ │ +0, 0, 0, 1, 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,
│ │ │ │ @@ -225,48 +225,48 @@
│ │ │ │  }
│ │ │ │                                                                 9
│ │ │ │                      LinearCode => LinearCode{AmbientModule => F
│ │ │ │  }
│ │ │ │                                               BaseField => F
│ │ │ │                                               cache => CacheTable{}
│ │ │ │                                               Code => image | 0   0   |
│ │ │ │ -                                                           | 0   0   |
│ │ │ │ -                                                           | a+1 1   |
│ │ │ │ -                                                           | a   a+1 |
│ │ │ │                                                             | 1   a+1 |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │                                                             | 0   0   |
│ │ │ │ +                                                           | a   a+1 |
│ │ │ │ +                                                           | 0   0   |
│ │ │ │ +                                                           | a+1 1   |
│ │ │ │                                                             | 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 |
│ │ │ │ +                                             GeneratorMatrix => | 0 1   0 0 0 a
│ │ │ │ +0 a+1 1 |
│ │ │ │ +                                                                | 0 a+1 0 0 0
│ │ │ │ +a+1 0 1   1 |
│ │ │ │ +                                             Generators => {{0, 1, 0, 0, 0, a,
│ │ │ │ +0, a + 1, 1}, {0, a + 1, 0, 0, 0, a + 1, 0, 1, 1}}
│ │ │ │ +                                             ParityCheckMatrix => | 1 0 0 0 0 0
│ │ │ │ +0 0   0 |
│ │ │ │ +                                                                  | 0 1 0 0 0 0
│ │ │ │ +0 1   a |
│ │ │ │ +                                                                  | 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 1
│ │ │ │ +0 a+1 0 |
│ │ │ │ +                                                                  | 0 0 0 0 0 0
│ │ │ │ +1 0   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, 1, 0, 0, 0, 0, 0, 1, a}, {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, 1, 0, a +
│ │ │ │ +1, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}}
│ │ │ │ +                    Points => {{0, a}, {a, a}, {a, 0}, {0, 0}, {0, 1}, {a, 1},
│ │ │ │ +{1, 0}, {1, a}, {1, 1}}
│ │ │ │                                           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/_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/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
│ │ │ + -- 3.14123s 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
│ │ │ + -- .538975s 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
│ │ │ + -- 10.6803s 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
│ │ │ + -- .518445s 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
│ │ │ + -- .40307s elapsed
│ │ │ + -- .403112s 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
│ │ │ + -- 3.14123s 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
│ │ │ + -- .538975s 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
│ │ │ + -- 10.6803s 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
│ │ │ + -- .518445s 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
│ │ │ + -- .40307s elapsed
│ │ │ + -- .403112s 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
│ │ │ │ + -- 3.14123s 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
│ │ │ │ + -- .538975s 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
│ │ │ │ + -- 10.6803s 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
│ │ │ │ + -- .518445s 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
│ │ │ │ + -- .40307s elapsed
│ │ │ │ + -- .403112s 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 10.7449s (cpu); 5.99443s (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 0.183193s (cpu); 0.110977s (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 1.68942s (cpu); 0.942159s (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 1.68684s (cpu); 0.880458s (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 0.782811s (cpu); 0.411027s (thread); 0s (gc)
│ │ │ + -- used 0.35246s (cpu); 0.16501s (thread); 0s (gc)
│ │ │ + -- used 0.000178728s (cpu); 2.826e-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 1.61711s (cpu); 0.769888s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.648202s (cpu); 0.372689s (thread); 0s (gc)
│ │ │ + -- used 0.94413s (cpu); 0.463723s (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 0.281777s (cpu); 0.144699s (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 10.7449s (cpu); 5.99443s (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 0.183193s (cpu); 0.110977s (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 1.68942s (cpu); 0.942159s (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 1.68684s (cpu); 0.880458s (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 10.7449s (cpu); 5.99443s (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 0.183193s (cpu); 0.110977s (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 1.68942s (cpu); 0.942159s (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 1.68684s (cpu); 0.880458s (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 0.782811s (cpu); 0.411027s (thread); 0s (gc)
│ │ │ + -- used 0.35246s (cpu); 0.16501s (thread); 0s (gc)
│ │ │ + -- used 0.000178728s (cpu); 2.826e-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 0.782811s (cpu); 0.411027s (thread); 0s (gc)
│ │ │ │ + -- used 0.35246s (cpu); 0.16501s (thread); 0s (gc)
│ │ │ │ + -- used 0.000178728s (cpu); 2.826e-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 1.61711s (cpu); 0.769888s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.648202s (cpu); 0.372689s (thread); 0s (gc)
│ │ │ + -- used 0.94413s (cpu); 0.463723s (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 0.281777s (cpu); 0.144699s (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 1.61711s (cpu); 0.769888s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{1, 1}} => 2        }
│ │ │ │        {{2, 2}, {1, 2}} => 4
│ │ │ │  
│ │ │ │ - -- used 0.648202s (cpu); 0.372689s (thread); 0s (gc)
│ │ │ │ + -- used 0.94413s (cpu); 0.463723s (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 0.281777s (cpu); 0.144699s (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 @@
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│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:version=1.1
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│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
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│ │ ├── ./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
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│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
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│ │ │  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 1.67653s (cpu); 1.01113s (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 1.23672s (cpu); 0.895897s (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 0.235935s (cpu); 0.16583s (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 0.0478578s (cpu); 0.0177279s (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 0.00391476s (cpu); 0.00501619s (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 0.0520121s (cpu); 0.0502269s (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 0.133529s (cpu); 0.0532624s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.506956s (cpu); 0.382073s (thread); 0s (gc)
│ │ │ + -- used 0.506407s (cpu); 0.434529s (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 0.0679985s (cpu); 0.0684068s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.000189065s (cpu); 0.00204299s (thread); 0s (gc)
│ │ │ + -- used 0.000181953s (cpu); 0.00240347s (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 0.513289s (cpu); 0.437916s (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 0.00799936s (cpu); 0.00960159s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.297917s (cpu); 0.233828s (thread); 0s (gc)
│ │ │ + -- used 0.280415s (cpu); 0.201278s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 4.99657s (cpu); 4.27726s (thread); 0s (gc)
│ │ │ + -- used 5.56409s (cpu); 5.19512s (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 0.000914347s (cpu); 0.00234961s (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 0.173339s (cpu); 0.096926s (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 0.460189s (cpu); 0.459117s (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 0.442528s (cpu); 0.354092s (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 0.0526343s (cpu); 0.0286878s (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 0.00353754s (cpu); 0.00469434s (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 0.0106964s (cpu); 0.0118743s (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 0.0066362s (cpu); 0.00998098s (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 1.34436s (cpu); 1.05269s (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 0.000191622s (cpu); 1.7917e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.000926578s (cpu); 0.00142987s (thread); 0s (gc)
│ │ │ + -- used 8.8576e-05s (cpu); 0.001817s (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 0.313154s (cpu); 0.160449s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ - -- used 0.010188s (cpu); 0.011592s (thread); 0s (gc)
│ │ │ + -- used 0.0251784s (cpu); 0.0130438s (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 0.202823s (cpu); 0.0866208s (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 0.0560332s (cpu); 0.0445508s (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 0.286483s (cpu); 0.205026s (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 0.630502s (cpu); 0.485306s (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 0.424047s (cpu); 0.359763s (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 0.0395858s (cpu); 0.0274959s (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 0.272228s (cpu); 0.144697s (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 0.0599731s (cpu); 0.0600981s (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 0.404235s (cpu); 0.256846s (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 0.346831s (cpu); 0.267674s (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 1.12492s (cpu); 0.843766s (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 0.0035798s (cpu); 0.000396927s (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 0.229144s (cpu); 0.155722s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.40278s (cpu); 0.331735s (thread); 0s (gc)
│ │ │ + -- used 0.418118s (cpu); 0.349079s (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 0.0479224s (cpu); 0.0487913s (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 3.40441s (cpu); 1.94829s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 0.000199554s (cpu); 2.7712e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000198431s (cpu); 2.8858e-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 5.38796s (cpu); 3.05731s (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 4.35536s (cpu); 2.58232s (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 0.302613s (cpu); 0.232862s (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 0.250059s (cpu); 0.17459s (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 2.09727s (cpu); 1.79077s (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 3.23221s (cpu); 2.7655s (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 0.143742s (cpu); 0.0704482s (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 0.649758s (cpu); 0.581574s (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 0.00170038s (cpu); 0.000657328s (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 0.0338999s (cpu); 0.0192598s (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 0.0961949s (cpu); 0.0397421s (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 0.00398732s (cpu); 0.00372748s (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 0.197652s (cpu); 0.12215s (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 0.182996s (cpu); 0.104471s (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 0.386452s (cpu); 0.224218s (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 0.1602s (cpu); 0.0880517s (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 0.114818s (cpu); 0.0469955s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ - -- used 0.0128589s (cpu); 0.0142427s (thread); 0s (gc)
│ │ │ + -- used 0.0282061s (cpu); 0.0175277s (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 2.24281s (cpu); 2.08813s (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 3.20606s (cpu); 2.47369s (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 0.01999s (cpu); 0.0192616s (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 0.468924s (cpu); 0.391353s (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 0.00800367s (cpu); 0.00646424s (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 0.617485s (cpu); 0.441421s (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 0.235019s (cpu); 0.166012s (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 0.207551s (cpu); 0.129157s (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 0.0301728s (cpu); 0.0164511s (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 0.0237843s (cpu); 0.0131322s (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 0.00261204s (cpu); 3.8137e-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 9.4412e-05s (cpu); 1.9161e-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 0.103983s (cpu); 0.10256s (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 0.0319987s (cpu); 0.031631s (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 1.60216s (cpu); 0.644944s (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 1.15182s (cpu); 0.995426s (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 0.092021s (cpu); 0.089201s (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 0.0200455s (cpu); 0.0198769s (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 0.070713s (cpu); 0.0698941s (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 0.00398198s (cpu); 0.0070835s (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 0.1167s (cpu); 0.0378465s (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 0.00401718s (cpu); 0.00572932s (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 0.00200616s (cpu); 0.00440771s (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 1.67653s (cpu); 1.01113s (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 1.23672s (cpu); 0.895897s (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 0.235935s (cpu); 0.16583s (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 0.0478578s (cpu); 0.0177279s (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 1.67653s (cpu); 1.01113s (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 1.23672s (cpu); 0.895897s (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 0.235935s (cpu); 0.16583s (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 0.0478578s (cpu); 0.0177279s (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 0.313154s (cpu); 0.160449s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │            
│ │ │            
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ - -- used 0.010188s (cpu); 0.011592s (thread); 0s (gc)
│ │ │ + -- used 0.0251784s (cpu); 0.0130438s (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 0.313154s (cpu); 0.160449s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │ - -- used 0.010188s (cpu); 0.011592s (thread); 0s (gc)
│ │ │ │ + -- used 0.0251784s (cpu); 0.0130438s (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 0.202823s (cpu); 0.0866208s (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 0.0560332s (cpu); 0.0445508s (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 0.202823s (cpu); 0.0866208s (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 0.0560332s (cpu); 0.0445508s (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 0.286483s (cpu); 0.205026s (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 0.286483s (cpu); 0.205026s (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 0.630502s (cpu); 0.485306s (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 0.424047s (cpu); 0.359763s (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 0.0395858s (cpu); 0.0274959s (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 0.272228s (cpu); 0.144697s (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 0.0599731s (cpu); 0.0600981s (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 0.404235s (cpu); 0.256846s (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 0.346831s (cpu); 0.267674s (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 1.12492s (cpu); 0.843766s (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 0.630502s (cpu); 0.485306s (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 0.424047s (cpu); 0.359763s (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 0.0395858s (cpu); 0.0274959s (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 0.272228s (cpu); 0.144697s (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 0.0599731s (cpu); 0.0600981s (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 0.404235s (cpu); 0.256846s (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 0.346831s (cpu); 0.267674s (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 1.12492s (cpu); 0.843766s (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 0.0035798s (cpu); 0.000396927s (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 0.229144s (cpu); 0.155722s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │      		          
                  
    i6 : time rationalMap psi
│ │ │ - -- used 0.40278s (cpu); 0.331735s (thread); 0s (gc)
│ │ │ + -- used 0.418118s (cpu); 0.349079s (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 0.0479224s (cpu); 0.0487913s (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 3.40441s (cpu); 1.94829s (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 0.000198431s (cpu); 2.8858e-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 5.38796s (cpu); 3.05731s (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 4.35536s (cpu); 2.58232s (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 0.0035798s (cpu); 0.000396927s (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 0.229144s (cpu); 0.155722s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.40278s (cpu); 0.331735s (thread); 0s (gc)
│ │ │ │ + -- used 0.418118s (cpu); 0.349079s (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 0.0479224s (cpu); 0.0487913s (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 3.40441s (cpu); 1.94829s (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 0.000198431s (cpu); 2.8858e-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 5.38796s (cpu); 3.05731s (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 4.35536s (cpu); 2.58232s (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 0.302613s (cpu); 0.232862s (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 0.250059s (cpu); 0.17459s (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 2.09727s (cpu); 1.79077s (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 3.23221s (cpu); 2.7655s (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 0.302613s (cpu); 0.232862s (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 0.250059s (cpu); 0.17459s (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 2.09727s (cpu); 1.79077s (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 3.23221s (cpu); 2.7655s (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 0.143742s (cpu); 0.0704482s (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 0.649758s (cpu); 0.581574s (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 0.143742s (cpu); 0.0704482s (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 0.649758s (cpu); 0.581574s (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 0.00170038s (cpu); 0.000657328s (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 0.00170038s (cpu); 0.000657328s (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 0.0338999s (cpu); 0.0192598s (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 0.0338999s (cpu); 0.0192598s (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 0.0961949s (cpu); 0.0397421s (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 0.182996s (cpu); 0.104471s (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 0.386452s (cpu); 0.224218s (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 0.0961949s (cpu); 0.0397421s (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 0.00398732s (cpu); 0.00372748s (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 0.197652s (cpu); 0.12215s (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 0.00398732s (cpu); 0.00372748s (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 0.197652s (cpu); 0.12215s (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 0.182996s (cpu); 0.104471s (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 0.386452s (cpu); 0.224218s (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 0.1602s (cpu); 0.0880517s (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 0.1602s (cpu); 0.0880517s (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 0.114818s (cpu); 0.0469955s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ 
    i4 : time isBirational(phi,Certify=>true)
│ │ │ - -- used 0.0128589s (cpu); 0.0142427s (thread); 0s (gc)
│ │ │ + -- used 0.0282061s (cpu); 0.0175277s (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 0.114818s (cpu); 0.0469955s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ i4 : time isBirational(phi,Certify=>true) │ │ │ │ - -- used 0.0128589s (cpu); 0.0142427s (thread); 0s (gc) │ │ │ │ + -- used 0.0282061s (cpu); 0.0175277s (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 2.24281s (cpu); 2.08813s (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 3.20606s (cpu); 2.47369s (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 2.24281s (cpu); 2.08813s (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 3.20606s (cpu); 2.47369s (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 0.01999s (cpu); 0.0192616s (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 0.468924s (cpu); 0.391353s (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 0.01999s (cpu); 0.0192616s (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 0.468924s (cpu); 0.391353s (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 0.00800367s (cpu); 0.00646424s (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 0.617485s (cpu); 0.441421s (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 0.00800367s (cpu); 0.00646424s (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 0.617485s (cpu); 0.441421s (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 0.235019s (cpu); 0.166012s (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 0.207551s (cpu); 0.129157s (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 0.235019s (cpu); 0.166012s (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 0.207551s (cpu); 0.129157s (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 0.0301728s (cpu); 0.0164511s (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 0.0237843s (cpu); 0.0131322s (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 0.00261204s (cpu); 3.8137e-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 9.4412e-05s (cpu); 1.9161e-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 0.0301728s (cpu); 0.0164511s (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 0.0237843s (cpu); 0.0131322s (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 0.00261204s (cpu); 3.8137e-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 9.4412e-05s (cpu); 1.9161e-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 0.103983s (cpu); 0.10256s (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 0.103983s (cpu); 0.10256s (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 0.0319987s (cpu); 0.031631s (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 1.60216s (cpu); 0.644944s (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 0.0319987s (cpu); 0.031631s (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 1.60216s (cpu); 0.644944s (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 1.15182s (cpu); 0.995426s (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 1.15182s (cpu); 0.995426s (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 0.092021s (cpu); 0.089201s (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 0.0200455s (cpu); 0.0198769s (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 0.092021s (cpu); 0.089201s (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 0.0200455s (cpu); 0.0198769s (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 0.070713s (cpu); 0.0698941s (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 0.00398198s (cpu); 0.0070835s (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 0.070713s (cpu); 0.0698941s (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 0.00398198s (cpu); 0.0070835s (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 0.1167s (cpu); 0.0378465s (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 0.00401718s (cpu); 0.00572932s (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 0.00200616s (cpu); 0.00440771s (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 0.1167s (cpu); 0.0378465s (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 0.00401718s (cpu); 0.00572932s (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 0.00200616s (cpu); 0.00440771s (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 0.00391476s (cpu); 0.00501619s (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 0.0520121s (cpu); 0.0502269s (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 0.133529s (cpu); 0.0532624s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │      		          
                  
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.506956s (cpu); 0.382073s (thread); 0s (gc)
│ │ │ + -- used 0.506407s (cpu); 0.434529s (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 0.0679985s (cpu); 0.0684068s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │      		          
                  
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.000189065s (cpu); 0.00204299s (thread); 0s (gc)
│ │ │ + -- used 0.000181953s (cpu); 0.00240347s (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 0.513289s (cpu); 0.437916s (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 0.00799936s (cpu); 0.00960159s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │      		          
                  
    i10 : time degreeMap psi
│ │ │ - -- used 0.297917s (cpu); 0.233828s (thread); 0s (gc)
│ │ │ + -- used 0.280415s (cpu); 0.201278s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │      		          
                  
    i11 : time projectiveDegrees psi
│ │ │ - -- used 4.99657s (cpu); 4.27726s (thread); 0s (gc)
│ │ │ + -- used 5.56409s (cpu); 5.19512s (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 0.000914347s (cpu); 0.00234961s (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 0.173339s (cpu); 0.096926s (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 0.460189s (cpu); 0.459117s (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 0.442528s (cpu); 0.354092s (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 0.0526343s (cpu); 0.0286878s (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 0.00353754s (cpu); 0.00469434s (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 0.0106964s (cpu); 0.0118743s (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 0.0066362s (cpu); 0.00998098s (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 1.34436s (cpu); 1.05269s (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 0.000191622s (cpu); 1.7917e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │      		          
                  
    i23 : time describe f
│ │ │ - -- used 0.000926578s (cpu); 0.00142987s (thread); 0s (gc)
│ │ │ + -- used 8.8576e-05s (cpu); 0.001817s (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 0.00391476s (cpu); 0.00501619s (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 0.0520121s (cpu); 0.0502269s (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 0.133529s (cpu); 0.0532624s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.506956s (cpu); 0.382073s (thread); 0s (gc)
│ │ │ │ + -- used 0.506407s (cpu); 0.434529s (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 0.0679985s (cpu); 0.0684068s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.000189065s (cpu); 0.00204299s (thread); 0s (gc)
│ │ │ │ + -- used 0.000181953s (cpu); 0.00240347s (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 0.513289s (cpu); 0.437916s (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 0.00799936s (cpu); 0.00960159s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.297917s (cpu); 0.233828s (thread); 0s (gc)
│ │ │ │ + -- used 0.280415s (cpu); 0.201278s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 4.99657s (cpu); 4.27726s (thread); 0s (gc)
│ │ │ │ + -- used 5.56409s (cpu); 5.19512s (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 0.000914347s (cpu); 0.00234961s (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 0.173339s (cpu); 0.096926s (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 0.460189s (cpu); 0.459117s (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 0.442528s (cpu); 0.354092s (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 0.0526343s (cpu); 0.0286878s (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 0.00353754s (cpu); 0.00469434s (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 0.0106964s (cpu); 0.0118743s (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 0.0066362s (cpu); 0.00998098s (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 1.34436s (cpu); 1.05269s (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 0.000191622s (cpu); 1.7917e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.000926578s (cpu); 0.00142987s (thread); 0s (gc)
│ │ │ │ + -- used 8.8576e-05s (cpu); 0.001817s (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 0.0496188s (cpu); 0.0187677s (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 0.0616345s (cpu); 0.0243433s (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 0.0329427s (cpu); 0.0204954s (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 0.199083s (cpu); 0.0409497s (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 0.104488s (cpu); 0.0354423s (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 0.745755s (cpu); 0.656212s (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 0.0532221s (cpu); 0.033334s (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 0.203629s (cpu); 0.0399022s (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 0.449047s (cpu); 0.136207s (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 0.14663s (cpu); 0.0690011s (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 0.042726s (cpu); 0.0236173s (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 0.0297058s (cpu); 0.0159538s (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 0.151228s (cpu); 0.131468s (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 0.760512s (cpu); 0.570769s (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 0.187049s (cpu); 0.169244s (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 0.829103s (cpu); 0.519474s (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 0.0496188s (cpu); 0.0187677s (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 0.0496188s (cpu); 0.0187677s (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 0.0616345s (cpu); 0.0243433s (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 0.0329427s (cpu); 0.0204954s (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 0.0616345s (cpu); 0.0243433s (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 0.0329427s (cpu); 0.0204954s (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 0.199083s (cpu); 0.0409497s (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 0.199083s (cpu); 0.0409497s (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 0.104488s (cpu); 0.0354423s (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 0.745755s (cpu); 0.656212s (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 0.104488s (cpu); 0.0354423s (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 0.745755s (cpu); 0.656212s (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 0.0532221s (cpu); 0.033334s (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 0.0532221s (cpu); 0.033334s (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 0.203629s (cpu); 0.0399022s (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 0.449047s (cpu); 0.136207s (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 0.14663s (cpu); 0.0690011s (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 0.042726s (cpu); 0.0236173s (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 0.203629s (cpu); 0.0399022s (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 0.449047s (cpu); 0.136207s (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 0.14663s (cpu); 0.0690011s (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 0.042726s (cpu); 0.0236173s (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 0.0297058s (cpu); 0.0159538s (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 0.0297058s (cpu); 0.0159538s (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 0.151228s (cpu); 0.131468s (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 0.151228s (cpu); 0.131468s (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 0.760512s (cpu); 0.570769s (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 0.760512s (cpu); 0.570769s (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 0.187049s (cpu); 0.169244s (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 0.187049s (cpu); 0.169244s (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 0.829103s (cpu); 0.519474s (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 0.829103s (cpu); 0.519474s (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=
│ │ │  #:len=338
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzcwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzb2x2ZURlY29tcG9zYWJsZVN5c3RlbSxMaXN0KSwi
│ │ ├── ./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
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTZWVkIiwiU2VlZCIsIkRlcHRoIn0sIFByaW1hcnlU
│ │ ├── ./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/_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) + -Div(z) + 2*Div(y)
│ │ │  
│ │ │  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 0.0519229s (cpu); 0.0519516s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.386445s (cpu); 0.388413s (thread); 0s (gc)
│ │ │ + -- used 0.463844s (cpu); 0.463568s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.516665s (cpu); 0.447697s (thread); 0s (gc)
│ │ │ + -- used 0.605642s (cpu); 0.537038s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00277537s (cpu); 0.00285669s (thread); 0s (gc)
│ │ │ + -- used 0.000372739s (cpu); 0.00345815s (thread); 0s (gc)
│ │ │  
│ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 3.8001e-05s (cpu); 0.00208153s (thread); 0s (gc)
│ │ │ + -- used 0.000702487s (cpu); 0.00272974s (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 0.175331s (cpu); 0.111951s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
│ │ │  
│ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00560499s (cpu); 0.00590381s (thread); 0s (gc)
│ │ │ + -- used 0.0031829s (cpu); 0.00678676s (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/_is__S__N__C.out
│ │ │ @@ -1,26 +1,26 @@
│ │ │  -- -*- M2-comint -*- hash: 2360371518304120718
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : isSNC( D )
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : R = QQ[x, y];
│ │ │  
│ │ │  i5 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o5 = Div(y) + Div(x) + Div(x+y)
│ │ │ +o5 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : isSNC( D )
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │ @@ -36,27 +36,27 @@
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o11 = -2*Div(y, z) + Div(x, z)
│ │ │ +o11 = Div(x, z) + -2*Div(y, z)
│ │ │  
│ │ │  o11 : WeilDivisor on R
│ │ │  
│ │ │  i12 : isSNC( D, IsGraded => true )
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : R = QQ[x, y];
│ │ │  
│ │ │  i14 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o14 : WeilDivisor on R
│ │ │  
│ │ │  i15 : isSNC( D, IsGraded => true )
│ │ │  
│ │ │  o15 = 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 0.271021s (cpu); 0.19832s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.41735s (cpu); 0.34655s (thread); 0s (gc)
│ │ │ + -- used 0.512253s (cpu); 0.446363s (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 0.295696s (cpu); 0.131635s (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 5.77645s (cpu); 4.54113s (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 6.05823s (cpu); 4.85428s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.579493s (cpu); 0.401292s (thread); 0s (gc)
│ │ │ + -- used 0.567285s (cpu); 0.381133s (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 1.15959s (cpu); 0.384356s (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 0.014388s (cpu); 0.0167698s (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 0.269651s (cpu); 0.0919033s (thread); 0s (gc)
│ │ │  
│ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00492897s (cpu); 0.00705635s (thread); 0s (gc)
│ │ │ + -- used 0.00736137s (cpu); 0.00749192s (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 0.115936s (cpu); 0.0492875s (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 0.151945s (cpu); 0.151798s (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 0.0386514s (cpu); 0.0379046s (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 0.170457s (cpu); 0.104393s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/example-output/_ring_lp__Basic__Divisor_rp.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 5006859181202351713
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = 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 : ring( D )
│ │ │  
│ │ │  o3 = R
│ │ ├── ./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/_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) + -Div(z) + 2*Div(y)
│ │ │  
│ │ │  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) + -Div(z) + 2*Div(y)
│ │ │ │  
│ │ │ │  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 0.0519229s (cpu); 0.0519516s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
    │ │ │ 
    i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.386445s (cpu); 0.388413s (thread); 0s (gc)
│ │ │ + -- used 0.463844s (cpu); 0.463568s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    │ │ │ 
    i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.516665s (cpu); 0.447697s (thread); 0s (gc)
    │ │ │ + -- used 0.605642s (cpu); 0.537038s (thread); 0s (gc) │ │ │ 
    i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00277537s (cpu); 0.00285669s (thread); 0s (gc)
    │ │ │ + -- used 0.000372739s (cpu); 0.00345815s (thread); 0s (gc) │ │ │ 
    i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 3.8001e-05s (cpu); 0.00208153s (thread); 0s (gc)
│ │ │ + -- used 0.000702487s (cpu); 0.00272974s (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 0.175331s (cpu); 0.111951s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00560499s (cpu); 0.00590381s (thread); 0s (gc)
│ │ │ + -- used 0.0031829s (cpu); 0.00678676s (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 0.0519229s (cpu); 0.0519516s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ i13 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.386445s (cpu); 0.388413s (thread); 0s (gc) │ │ │ │ + -- used 0.463844s (cpu); 0.463568s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time dualize(M, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.516665s (cpu); 0.447697s (thread); 0s (gc) │ │ │ │ + -- used 0.605642s (cpu); 0.537038s (thread); 0s (gc) │ │ │ │ i15 : time dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00277537s (cpu); 0.00285669s (thread); 0s (gc) │ │ │ │ + -- used 0.000372739s (cpu); 0.00345815s (thread); 0s (gc) │ │ │ │ i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 3.8001e-05s (cpu); 0.00208153s (thread); 0s (gc) │ │ │ │ + -- used 0.000702487s (cpu); 0.00272974s (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 0.175331s (cpu); 0.111951s (thread); 0s (gc) │ │ │ │ │ │ │ │ o20 : Ideal of R │ │ │ │ i21 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00560499s (cpu); 0.00590381s (thread); 0s (gc) │ │ │ │ + -- used 0.0031829s (cpu); 0.00678676s (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 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    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/_is__S__N__C.html │ │ │ @@ -79,15 +79,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    │ │ │ 
    i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ 
    i3 : isSNC( D )
│ │ │  
│ │ │  o3 = false
    │ │ │ @@ -96,15 +96,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── 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 0.000321904s (cpu); 0.0029468s (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 0.000731504s (cpu); 0.00171919s (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 0.0812921s (cpu); 0.0168681s (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 8.5582e-05s (cpu); 0.00154131s (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 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : R = QQ[x, y];
    │ │ │ 
    i5 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o5 = Div(y) + Div(x) + Div(x+y)
│ │ │ +o5 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    │ │ │ 
    i6 : isSNC( D )
│ │ │  
│ │ │  o6 = false
    │ │ │ @@ -133,15 +133,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    │ │ │ 
    i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o11 = -2*Div(y, z) + Div(x, z)
│ │ │ +o11 = Div(x, z) + -2*Div(y, z)
│ │ │  
│ │ │  o11 : WeilDivisor on R
    │ │ │ 
    i12 : isSNC( D, IsGraded => true )
│ │ │  
│ │ │  o12 = true
    │ │ │ @@ -150,15 +150,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : R = QQ[x, y];
    │ │ │ 
    i14 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o14 : WeilDivisor on R
    │ │ │ 
    i15 : isSNC( D, IsGraded => true )
│ │ │  
│ │ │  o15 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -16,24 +16,24 @@ │ │ │ │ o a _B_o_o_l_e_a_n_ _v_a_l_u_e, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ This function returns true if the divisor is simple normal crossings, this │ │ │ │ includes checking that the ambient ring is regular. │ │ │ │ i1 : R = QQ[x, y, z] / ideal(x * y - z^2 ); │ │ │ │ i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ │ │ -o2 = Div(x, z) + -2*Div(y, z) │ │ │ │ +o2 = -2*Div(y, z) + Div(x, z) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : isSNC( D ) │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : R = QQ[x, y]; │ │ │ │ i5 : D = divisor(x*y*(x+y)) │ │ │ │ │ │ │ │ -o5 = Div(y) + Div(x) + Div(x+y) │ │ │ │ +o5 = Div(x+y) + Div(y) + Div(x) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ i6 : isSNC( D ) │ │ │ │ │ │ │ │ o6 = false │ │ │ │ i7 : R = QQ[x, y]; │ │ │ │ i8 : D = divisor(x*y*(x+1)) │ │ │ │ @@ -46,24 +46,24 @@ │ │ │ │ o9 = true │ │ │ │ If IsGraded is set to true (default false), then the divisor is treated as if │ │ │ │ it is on the $Proj$ of the ambient ring. In particular, non-SNC behavior at the │ │ │ │ origin is ignored. │ │ │ │ i10 : R = QQ[x, y, z] / ideal(x * y - z^2 ); │ │ │ │ i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ │ │ -o11 = -2*Div(y, z) + Div(x, z) │ │ │ │ +o11 = Div(x, z) + -2*Div(y, z) │ │ │ │ │ │ │ │ o11 : WeilDivisor on R │ │ │ │ i12 : isSNC( D, IsGraded => true ) │ │ │ │ │ │ │ │ o12 = true │ │ │ │ i13 : R = QQ[x, y]; │ │ │ │ i14 : D = divisor(x*y*(x+y)) │ │ │ │ │ │ │ │ -o14 = Div(y) + Div(x) + Div(x+y) │ │ │ │ +o14 = Div(x+y) + Div(y) + Div(x) │ │ │ │ │ │ │ │ o14 : WeilDivisor on R │ │ │ │ i15 : isSNC( D, IsGraded => true ) │ │ │ │ │ │ │ │ o15 = true │ │ │ │ i16 : R = QQ[x,y,z]; │ │ │ │ i17 : D = divisor(x*y*(x+y)) │ │ ├── ./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 0.271021s (cpu); 0.19832s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
    │ │ │ 
    i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.41735s (cpu); 0.34655s (thread); 0s (gc)
    │ │ │ + -- used 0.512253s (cpu); 0.446363s (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 0.295696s (cpu); 0.131635s (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 5.77645s (cpu); 4.54113s (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 6.05823s (cpu); 4.85428s (thread); 0s (gc) │ │ │ 
    i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.579493s (cpu); 0.401292s (thread); 0s (gc)
    │ │ │ + -- used 0.567285s (cpu); 0.381133s (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 1.15959s (cpu); 0.384356s (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 0.014388s (cpu); 0.0167698s (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 0.269651s (cpu); 0.0919033s (thread); 0s (gc) │ │ │ 
    i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00492897s (cpu); 0.00705635s (thread); 0s (gc)
    │ │ │ + -- used 0.00736137s (cpu); 0.00749192s (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 0.271021s (cpu); 0.19832s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 : Ideal of R │ │ │ │ i24 : time reflexify(J*R^1); │ │ │ │ - -- used 0.41735s (cpu); 0.34655s (thread); 0s (gc) │ │ │ │ + -- used 0.512253s (cpu); 0.446363s (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 0.295696s (cpu); 0.131635s (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 5.77645s (cpu); 4.54113s (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 6.05823s (cpu); 4.85428s (thread); 0s (gc) │ │ │ │ i33 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ │ - -- used 0.579493s (cpu); 0.401292s (thread); 0s (gc) │ │ │ │ + -- used 0.567285s (cpu); 0.381133s (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 1.15959s (cpu); 0.384356s (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 0.014388s (cpu); 0.0167698s (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 0.269651s (cpu); 0.0919033s (thread); 0s (gc) │ │ │ │ i41 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ │ - -- used 0.00492897s (cpu); 0.00705635s (thread); 0s (gc) │ │ │ │ + -- used 0.00736137s (cpu); 0.00749192s (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 0.115936s (cpu); 0.0492875s (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 0.151945s (cpu); 0.151798s (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 0.0386514s (cpu); 0.0379046s (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 0.170457s (cpu); 0.104393s (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 0.115936s (cpu); 0.0492875s (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 0.151945s (cpu); 0.151798s (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 0.0386514s (cpu); 0.0379046s (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 0.170457s (cpu); 0.104393s (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/_ring_lp__Basic__Divisor_rp.html │ │ │ @@ -75,15 +75,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    │ │ │ 
    i2 : D = 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 : ring( D )
│ │ │  
│ │ │  o3 = R
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,15 +13,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _r_i_n_g,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns the ambient ring of a divisor.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D = 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 : ring( D )
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/Divisor/html/_to__R__Weil__Divisor.html
│ │ │ @@ -78,36 +78,36 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ -
│ │ │            
│ │ │  
│ │ │          
                  
    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
│ │ │ @@ -3,27 +3,19 @@
│ │ │  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
│ │ │ +                              1   2   3     1   5     2   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
│ │ │ -                "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
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
│ │ │ @@ -87,31 +87,23 @@
│ │ │            
                  
    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
│ │ │ +                              1   2   3     1   5     2   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
│ │ │ -                "ring" => R
│ │ │ -                "vertices" => {x , x , x , x , x }
│ │ │ -                                1   2   3   4   5
│ │ │ -
│ │ │ -o4 : HyperGraph
    
│ │ │ +    		              
    i4 : randomHyperGraph(R,{3,2,4})
    
│ │ │      		          
                  
    i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch limit reached
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    The randomHyperGraph method will return null immediately if the sizes of the edges fail to pass the LYM-inequality: $1/(n choose D_1) + 1/(n choose D_2) + ... + 1/(n choose D_m) \leq 1$ where $n$ is the number of variables in R and $m$ is the length of D. Note that even if D passes this inequality, it is not necessarily true that there is some hypergraph with edge sizes given by D. See D. Lubell's "A short proof of Sperner's lemma," J. Combin. Theory, 1:299 (1966).
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,29 +26,21 @@
│ │ │ │  _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
│ │ │ │ +                              1   2   3     1   5     2   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
│ │ │ │ -                "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
│ │ │ │  The randomHyperGraph method will return null immediately if the sizes of the
│ │ │ │  edges fail to pass the LYM-inequality: $1/(n choose D_1) + 1/(n choose D_2) +
│ │ │ │  ... + 1/(n choose D_m) \leq 1$ where $n$ is the number of variables in R and
│ │ │ │  $m$ is the length of D. Note that even if D passes this inequality, it is not
│ │ │ │  necessarily true that there is some hypergraph with edge sizes given by D. See
│ │ ├── ./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
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│ │ ├── ./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
│ │ │ + -- .253943s 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]
    │ │ │ 
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .304758s elapsed
    │ │ │ + -- .253943s 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 │ │ │ │ + -- .253943s 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 0.000321904s (cpu); 0.0029468s (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 0.000731504s (cpu); 0.00171919s (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 0.0812921s (cpu); 0.0168681s (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 8.5582e-05s (cpu); 0.00154131s (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 1.64414s (cpu); 1.50525s (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 0.0159832s (cpu); 0.0171587s (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 1.71891s (cpu); 1.50291s (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 0.0200004s (cpu); 0.0167791s (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 0.000321904s (cpu); 0.0029468s (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 0.000731504s (cpu); 0.00171919s (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 0.0812921s (cpu); 0.0168681s (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 8.5582e-05s (cpu); 0.00154131s (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 1.64414s (cpu); 1.50525s (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 0.0159832s (cpu); 0.0171587s (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 1.64414s (cpu); 1.50525s (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 0.0159832s (cpu); 0.0171587s (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 1.71891s (cpu); 1.50291s (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 0.0200004s (cpu); 0.0167791s (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 1.71891s (cpu); 1.50291s (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 0.0200004s (cpu); 0.0167791s (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
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│ │ ├── ./usr/share/doc/Macaulay2/EllipticCurves/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
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│ │ ├── ./usr/share/doc/Macaulay2/EllipticIntegrals/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
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│ │ ├── ./usr/share/doc/Macaulay2/EngineTests/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,user=buildd,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
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│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:len=17
│ │ │  RW51bWVyYXRpb25DdXJ2ZXM=
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│ │ ├── ./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 0.0359259s (cpu); 0.0360255s (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 0.332002s (cpu); 0.278165s (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 0.12325s (cpu); 0.123198s (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 4.78929s (cpu); 4.20224s (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 0.126345s (cpu); 0.127092s (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 4.83105s (cpu); 4.25948s (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 1.63332s (cpu); 1.42721s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.15585s (cpu); 5.41896s (thread); 0s (gc)
│ │ │ + -- used 6.3793s (cpu); 5.18952s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.5404s (cpu); 1.29761s (thread); 0s (gc)
│ │ │ + -- used 1.54835s (cpu); 1.34088s (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 6.41281s (cpu); 5.231s (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 6.53465s (cpu); 5.36443s (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 0.0359259s (cpu); 0.0360255s (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 0.0359259s (cpu); 0.0360255s (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 0.332002s (cpu); 0.278165s (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 0.12325s (cpu); 0.123198s (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 4.78929s (cpu); 4.20224s (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 0.126345s (cpu); 0.127092s (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 4.83105s (cpu); 4.25948s (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 1.63332s (cpu); 1.42721s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │      		          
                  
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.15585s (cpu); 5.41896s (thread); 0s (gc)
│ │ │ + -- used 6.3793s (cpu); 5.18952s (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 1.54835s (cpu); 1.34088s (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 6.41281s (cpu); 5.231s (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 6.53465s (cpu); 5.36443s (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 0.332002s (cpu); 0.278165s (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 0.12325s (cpu); 0.123198s (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 4.78929s (cpu); 4.20224s (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 0.126345s (cpu); 0.127092s (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 4.83105s (cpu); 4.25948s (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 1.63332s (cpu); 1.42721s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 7.15585s (cpu); 5.41896s (thread); 0s (gc)
│ │ │ │ + -- used 6.3793s (cpu); 5.18952s (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 1.54835s (cpu); 1.34088s (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 6.41281s (cpu); 5.231s (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 6.53465s (cpu); 5.36443s (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 .00491035 seconds
│ │ │       -- used 0 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00315213 seconds
│ │ │ -     -- used .00397295 seconds
│ │ │ +     -- used .00565381 seconds
│ │ │ +     -- used .00584259 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00917258 seconds
│ │ │ -     -- used .023553 seconds
│ │ │ +     -- used .00917477 seconds
│ │ │ +     -- used .0240093 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0161827 seconds
│ │ │ -     -- used .187147 seconds
│ │ │ +     -- used .0174032 seconds
│ │ │ +     -- used .203372 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0342961 seconds
│ │ │ -     -- used .7618 seconds
│ │ │ +     -- used .0336706 seconds
│ │ │ +     -- used .84549 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 .00491035 seconds
│ │ │       -- used 0 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00315213 seconds
│ │ │ -     -- used .00397295 seconds
│ │ │ +     -- used .00565381 seconds
│ │ │ +     -- used .00584259 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00917258 seconds
│ │ │ -     -- used .023553 seconds
│ │ │ +     -- used .00917477 seconds
│ │ │ +     -- used .0240093 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0161827 seconds
│ │ │ -     -- used .187147 seconds
│ │ │ +     -- used .0174032 seconds
│ │ │ +     -- used .203372 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0342961 seconds
│ │ │ -     -- used .7618 seconds
│ │ │ +     -- used .0336706 seconds
│ │ │ +     -- used .84549 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 .00491035 seconds
│ │ │ │       -- used 0 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .00315213 seconds
│ │ │ │ -     -- used .00397295 seconds
│ │ │ │ +     -- used .00565381 seconds
│ │ │ │ +     -- used .00584259 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00917258 seconds
│ │ │ │ -     -- used .023553 seconds
│ │ │ │ +     -- used .00917477 seconds
│ │ │ │ +     -- used .0240093 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0161827 seconds
│ │ │ │ -     -- used .187147 seconds
│ │ │ │ +     -- used .0174032 seconds
│ │ │ │ +     -- used .203372 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .0342961 seconds
│ │ │ │ -     -- used .7618 seconds
│ │ │ │ +     -- used .0336706 seconds
│ │ │ │ +     -- used .84549 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
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gOC1kaW1lbnNpb25hbCBlY29ub21p
│ │ │  Y3MgcHJvYmxlbSIsICJsaW5lbnVtIiA9PiA0OCwgSW5wdXRzID0+IHtTUEFOe1RUeyJrayJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/ExteriorIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,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 0.151725s (cpu); 0.148961s (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 0.386443s (cpu); 0.262234s (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 0.425193s (cpu); 0.300296s (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 0.328392s (cpu); 0.203597s (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 0.304s (cpu); 0.183552s (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 0.421572s (cpu); 0.297166s (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 0.410404s (cpu); 0.272671s (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 14.7048s (cpu); 8.94793s (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 0.582034s (cpu); 0.37585s (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 1.03472s (cpu); 0.643777s (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 0.759368s (cpu); 0.431094s (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 4.93956s (cpu); 3.58063s (thread); 0s (gc)
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.03145s (cpu); 0.679038s (thread); 0s (gc)
│ │ │ + -- used 1.14852s (cpu); 0.780022s (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 3.70244s (cpu); 3.28011s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 3.40978s (cpu); 2.04641s (thread); 0s (gc)
│ │ │ + -- used 3.59863s (cpu); 2.21983s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 1.07836s (cpu); 0.777656s (thread); 0s (gc)
│ │ │ + -- used 1.25942s (cpu); 0.949095s (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 4.33102s (cpu); 2.60006s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 4.46603s (cpu); 2.88283s (thread); 0s (gc)
│ │ │ + -- used 4.77595s (cpu); 3.10063s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 14.8083s (cpu); 8.44014s (thread); 0s (gc)
│ │ │ + -- used 16.0676s (cpu); 9.70031s (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 12.1736s (cpu); 7.17691s (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 1.01858s (cpu); 0.692147s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 10.1165s (cpu); 7.01836s (thread); 0s (gc)
│ │ │ + -- used 11.4272s (cpu); 8.16519s (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 1.48057s (cpu); 1.0271s (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 0.189984s (cpu); 0.128483s (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 0.187338s (cpu); 0.129729s (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 0.679307s (cpu); 0.493977s (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 0.780473s (cpu); 0.527543s (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 0.51371s (cpu); 0.330287s (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
│ │ │ + -- 1.52755s 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
│ │ │ + -- .873203s 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 6.852e-05s (cpu); 0.00296429s (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 0.000980439s (cpu); 0.00275931s (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 0.000259643s (cpu); 0.00265577s (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 0.284879s (cpu); 0.170149s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00883907s (cpu); 0.0114568s (thread); 0s (gc)
│ │ │ + -- used 0.013753s (cpu); 0.0153648s (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 0.484212s (cpu); 0.485419s (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 1.52745s (cpu); 1.40994s (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 1.75085s (cpu); 1.31423s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.53229s (cpu); 4.74206s (thread); 0s (gc)
│ │ │ + -- used 7.42791s (cpu); 5.70944s (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 0.0240011s (cpu); 0.024977s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.493224s (cpu); 0.310068s (thread); 0s (gc)
│ │ │ + -- used 0.641285s (cpu); 0.441222s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.852714s (cpu); 0.524319s (thread); 0s (gc)
│ │ │ + -- used 0.989714s (cpu); 0.681822s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.322571s (cpu); 0.208202s (thread); 0s (gc)
│ │ │ + -- used 0.372437s (cpu); 0.248959s (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 7.42062s (cpu); 5.54373s (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 1.52204s (cpu); 1.19781s (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 0.159078s (cpu); 0.0968386s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.271628s (cpu); 0.149929s (thread); 0s (gc)
│ │ │ + -- used 0.299925s (cpu); 0.175435s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.725s (cpu); 1.31345s (thread); 0s (gc)
│ │ │ + -- used 1.68002s (cpu); 1.25081s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.13749s (cpu); 0.85499s (thread); 0s (gc)
│ │ │ + -- used 1.33178s (cpu); 1.01938s (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 0.539347s (cpu); 0.36151s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = true
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.17628s (cpu); 1.4555s (thread); 0s (gc)
│ │ │ + -- used 2.43019s (cpu); 1.67784s (thread); 0s (gc)
│ │ │  
│ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.165936s (cpu); 0.105402s (thread); 0s (gc)
│ │ │ + -- used 0.190522s (cpu); 0.13357s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.211774s (cpu); 0.158759s (thread); 0s (gc)
│ │ │ + -- used 0.237232s (cpu); 0.177104s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.48716s (cpu); 1.11753s (thread); 0s (gc)
│ │ │ + -- used 1.75319s (cpu); 1.38235s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.52406s (cpu); 1.15802s (thread); 0s (gc)
│ │ │ + -- used 1.87181s (cpu); 1.49748s (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 0.151725s (cpu); 0.148961s (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 0.386443s (cpu); 0.262234s (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 0.425193s (cpu); 0.300296s (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 0.328392s (cpu); 0.203597s (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 0.304s (cpu); 0.183552s (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 0.421572s (cpu); 0.297166s (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 0.410404s (cpu); 0.272671s (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 14.7048s (cpu); 8.94793s (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 0.582034s (cpu); 0.37585s (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 1.03472s (cpu); 0.643777s (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 0.759368s (cpu); 0.431094s (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 4.93956s (cpu); 3.58063s (thread); 0s (gc) │ │ │ 
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.03145s (cpu); 0.679038s (thread); 0s (gc)
│ │ │ + -- used 1.14852s (cpu); 0.780022s (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 3.70244s (cpu); 3.28011s (thread); 0s (gc) │ │ │ 
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 3.40978s (cpu); 2.04641s (thread); 0s (gc)
    │ │ │ + -- used 3.59863s (cpu); 2.21983s (thread); 0s (gc) │ │ │ 
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 1.07836s (cpu); 0.777656s (thread); 0s (gc)
│ │ │ + -- used 1.25942s (cpu); 0.949095s (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 4.33102s (cpu); 2.60006s (thread); 0s (gc) │ │ │ 
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 4.46603s (cpu); 2.88283s (thread); 0s (gc)
    │ │ │ + -- used 4.77595s (cpu); 3.10063s (thread); 0s (gc) │ │ │ 
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 14.8083s (cpu); 8.44014s (thread); 0s (gc)
│ │ │ + -- used 16.0676s (cpu); 9.70031s (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 12.1736s (cpu); 7.17691s (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 0.151725s (cpu); 0.148961s (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 0.386443s (cpu); 0.262234s (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 0.425193s (cpu); 0.300296s (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 0.328392s (cpu); 0.203597s (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 0.304s (cpu); 0.183552s (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 0.421572s (cpu); 0.297166s (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 0.410404s (cpu); 0.272671s (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 14.7048s (cpu); 8.94793s (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 0.582034s (cpu); 0.37585s (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 1.03472s (cpu); 0.643777s (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 0.759368s (cpu); 0.431094s (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 4.93956s (cpu); 3.58063s (thread); 0s (gc) │ │ │ │ i48 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>StrategyDefaultNonRandom) │ │ │ │ - -- used 1.03145s (cpu); 0.679038s (thread); 0s (gc) │ │ │ │ + -- used 1.14852s (cpu); 0.780022s (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 3.70244s (cpu); 3.28011s (thread); 0s (gc) │ │ │ │ i50 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>LexSmallest) │ │ │ │ - -- used 3.40978s (cpu); 2.04641s (thread); 0s (gc) │ │ │ │ + -- used 3.59863s (cpu); 2.21983s (thread); 0s (gc) │ │ │ │ i51 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>LexSmallestTerm) │ │ │ │ - -- used 1.07836s (cpu); 0.777656s (thread); 0s (gc) │ │ │ │ + -- used 1.25942s (cpu); 0.949095s (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 4.33102s (cpu); 2.60006s (thread); 0s (gc) │ │ │ │ i53 : time regularInCodimension(1, S/J, MaxMinors => 100, │ │ │ │ Strategy=>GRevLexSmallestTerm) │ │ │ │ - -- used 4.46603s (cpu); 2.88283s (thread); 0s (gc) │ │ │ │ + -- used 4.77595s (cpu); 3.10063s (thread); 0s (gc) │ │ │ │ i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points) │ │ │ │ - -- used 14.8083s (cpu); 8.44014s (thread); 0s (gc) │ │ │ │ + -- used 16.0676s (cpu); 9.70031s (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 12.1736s (cpu); 7.17691s (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 1.01858s (cpu); 0.692147s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    │ │ │ 
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 10.1165s (cpu); 7.01836s (thread); 0s (gc)
    │ │ │ + -- used 11.4272s (cpu); 8.16519s (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 1.48057s (cpu); 1.0271s (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 0.189984s (cpu); 0.128483s (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 0.187338s (cpu); 0.129729s (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 0.679307s (cpu); 0.493977s (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 0.780473s (cpu); 0.527543s (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 0.51371s (cpu); 0.330287s (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 1.01858s (cpu); 0.692147s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 10.1165s (cpu); 7.01836s (thread); 0s (gc)
│ │ │ │ + -- used 11.4272s (cpu); 8.16519s (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 1.48057s (cpu); 1.0271s (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 0.189984s (cpu); 0.128483s (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 0.187338s (cpu); 0.129729s (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 0.679307s (cpu); 0.493977s (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 0.780473s (cpu); 0.527543s (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 0.51371s (cpu); 0.330287s (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
│ │ │ + -- 1.52755s 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
│ │ │ + -- .873203s 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
│ │ │ │ + -- 1.52755s 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
│ │ │ │ + -- .873203s 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 6.852e-05s (cpu); 0.00296429s (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 0.000980439s (cpu); 0.00275931s (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 0.000259643s (cpu); 0.00265577s (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 6.852e-05s (cpu); 0.00296429s (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 0.000980439s (cpu); 0.00275931s (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 0.000259643s (cpu); 0.00265577s (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 0.284879s (cpu); 0.170149s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │      		          
                  
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00883907s (cpu); 0.0114568s (thread); 0s (gc)
│ │ │ + -- used 0.013753s (cpu); 0.0153648s (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 0.284879s (cpu); 0.170149s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ │ - -- used 0.00883907s (cpu); 0.0114568s (thread); 0s (gc)
│ │ │ │ + -- used 0.013753s (cpu); 0.0153648s (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 0.484212s (cpu); 0.485419s (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 1.52745s (cpu); 1.40994s (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 0.484212s (cpu); 0.485419s (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 1.52745s (cpu); 1.40994s (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 1.75085s (cpu); 1.31423s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    │ │ │ 
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.53229s (cpu); 4.74206s (thread); 0s (gc)
    │ │ │ + -- used 7.42791s (cpu); 5.70944s (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 0.0240011s (cpu); 0.024977s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    │ │ │ 
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.493224s (cpu); 0.310068s (thread); 0s (gc)
│ │ │ + -- used 0.641285s (cpu); 0.441222s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    │ │ │ 
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.852714s (cpu); 0.524319s (thread); 0s (gc)
│ │ │ + -- used 0.989714s (cpu); 0.681822s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    │ │ │ 
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.322571s (cpu); 0.208202s (thread); 0s (gc)
│ │ │ + -- used 0.372437s (cpu); 0.248959s (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 7.42062s (cpu); 5.54373s (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 1.52204s (cpu); 1.19781s (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 0.159078s (cpu); 0.0968386s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    │ │ │ 
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.271628s (cpu); 0.149929s (thread); 0s (gc)
│ │ │ + -- used 0.299925s (cpu); 0.175435s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    │ │ │ 
    i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.725s (cpu); 1.31345s (thread); 0s (gc)
│ │ │ + -- used 1.68002s (cpu); 1.25081s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    │ │ │ 
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.13749s (cpu); 0.85499s (thread); 0s (gc)
│ │ │ + -- used 1.33178s (cpu); 1.01938s (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 0.539347s (cpu); 0.36151s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = true
    │ │ │ 
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.17628s (cpu); 1.4555s (thread); 0s (gc)
    │ │ │ + -- used 2.43019s (cpu); 1.67784s (thread); 0s (gc) │ │ │ 
    i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.165936s (cpu); 0.105402s (thread); 0s (gc)
│ │ │ + -- used 0.190522s (cpu); 0.13357s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
    │ │ │ 
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.211774s (cpu); 0.158759s (thread); 0s (gc)
│ │ │ + -- used 0.237232s (cpu); 0.177104s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    │ │ │ 
    i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.48716s (cpu); 1.11753s (thread); 0s (gc)
│ │ │ + -- used 1.75319s (cpu); 1.38235s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    │ │ │ 
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.52406s (cpu); 1.15802s (thread); 0s (gc)
│ │ │ + -- used 1.87181s (cpu); 1.49748s (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 1.75085s (cpu); 1.31423s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = true │ │ │ │ i9 : time regularInCodimension(2, S) │ │ │ │ - -- used 6.53229s (cpu); 4.74206s (thread); 0s (gc) │ │ │ │ + -- used 7.42791s (cpu); 5.70944s (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 0.0240011s (cpu); 0.024977s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 = -1 │ │ │ │ i13 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.493224s (cpu); 0.310068s (thread); 0s (gc) │ │ │ │ + -- used 0.641285s (cpu); 0.441222s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 = true │ │ │ │ i14 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.852714s (cpu); 0.524319s (thread); 0s (gc) │ │ │ │ + -- used 0.989714s (cpu); 0.681822s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.322571s (cpu); 0.208202s (thread); 0s (gc) │ │ │ │ + -- used 0.372437s (cpu); 0.248959s (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) │ │ │ │ @@ -462,15 +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 │ │ │ │ +internalChooseMinor: Choosing -- used 7.42062s (cpu); 5.54373s (thread); 0s │ │ │ │ (gc) │ │ │ │ LexSmallestTerm │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing Random │ │ │ │ internalChooseMinor: Choosing GRevLexSmallest │ │ │ │ internalChooseMinor: Choosing RandomNonZero │ │ │ │ internalChooseMinor: Choosing GRevLexSmallest │ │ │ │ @@ -517,15 +517,15 @@ │ │ │ │ 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 1.52204s (cpu); 1.19781s (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 +592,51 @@ │ │ │ │ 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 0.159078s (cpu); 0.0968386s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 = true │ │ │ │ i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.271628s (cpu); 0.149929s (thread); 0s (gc) │ │ │ │ + -- used 0.299925s (cpu); 0.175435s (thread); 0s (gc) │ │ │ │ │ │ │ │ o22 = true │ │ │ │ i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 1.725s (cpu); 1.31345s (thread); 0s (gc) │ │ │ │ + -- used 1.68002s (cpu); 1.25081s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = true │ │ │ │ i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 1.13749s (cpu); 0.85499s (thread); 0s (gc) │ │ │ │ + -- used 1.33178s (cpu); 1.01938s (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 0.539347s (cpu); 0.36151s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = true │ │ │ │ i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 2.17628s (cpu); 1.4555s (thread); 0s (gc) │ │ │ │ + -- used 2.43019s (cpu); 1.67784s (thread); 0s (gc) │ │ │ │ i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.165936s (cpu); 0.105402s (thread); 0s (gc) │ │ │ │ + -- used 0.190522s (cpu); 0.13357s (thread); 0s (gc) │ │ │ │ │ │ │ │ o29 = true │ │ │ │ i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.211774s (cpu); 0.158759s (thread); 0s (gc) │ │ │ │ + -- used 0.237232s (cpu); 0.177104s (thread); 0s (gc) │ │ │ │ │ │ │ │ o30 = true │ │ │ │ i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 1.48716s (cpu); 1.11753s (thread); 0s (gc) │ │ │ │ + -- used 1.75319s (cpu); 1.38235s (thread); 0s (gc) │ │ │ │ │ │ │ │ o31 = true │ │ │ │ i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 1.52406s (cpu); 1.15802s (thread); 0s (gc) │ │ │ │ + -- used 1.87181s (cpu); 1.49748s (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 0.00409759s (cpu); 0.00280852s (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 0.00579087s (cpu); 0.00639687s (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 0.00409759s (cpu); 0.00280852s (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 0.00579087s (cpu); 0.00639687s (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 0.00409759s (cpu); 0.00280852s (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 0.00579087s (cpu); 0.00639687s (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 => 0x7f9796fc4610} │ │ │ │ │ │ i3 : address x │ │ │ │ │ │ -o3 = 0x7f5daa523ec0 │ │ │ +o3 = 0x7f9796fc4610 │ │ │ │ │ │ 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 = {0x7f75535b4440, 0x7f75535b4430, 0x7f75535b4420} │ │ │ │ │ │ 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 = {0x7fd11da292a0, 0x7fd11da29280, 0x7fd11da29270} │ │ │ │ │ │ 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 = 0x7f615a9ef7f0 │ │ │ │ │ │ o1 : Pointer │ │ │ │ │ │ i2 : voidstar ptr │ │ │ │ │ │ -o2 = 0x7f3cb62b2350 │ │ │ +o2 = 0x7f615a9ef7f0 │ │ │ │ │ │ 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 = 0x7f24f30d5860 │ │ │ │ │ │ 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 = 0x7fecd41edd60 │ │ │ │ │ │ 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" => 6.90582e-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 => 0x7fb762630740} │ │ │ │ │ │ i3 : ptr = address x │ │ │ │ │ │ -o3 = 0x7f9390dc9080 │ │ │ +o3 = 0x7fb762630740 │ │ │ │ │ │ o3 : Pointer │ │ │ │ │ │ i4 : ptr + 5 │ │ │ │ │ │ -o4 = 0x7f9390dc9085 │ │ │ +o4 = 0x7fb762630745 │ │ │ │ │ │ o4 : Pointer │ │ │ │ │ │ i5 : ptr - 3 │ │ │ │ │ │ -o5 = 0x7f9390dc907d │ │ │ +o5 = 0x7fb76263073d │ │ │ │ │ │ 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{0x7f0fc49a1550, 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 = 0x7f7cf519a990 │ │ │ │ │ │ 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 = 0x560d49b33bc0 │ │ │ │ │ │ o1 : Pointer │ │ │ │ │ │ i2 : address int 5 │ │ │ │ │ │ -o2 = 0x7f0602c6c0f0 │ │ │ +o2 = 0x7f134cf75d20 │ │ │ │ │ │ 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 = 0x7f59040639d0 │ │ │ │ │ │ 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 = 0x7f88f1b1a9a0 │ │ │ │ │ │ o1 : ForeignObject of type void* │ │ │ │ │ │ i2 : ptr = getMemory(8, Atomic => true) │ │ │ │ │ │ -o2 = 0x7f39d4990d40 │ │ │ +o2 = 0x7f88ee9884e0 │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ │ │ i3 : ptr = getMemory int │ │ │ │ │ │ -o3 = 0x7f39d4990c50 │ │ │ +o3 = 0x7f88ee9883f0 │ │ │ │ │ │ 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 0x7f0d080639d0 │ │ │ +freeing memory at 0x7f0d08066c50 │ │ │ +freeing memory at 0x7f0d08069180 │ │ │ +freeing memory at 0x7f0d08063bc0 │ │ │ +freeing memory at 0x7f0d0806a210 │ │ │ +freeing memory at 0x7f0d080639f0 │ │ │ +freeing memory at 0x7f0d0806a230 │ │ │ +freeing memory at 0x7f0d0805f3f0 │ │ │ +freeing memory at 0x7f0d08063bf0 │ │ │ │ │ │ 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 = 0x7f2f56046c00 │ │ │ │ │ │ o5 : ForeignObject of type void* │ │ │ │ │ │ i6 : value x │ │ │ │ │ │ -o6 = 0x7f769ebd8d90 │ │ │ +o6 = 0x7f2f56046c00 │ │ │ │ │ │ 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 => 0x7f9796fc4610} │ │ │
    │ │ │
    │ │ │

    To get this, use address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f5daa523ec0
│ │ │ +o3 = 0x7f9796fc4610
│ │ │  
│ │ │  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 => 0x7f9796fc4610} │ │ │ │ To get this, use _a_d_d_r_e_s_s. │ │ │ │ i3 : address x │ │ │ │ │ │ │ │ -o3 = 0x7f5daa523ec0 │ │ │ │ +o3 = 0x7f9796fc4610 │ │ │ │ │ │ │ │ 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 = {0x7f75535b4440, 0x7f75535b4430, 0x7f75535b4420}
│ │ │  
│ │ │  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 = {0x7f75535b4440, 0x7f75535b4430, 0x7f75535b4420} │ │ │ │ │ │ │ │ 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 = {0x7fd11da292a0, 0x7fd11da29280, 0x7fd11da29270}
│ │ │  
│ │ │  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 = {0x7fd11da292a0, 0x7fd11da29280, 0x7fd11da29270}
│ │ │ │  
│ │ │ │  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 = 0x7f615a9ef7f0
│ │ │  
│ │ │  o1 : Pointer
    
│ │ │      		          
                  
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f3cb62b2350
│ │ │ +o2 = 0x7f615a9ef7f0
│ │ │  
│ │ │  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 = 0x7f615a9ef7f0
│ │ │ │  
│ │ │ │  o1 : Pointer
│ │ │ │  i2 : voidstar ptr
│ │ │ │  
│ │ │ │ -o2 = 0x7f3cb62b2350
│ │ │ │ +o2 = 0x7f615a9ef7f0
│ │ │ │  
│ │ │ │  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 = 0x7f24f30d5860
│ │ │  
│ │ │  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 = 0x7f24f30d5860
│ │ │ │  
│ │ │ │  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 = 0x7fecd41edd60
│ │ │  
│ │ │  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 = 0x7fecd41edd60
│ │ │ │  
│ │ │ │  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" => 6.90582e-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" => 6.90582e-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 => 0x7fb762630740}
│ │ │      		          
    
│ │ │          
    
│ │ │            
    These pointers can be accessed using address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │          
                  
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f9390dc9080
│ │ │ +o3 = 0x7fb762630740
│ │ │  
│ │ │  o3 : Pointer
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    Simple arithmetic can be performed on pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f9390dc9085
│ │ │ +o4 = 0x7fb762630745
│ │ │  
│ │ │  o4 : Pointer
    
│ │ │      		          
                  
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f9390dc907d
│ │ │ +o5 = 0x7fb76263073d
│ │ │  
│ │ │  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 => 0x7fb762630740}
│ │ │ │  These pointers can be accessed using _a_d_d_r_e_s_s.
│ │ │ │  i3 : ptr = address x
│ │ │ │  
│ │ │ │ -o3 = 0x7f9390dc9080
│ │ │ │ +o3 = 0x7fb762630740
│ │ │ │  
│ │ │ │  o3 : Pointer
│ │ │ │  Simple arithmetic can be performed on pointers.
│ │ │ │  i4 : ptr + 5
│ │ │ │  
│ │ │ │ -o4 = 0x7f9390dc9085
│ │ │ │ +o4 = 0x7fb762630745
│ │ │ │  
│ │ │ │  o4 : Pointer
│ │ │ │  i5 : ptr - 3
│ │ │ │  
│ │ │ │ -o5 = 0x7f9390dc907d
│ │ │ │ +o5 = 0x7fb76263073d
│ │ │ │  
│ │ │ │  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{0x7f0fc49a1550, mpfr} │ │ │
    │ │ │ │ │ │
    │ │ │

    Functions and methods returning a shared library:

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -12,15 +12,15 @@ │ │ │ │ i1 : mpfr = openSharedLibrary "mpfr" │ │ │ │ │ │ │ │ o1 = mpfr │ │ │ │ │ │ │ │ o1 : SharedLibrary │ │ │ │ i2 : peek mpfr │ │ │ │ │ │ │ │ -o2 = SharedLibrary{0x7f2afa172550, mpfr} │ │ │ │ +o2 = SharedLibrary{0x7f0fc49a1550, mpfr} │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa sshhaarreedd lliibbrraarryy:: ********** │ │ │ │ * _o_p_e_n_S_h_a_r_e_d_L_i_b_r_a_r_y -- open a shared library │ │ │ │ ********** MMeetthhooddss tthhaatt uussee aa sshhaarreedd lliibbrraarryy:: ********** │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) -- see │ │ │ │ _f_o_r_e_i_g_n_F_u_n_c_t_i_o_n -- construct a foreign function │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,VisibleList) -- see │ │ │ │ _f_o_r_e_i_g_n_F_u_n_c_t_i_o_n -- construct a foreign function │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/__st_spvoidstar_sp_eq_sp__Thing.html │ │ │ @@ -75,15 +75,15 @@ │ │ │ o1 = 5 │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ 
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f7a7a509710
│ │ │ +o2 = 0x7f7cf519a990
│ │ │  
│ │ │  o2 : Pointer
    │ │ │ 
    i3 : *ptr = int 6
│ │ │  
│ │ │  o3 = 6
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7f7a7a509710
│ │ │ │ +o2 = 0x7f7cf519a990
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : *ptr = int 6
│ │ │ │  
│ │ │ │  o3 = 6
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_address.html
│ │ │ @@ -70,27 +70,27 @@
│ │ │          
    
│ │ │            
    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 = 0x560d49b33bc0
│ │ │  
│ │ │  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 = 0x7f134cf75d20
│ │ │  
│ │ │  o2 : Pointer
    
│ │ │      		          
    
│ │ │        
│ │ │        
    
│ │ │          
    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 = 0x560d49b33bc0
│ │ │ │  
│ │ │ │  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 = 0x7f134cf75d20
│ │ │ │  
│ │ │ │  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 = 0x7f59040639d0
│ │ │  
│ │ │  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 = 0x7f59040639d0 │ │ │ │ │ │ │ │ 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 = 0x7f88f1b1a9a0
│ │ │  
│ │ │  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 = 0x7f88ee9884e0
│ │ │  
│ │ │  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 = 0x7f88ee9883f0
│ │ │  
│ │ │  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 = 0x7f88f1b1a9a0 │ │ │ │ │ │ │ │ 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 = 0x7f88ee9884e0 │ │ │ │ │ │ │ │ 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 = 0x7f88ee9883f0 │ │ │ │ │ │ │ │ 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 0x7f0d080639d0 │ │ │ +freeing memory at 0x7f0d08066c50 │ │ │ +freeing memory at 0x7f0d08069180 │ │ │ +freeing memory at 0x7f0d08063bc0 │ │ │ +freeing memory at 0x7f0d0806a210 │ │ │ +freeing memory at 0x7f0d080639f0 │ │ │ +freeing memory at 0x7f0d0806a230 │ │ │ +freeing memory at 0x7f0d0805f3f0 │ │ │ +freeing memory at 0x7f0d08063bf0 │ │ │
    │ │ │ │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │
    • │ │ │ ├── html2text {} │ │ │ │ @@ -32,21 +32,21 @@ │ │ │ │ i3 : finalizer = x -> (print("freeing memory at " | net x); free x) │ │ │ │ │ │ │ │ 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 0x7f0d080639d0 │ │ │ │ +freeing memory at 0x7f0d08066c50 │ │ │ │ +freeing memory at 0x7f0d08069180 │ │ │ │ +freeing memory at 0x7f0d08063bc0 │ │ │ │ +freeing memory at 0x7f0d0806a210 │ │ │ │ +freeing memory at 0x7f0d080639f0 │ │ │ │ +freeing memory at 0x7f0d0806a230 │ │ │ │ +freeing memory at 0x7f0d0805f3f0 │ │ │ │ +freeing memory at 0x7f0d08063bf0 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_t_M_e_m_o_r_y -- allocate memory using the garbage collector │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _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 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_value_lp__Foreign__Object_rp.html │ │ │ @@ -108,22 +108,22 @@ │ │ │
      │ │ │

      Foreign pointer objects are converted to Pointer objects.

      │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7f769ebd8d90
│ │ │ +o5 = 0x7f2f56046c00
│ │ │  
│ │ │  o5 : ForeignObject of type void*
      │ │ │ 
      i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f769ebd8d90
│ │ │ +o6 = 0x7f2f56046c00
│ │ │  
│ │ │  o6 : Pointer
      │ │ │
      │ │ │
      │ │ │

      Foreign string objects are converted to strings.

      │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -35,20 +35,20 @@ │ │ │ │ │ │ │ │ o4 = 5 │ │ │ │ │ │ │ │ o4 : RR (of precision 53) │ │ │ │ Foreign pointer objects are converted to _P_o_i_n_t_e_r objects. │ │ │ │ i5 : x = voidstar address int 5 │ │ │ │ │ │ │ │ -o5 = 0x7f769ebd8d90 │ │ │ │ +o5 = 0x7f2f56046c00 │ │ │ │ │ │ │ │ o5 : ForeignObject of type void* │ │ │ │ i6 : value x │ │ │ │ │ │ │ │ -o6 = 0x7f769ebd8d90 │ │ │ │ +o6 = 0x7f2f56046c00 │ │ │ │ │ │ │ │ o6 : Pointer │ │ │ │ Foreign string objects are converted to strings. │ │ │ │ i7 : x = charstar "Hello, world!" │ │ │ │ │ │ │ │ o7 = Hello, world! │ │ ├── ./usr/share/doc/Macaulay2/FormalGroupLaws/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,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 │ │ │ c2VyaWVzKFJpbmdFbGVtZW50LFpaKQ== │ │ │ #:len=1234 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29uc3RydWN0aW5nIGEgZm9ybWFsIHNl │ │ │ cmllcyIsICJsaW5lbnVtIiA9PiAzNzIsIElucHV0cyA9PiB7U1BBTntUVHsicyJ9LCIsICIsU1BB │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,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 │ │ │ dG9yaWNHcm9lYm5lcg== │ │ │ #:len=2522 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2FsY3VsYXRlcyBhIEdyb2VibmVyIGJh │ │ │ c2lzIG9mIHRoZSB0b3JpYyBpZGVhbCBJX0EsIGdpdmVuIEE7IGludm9rZXMgXCJncm9lYm5lclwi │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/example-output/_put__Matrix.out │ │ │ @@ -6,27 +6,27 @@ │ │ │ | 1 2 3 4 | │ │ │ │ │ │ 2 4 │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i2 : s = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-26596-0/0 │ │ │ +o2 = /tmp/M2-42931-0/0 │ │ │ │ │ │ i3 : F = openOut(s) │ │ │ │ │ │ -o3 = /tmp/M2-26596-0/0 │ │ │ +o3 = /tmp/M2-42931-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : putMatrix(F,A) │ │ │ │ │ │ i5 : close(F) │ │ │ │ │ │ -o5 = /tmp/M2-26596-0/0 │ │ │ +o5 = /tmp/M2-42931-0/0 │ │ │ │ │ │ o5 : File │ │ │ │ │ │ i6 : getMatrix(s) │ │ │ │ │ │ o6 = | 1 1 1 1 | │ │ │ | 1 2 3 4 | │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/html/_put__Matrix.html │ │ │ @@ -75,30 +75,30 @@ │ │ │ │ │ │ 2 4 │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ 
    i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-26596-0/0
    │ │ │ +o2 = /tmp/M2-42931-0/0 │ │ │ 
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-26596-0/0
│ │ │ +o3 = /tmp/M2-42931-0/0
│ │ │  
│ │ │  o3 : File
    │ │ │ 
    i4 : putMatrix(F,A)
    │ │ │ 
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-26596-0/0
│ │ │ +o5 = /tmp/M2-42931-0/0
│ │ │  
│ │ │  o5 : File
    │ │ │ 
    i6 : getMatrix(s)
│ │ │  
│ │ │  o6 = | 1 1 1 1 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,24 +17,24 @@
│ │ │ │  o1 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ │ │  
│ │ │ │                2       4
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  i2 : s = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-26596-0/0
│ │ │ │ +o2 = /tmp/M2-42931-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-26596-0/0
│ │ │ │ +o3 = /tmp/M2-42931-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-26596-0/0
│ │ │ │ +o5 = /tmp/M2-42931-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : getMatrix(s)
│ │ │ │  
│ │ │ │  o6 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FourierMotzkin/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,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
│ │ │  Zm91cmllck1vdHpraW4=
│ │ │  #:len=3383
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW50ZXJjaGFuZ2UgaW5lcXVhbGl0eS9n
│ │ │  ZW5lcmF0b3IgcmVwcmVzZW50YXRpb24gb2YgYSBwb2x5aGVkcmFsIGNvbmUiLCAibGluZW51bSIg
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb  9 22:54:37 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=998,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
│ │ │  aXNGUFQoLi4uLFFHb3JlbnN0ZWluSW5kZXg9Pi4uLik=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTI0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tpc0ZQVCxRR29yZW5zdGVpbkluZGV4XSwiaXNGUFQo
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out
│ │ │ @@ -155,31 +155,31 @@
│ │ │  i26 : numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true) -- FinalAttempt improves the estimate slightly
│ │ │  
│ │ │  o26 = {.142067, .144}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 2.18823s (cpu); 1.1308s (thread); 0s (gc)
│ │ │ + -- used 2.49829s (cpu); 1.34586s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.28867s (cpu); 0.680061s (thread); 0s (gc)
│ │ │ + -- used 1.46692s (cpu); 0.866239s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.02845s (cpu); 0.545149s (thread); 0s (gc)
│ │ │ + -- used 1.21568s (cpu); 0.724187s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out
│ │ │ @@ -43,34 +43,34 @@
│ │ │  o12 = 220
│ │ │  
│ │ │  i13 : R = ZZ/17[x,y,z];
│ │ │  
│ │ │  i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
│ │ │  
│ │ │  i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00824026s (cpu); 0.00665598s (thread); 0s (gc)
│ │ │ + -- used 0.00798928s (cpu); 0.00526796s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.504015s (cpu); 0.377564s (thread); 0s (gc)
│ │ │ + -- used 0.436861s (cpu); 0.314485s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
│ │ │  
│ │ │  i17 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i18 : f = x^3 + y^3 + z^3 + x*y*z;
│ │ │  
│ │ │  i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.438746s (cpu); 0.246817s (thread); 0s (gc)
│ │ │ + -- used 0.427572s (cpu); 0.229273s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.52422s (cpu); 1.11548s (thread); 0s (gc)
│ │ │ + -- used 1.45998s (cpu); 1.26371s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
│ │ │  
│ │ │  i21 : R = ZZ/3[x,y];
│ │ │  
│ │ │  i22 : M = ideal(x, y);
│ │ │  
│ │ │ @@ -85,34 +85,34 @@
│ │ │  o24 = 8
│ │ │  
│ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │  
│ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.01826s (cpu); 0.572049s (thread); 0s (gc)
│ │ │ + -- used 1.10417s (cpu); 0.678721s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.37637s (cpu); 0.772844s (thread); 0s (gc)
│ │ │ + -- used 1.53838s (cpu); 0.940415s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
│ │ │  
│ │ │  i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 1.72392s (cpu); 1.38073s (thread); 0s (gc)
│ │ │ + -- used 1.74101s (cpu); 1.45864s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.60114s (cpu); 0.47435s (thread); 0s (gc)
│ │ │ + -- used 0.61699s (cpu); 0.564112s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
│ │ │  
│ │ │  i32 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html
│ │ │ @@ -324,33 +324,33 @@
│ │ │
    │ │ │
    │ │ │

    The computations performed when FinalAttempt is set to true are often slow, and often fail to improve the estimate, and for this reason, this option should be used sparingly. It is often more effective to increase the values of Attempts or DepthOfSearch, instead.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -229,29 +229,29 @@ │ │ │ │ │ │ │ │ o26 : List │ │ │ │ The computations performed when FinalAttempt is set to true are often slow, and │ │ │ │ often fail to improve the estimate, and for this reason, this option should be │ │ │ │ used sparingly. It is often more effective to increase the values of Attempts │ │ │ │ or DepthOfSearch, instead. │ │ │ │ i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true) │ │ │ │ - -- used 2.18823s (cpu); 1.1308s (thread); 0s (gc) │ │ │ │ + -- used 2.49829s (cpu); 1.34586s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = {.142067, .144} │ │ │ │ │ │ │ │ o27 : List │ │ │ │ i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7) │ │ │ │ - -- used 1.28867s (cpu); 0.680061s (thread); 0s (gc) │ │ │ │ + -- used 1.46692s (cpu); 0.866239s (thread); 0s (gc) │ │ │ │ │ │ │ │ 1 │ │ │ │ o28 = - │ │ │ │ 7 │ │ │ │ │ │ │ │ o28 : QQ │ │ │ │ i29 : time fpt(f, DepthOfSearch => 4) │ │ │ │ - -- used 1.02845s (cpu); 0.545149s (thread); 0s (gc) │ │ │ │ + -- used 1.21568s (cpu); 0.724187s (thread); 0s (gc) │ │ │ │ │ │ │ │ 1 │ │ │ │ o29 = - │ │ │ │ 7 │ │ │ │ │ │ │ │ o29 : QQ │ │ │ │ As seen in several examples above, when the exact answer is not found, a list │ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ @@ -175,21 +175,21 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 2.18823s (cpu); 1.1308s (thread); 0s (gc)
│ │ │ + -- used 2.49829s (cpu); 1.34586s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    │ │ │ 
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.28867s (cpu); 0.680061s (thread); 0s (gc)
│ │ │ + -- used 1.46692s (cpu); 0.866239s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    │ │ │ 
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.02845s (cpu); 0.545149s (thread); 0s (gc)
│ │ │ + -- used 1.21568s (cpu); 0.724187s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
    │ │ │ 
    i13 : R = ZZ/17[x,y,z];
    │ │ │ 
    i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
    │ │ │ 
    i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00824026s (cpu); 0.00665598s (thread); 0s (gc)
│ │ │ + -- used 0.00798928s (cpu); 0.00526796s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    │ │ │ 
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.504015s (cpu); 0.377564s (thread); 0s (gc)
│ │ │ + -- used 0.436861s (cpu); 0.314485s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
    │ │ │
    │ │ │
    │ │ │

    The valid values for the option ContainmentTest are FrobeniusPower, FrobeniusRoot, and StandardPower. The default value of this option depends on what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to StandardPower if frobeniusNu is passed an ideal $I$. We describe the consequences of setting ContainmentTest to each of these values below.

    │ │ │

    First, if ContainmentTest is set to StandardPower, then the ideal containments checked when computing frobeniusNu(e,I,J) are verified directly. That is, the standard power $I^n$ is first computed, and a check is then run to see if it is contained in the $p^e$-th Frobenius power of $J$.

    │ │ │ @@ -200,21 +200,21 @@ │ │ │ 
    i17 : R = ZZ/5[x,y,z];
    │ │ │ │ │ │ │ │ │ 
    i18 : f = x^3 + y^3 + z^3 + x*y*z;
    │ │ │ │ │ │ │ │ │ 
    i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.438746s (cpu); 0.246817s (thread); 0s (gc)
│ │ │ + -- used 0.427572s (cpu); 0.229273s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    │ │ │ │ │ │ │ │ │ 
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.52422s (cpu); 1.11548s (thread); 0s (gc)
│ │ │ + -- used 1.45998s (cpu); 1.26371s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
    │ │ │ │ │ │ │ │ │
    │ │ │

    Finally, when ContainmentTest is set to FrobeniusPower, then instead of producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as defined in the paper Frobenius Powers by Hernández, Teixeira, and Witt, which can be computed with the function frobeniusPower, from the TestIdeals package. In particular, frobeniusNu(e,I,J) and frobeniusNu(e,I,J,ContainmentTest=>FrobeniusPower) need not agree. However, they will agree when $I$ is a principal ideal.

    │ │ │
    │ │ │ @@ -246,38 +246,38 @@ │ │ │ 
    i25 : R = ZZ/5[x,y,z];
    │ │ │ │ │ │ │ │ │ 
    i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
    │ │ │ │ │ │ │ │ │ 
    i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.01826s (cpu); 0.572049s (thread); 0s (gc)
│ │ │ + -- used 1.10417s (cpu); 0.678721s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    │ │ │ │ │ │ │ │ │ 
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.37637s (cpu); 0.772844s (thread); 0s (gc)
│ │ │ + -- used 1.53838s (cpu); 0.940415s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
    │ │ │ │ │ │ │ │ │ 
    i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
    │ │ │ │ │ │ │ │ │ 
    i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 1.72392s (cpu); 1.38073s (thread); 0s (gc)
│ │ │ + -- used 1.74101s (cpu); 1.45864s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    │ │ │ │ │ │ │ │ │ 
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.60114s (cpu); 0.47435s (thread); 0s (gc)
│ │ │ + -- used 0.61699s (cpu); 0.564112s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
    │ │ │ │ │ │ │ │ │
    │ │ │

    The option AtOrigin (default value true) can be turned off to tell frobeniusNu to effectively do the computation over all possible maximal ideals $J$ and take the minimum.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -107,19 +107,19 @@ │ │ │ │ algorithms, namely diagonal polynomials, binomials, forms in two variables, and │ │ │ │ polynomials whose factors are in simple normal crossing. This feature can be │ │ │ │ disabled by setting the option UseSpecialAlgorithms (default value true) to │ │ │ │ false. │ │ │ │ i13 : R = ZZ/17[x,y,z]; │ │ │ │ i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial │ │ │ │ i15 : time frobeniusNu(3, f) │ │ │ │ - -- used 0.00824026s (cpu); 0.00665598s (thread); 0s (gc) │ │ │ │ + -- used 0.00798928s (cpu); 0.00526796s (thread); 0s (gc) │ │ │ │ │ │ │ │ o15 = 3756 │ │ │ │ i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false) │ │ │ │ - -- used 0.504015s (cpu); 0.377564s (thread); 0s (gc) │ │ │ │ + -- used 0.436861s (cpu); 0.314485s (thread); 0s (gc) │ │ │ │ │ │ │ │ o16 = 3756 │ │ │ │ The valid values for the option ContainmentTest are FrobeniusPower, │ │ │ │ FrobeniusRoot, and StandardPower. The default value of this option depends on │ │ │ │ what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to │ │ │ │ FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to │ │ │ │ StandardPower if frobeniusNu is passed an ideal $I$. We describe the │ │ │ │ @@ -134,19 +134,19 @@ │ │ │ │ is contained in $J$. The output is unaffected, but this option often speeds up │ │ │ │ computations, specially when a polynomial or principal ideal is passed as the │ │ │ │ second argument. │ │ │ │ i17 : R = ZZ/5[x,y,z]; │ │ │ │ i18 : f = x^3 + y^3 + z^3 + x*y*z; │ │ │ │ i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by │ │ │ │ default │ │ │ │ - -- used 0.438746s (cpu); 0.246817s (thread); 0s (gc) │ │ │ │ + -- used 0.427572s (cpu); 0.229273s (thread); 0s (gc) │ │ │ │ │ │ │ │ o19 = 499 │ │ │ │ i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower) │ │ │ │ - -- used 1.52422s (cpu); 1.11548s (thread); 0s (gc) │ │ │ │ + -- used 1.45998s (cpu); 1.26371s (thread); 0s (gc) │ │ │ │ │ │ │ │ o20 = 499 │ │ │ │ Finally, when ContainmentTest is set to FrobeniusPower, then instead of │ │ │ │ producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead │ │ │ │ outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius │ │ │ │ power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the │ │ │ │ $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as │ │ │ │ @@ -168,31 +168,31 @@ │ │ │ │ The function frobeniusNu works by searching through the list of potential │ │ │ │ integers $n$ and checking containments of $I^n$ in the specified Frobenius │ │ │ │ power of $J$. The way this search is approached is specified by the option │ │ │ │ Search, which can be set to Binary (the default value) or Linear. │ │ │ │ i25 : R = ZZ/5[x,y,z]; │ │ │ │ i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8; │ │ │ │ i27 : time frobeniusNu(5, f) -- uses binary search (default) │ │ │ │ - -- used 1.01826s (cpu); 0.572049s (thread); 0s (gc) │ │ │ │ + -- used 1.10417s (cpu); 0.678721s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = 1124 │ │ │ │ i28 : time frobeniusNu(5, f, Search => Linear) │ │ │ │ - -- used 1.37637s (cpu); 0.772844s (thread); 0s (gc) │ │ │ │ + -- used 1.53838s (cpu); 0.940415s (thread); 0s (gc) │ │ │ │ │ │ │ │ o28 = 1124 │ │ │ │ i29 : M = ideal(x, y, z); │ │ │ │ │ │ │ │ o29 : Ideal of R │ │ │ │ i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default) │ │ │ │ - -- used 1.72392s (cpu); 1.38073s (thread); 0s (gc) │ │ │ │ + -- used 1.74101s (cpu); 1.45864s (thread); 0s (gc) │ │ │ │ │ │ │ │ o30 = 97 │ │ │ │ i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets │ │ │ │ luckier │ │ │ │ - -- used 0.60114s (cpu); 0.47435s (thread); 0s (gc) │ │ │ │ + -- used 0.61699s (cpu); 0.564112s (thread); 0s (gc) │ │ │ │ │ │ │ │ o31 = 97 │ │ │ │ The option AtOrigin (default value true) can be turned off to tell frobeniusNu │ │ │ │ to effectively do the computation over all possible maximal ideals $J$ and take │ │ │ │ the minimum. │ │ │ │ i32 : R = ZZ/7[x,y]; │ │ │ │ i33 : f = (x - 1)^3 - (y - 2)^2; │ │ ├── ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,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 │ │ │ YXJjcw== │ │ │ #:len=3082 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJpbnRzIG5vZGUgbGFiZWxzIGZvciB0 │ │ │ aGUgZGVzaW5ndWxhcml6YXRpb24gdHJlZSIsICJsaW5lbnVtIiA9PiA2NTIsIElucHV0cyA9PiB7 │ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.24. 02/07/2024 on Sun Feb 9 22:54:37 2025 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=998,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 │ │ │ UlJFRk1ldGhvZA== │ │ │ #:len=209 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjEzMCwgInVuZG9jdW1lbnRlZCIgPT4g │ │ │ dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiUlJFRk1l │ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ @@ -208,21 +208,21 @@ │ │ │ | 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 1.7139s (cpu); 0.46517s (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 3.43938s (cpu); 1.08827s (thread); 0s (gc) │ │ │ │ │ │ o28 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ o28 : KClass │ │ │ │ │ │ i29 : │ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ @@ -336,23 +336,23 @@ │ │ │ | 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 1.7139s (cpu); 0.46517s (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 3.43938s (cpu); 1.08827s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
    │ │ │ │ │ │ │ │ │
    │ │ │ ├── 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 1.7139s (cpu); 0.46517s (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 3.43938s (cpu); 1.08827s (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 = {newDigraph, map, 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 = {newDigraph, map, digraph}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 4 => {} │ │ │ │ 5 => {6} │ │ │ │ 6 => {} │ │ │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ i3 : keys H │ │ │ │ │ │ │ │ -o3 = {digraph, map, newDigraph} │ │ │ │ +o3 = {newDigraph, map, 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 │ │ │ @@ -200,72 +200,72 @@ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8 │ │ │ +o13 = | -42 12 -9 -46 -17 -41 -23 -6 12 -38 21 -11 19 -29 -3 -22 -16 -13 -8 │ │ │ ----------------------------------------------------------------------- │ │ │ - -29 -22 -29 -24 | │ │ │ + 19 -10 -29 -29 -24 | │ │ │ │ │ │ 1 24 │ │ │ o13 : Matrix kk <-- kk │ │ │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o14 = ideal (a - 30b*c + 23c - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c - │ │ │ + 2 2 2 │ │ │ +o14 = ideal (a + 12b*c - 46c - 38a*d - 17b*d + 12c*d - 42d , a*b - 3b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 33c - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c - 22a*d + 32b*d │ │ │ + 2 2 2 │ │ │ + 21c - 22a*d - 11b*d - 41c*d - 9d , a*c - 10b*c - 16c - 29a*d - 13b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - + 19c*d - 4d , b - 29b*c - 29c - 24a*d - 8b*d + 19c*d - 12d ) │ │ │ + 2 2 2 2 │ │ │ + + 19c*d - 23d , b - 29b*c - 8c - 24a*d + 19b*d - 29c*d - 6d ) │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ - 2 2 2 │ │ │ -o15 = {ideal (a - 29b - 38c - 9d, b - 29b*c - 29c + 3b*d + 16c*d - 26d ), │ │ │ + 2 2 2 │ │ │ +o15 = {ideal (a - 10b - 16c - 41d, b - 29b*c - 8c - 19b*d - 9c*d + 20d ), │ │ │ ----------------------------------------------------------------------- │ │ │ - ideal (c - 22d, b - 21d, a + 8d)} │ │ │ + ideal (c - 29d, b + 16d, a - 20d)} │ │ │ │ │ │ o15 : List │ │ │ │ │ │ i16 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o16 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47 │ │ │ +o16 = | -6 -13 21 -15 -31 6 -9 -1 50 -39 -18 21 -39 31 -31 44 -18 -13 -47 21 │ │ │ ----------------------------------------------------------------------- │ │ │ - 34 19 21 39 0 | │ │ │ + 34 19 39 0 | │ │ │ │ │ │ 1 24 │ │ │ o16 : Matrix kk <-- kk │ │ │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 2 │ │ │ -o17 = ideal (a + 17b*c - 20c - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c │ │ │ + 2 2 2 │ │ │ +o17 = ideal (a + 50b*c - 15c - 39a*d - 31b*d - 13c*d - 6d , a*b - 31b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 │ │ │ - - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c + 21a*d - 13b*d - │ │ │ + 2 2 2 │ │ │ + 18c + 44a*d + 21b*d + 6c*d + 21d , a*c + 34b*c - 18c + 19a*d - 13b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 39c*d - 43d , b + 39b*c - 47c + 34b*d + 40c*d - 41d ) │ │ │ + 2 2 2 2 │ │ │ + - 39c*d - 9d , b + 39b*c - 47c + 21b*d + 31c*d - d ) │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ - 2 2 2 │ │ │ -o18 = {ideal (a*c + 19b*c - 18c + 21a*d - 13b*d - 39c*d - 43d , b + 39b*c - │ │ │ + 2 2 2 │ │ │ +o18 = {ideal (a*c + 34b*c - 18c + 19a*d - 13b*d - 39c*d - 9d , b + 39b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 47c + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c - 39a*d - 29b*d - 30c*d │ │ │ + 2 2 2 │ │ │ + 47c + 21b*d + 31c*d - d , a*b - 31b*c - 18c + 44a*d + 21b*d + 6c*d + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - + 16d , a + 17b*c - 20c - 4a*d - b*d - 2c*d + 46d )} │ │ │ + 2 2 2 2 │ │ │ + 21d , a + 50b*c - 15c - 39a*d - 31b*d - 13c*d - 6d )} │ │ │ │ │ │ o18 : List │ │ │ │ │ │ i19 : │ │ ├── ./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 │ │ │ @@ -294,85 +294,85 @@ │ │ │
    │ │ │

    There are 2 components. We attempt to find a point on the first component

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8
│ │ │ +o13 = | -42 12 -9 -46 -17 -41 -23 -6 12 -38 21 -11 19 -29 -3 -22 -16 -13 -8
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -29 -22 -29 -24 |
│ │ │ +      19 -10 -29 -29 -24 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    │ │ │ 
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o14 = ideal (a  - 30b*c + 23c  - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c -
│ │ │ +              2              2                              2              
│ │ │ +o14 = ideal (a  + 12b*c - 46c  - 38a*d - 17b*d + 12c*d - 42d , a*b - 3b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                   2                
│ │ │ -      33c  - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c  - 22a*d + 32b*d
│ │ │ +         2                             2                   2                
│ │ │ +      21c  - 22a*d - 11b*d - 41c*d - 9d , a*c - 10b*c - 16c  - 29a*d - 13b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2   2              2                             2
│ │ │ -      + 19c*d - 4d , b  - 29b*c - 29c  - 24a*d - 8b*d + 19c*d - 12d )
│ │ │ +                   2   2             2                             2
│ │ │ +      + 19c*d - 23d , b  - 29b*c - 8c  - 24a*d + 19b*d - 29c*d - 6d )
│ │ │  
│ │ │  o14 : Ideal of S
    │ │ │ 
    i15 : decompose F1
│ │ │  
│ │ │ -                                   2              2                     2
│ │ │ -o15 = {ideal (a - 29b - 38c - 9d, b  - 29b*c - 29c  + 3b*d + 16c*d - 26d ),
│ │ │ +                                    2             2                     2
│ │ │ +o15 = {ideal (a - 10b - 16c - 41d, b  - 29b*c - 8c  - 19b*d - 9c*d + 20d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 22d, b - 21d, a + 8d)}
│ │ │ +      ideal (c - 29d, b + 16d, a - 20d)}
│ │ │  
│ │ │  o15 : List
    │ │ │
    │ │ │
    │ │ │

    We attempt to find a point on the second component in parameter space, and its corresponding ideal.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o16 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47
│ │ │ +o16 = | -6 -13 21 -15 -31 6 -9 -1 50 -39 -18 21 -39 31 -31 44 -18 -13 -47 21
│ │ │        -----------------------------------------------------------------------
│ │ │ -      34 19 21 39 0 |
│ │ │ +      34 19 39 0 |
│ │ │  
│ │ │                 1       24
│ │ │  o16 : Matrix kk  <-- kk
    │ │ │ 
    i17 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                          2                   2
│ │ │ -o17 = ideal (a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c 
│ │ │ +              2              2                             2               
│ │ │ +o17 = ideal (a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d , a*b - 31b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                                   2                   2                  
│ │ │ -      - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c  + 21a*d - 13b*d -
│ │ │ +         2                             2                   2                
│ │ │ +      18c  + 44a*d + 21b*d + 6c*d + 21d , a*c + 34b*c - 18c  + 19a*d - 13b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                      2
│ │ │ -      39c*d - 43d , b  + 39b*c - 47c  + 34b*d + 40c*d - 41d )
│ │ │ +                  2   2              2                    2
│ │ │ +      - 39c*d - 9d , b  + 39b*c - 47c  + 21b*d + 31c*d - d )
│ │ │  
│ │ │  o17 : Ideal of S
    │ │ │ 
    i18 : decompose F2
│ │ │  
│ │ │ -                               2                              2   2          
│ │ │ -o18 = {ideal (a*c + 19b*c - 18c  + 21a*d - 13b*d - 39c*d - 43d , b  + 39b*c -
│ │ │ +                               2                             2   2          
│ │ │ +o18 = {ideal (a*c + 34b*c - 18c  + 19a*d - 13b*d - 39c*d - 9d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                      2                   2                        
│ │ │ -      47c  + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c  - 39a*d - 29b*d - 30c*d
│ │ │ +         2                    2                   2                         
│ │ │ +      47c  + 21b*d + 31c*d - d , a*b - 31b*c - 18c  + 44a*d + 21b*d + 6c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2   2              2                          2
│ │ │ -      + 16d , a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d )}
│ │ │ +         2   2              2                             2
│ │ │ +      21d , a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d )}
│ │ │  
│ │ │  o18 : List
    │ │ │
    │ │ │
    │ │ │

    It turns out that this is the ideal of 2 skew lines, just not defined over this field.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -213,72 +213,72 @@ │ │ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ There are 2 components. We attempt to find a point on the first component │ │ │ │ i13 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8 │ │ │ │ +o13 = | -42 12 -9 -46 -17 -41 -23 -6 12 -38 21 -11 19 -29 -3 -22 -16 -13 -8 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -29 -22 -29 -24 | │ │ │ │ + 19 -10 -29 -29 -24 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o13 : Matrix kk <-- kk │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ -o14 = ideal (a - 30b*c + 23c - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c - │ │ │ │ +o14 = ideal (a + 12b*c - 46c - 38a*d - 17b*d + 12c*d - 42d , a*b - 3b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 33c - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c - 22a*d + 32b*d │ │ │ │ + 2 2 2 │ │ │ │ + 21c - 22a*d - 11b*d - 41c*d - 9d , a*c - 10b*c - 16c - 29a*d - 13b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - + 19c*d - 4d , b - 29b*c - 29c - 24a*d - 8b*d + 19c*d - 12d ) │ │ │ │ + 2 2 2 2 │ │ │ │ + + 19c*d - 23d , b - 29b*c - 8c - 24a*d + 19b*d - 29c*d - 6d ) │ │ │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o15 = {ideal (a - 29b - 38c - 9d, b - 29b*c - 29c + 3b*d + 16c*d - 26d ), │ │ │ │ + 2 2 2 │ │ │ │ +o15 = {ideal (a - 10b - 16c - 41d, b - 29b*c - 8c - 19b*d - 9c*d + 20d ), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - ideal (c - 22d, b - 21d, a + 8d)} │ │ │ │ + ideal (c - 29d, b + 16d, a - 20d)} │ │ │ │ │ │ │ │ o15 : List │ │ │ │ We attempt to find a point on the second component in parameter space, and its │ │ │ │ corresponding ideal. │ │ │ │ i16 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o16 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47 │ │ │ │ +o16 = | -6 -13 21 -15 -31 6 -9 -1 50 -39 -18 21 -39 31 -31 44 -18 -13 -47 21 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 34 19 21 39 0 | │ │ │ │ + 34 19 39 0 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o16 : Matrix kk <-- kk │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 2 │ │ │ │ -o17 = ideal (a + 17b*c - 20c - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c │ │ │ │ + 2 2 2 │ │ │ │ +o17 = ideal (a + 50b*c - 15c - 39a*d - 31b*d - 13c*d - 6d , a*b - 31b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 │ │ │ │ - - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c + 21a*d - 13b*d - │ │ │ │ + 2 2 2 │ │ │ │ + 18c + 44a*d + 21b*d + 6c*d + 21d , a*c + 34b*c - 18c + 19a*d - 13b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 39c*d - 43d , b + 39b*c - 47c + 34b*d + 40c*d - 41d ) │ │ │ │ + 2 2 2 2 │ │ │ │ + - 39c*d - 9d , b + 39b*c - 47c + 21b*d + 31c*d - d ) │ │ │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o18 = {ideal (a*c + 19b*c - 18c + 21a*d - 13b*d - 39c*d - 43d , b + 39b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o18 = {ideal (a*c + 34b*c - 18c + 19a*d - 13b*d - 39c*d - 9d , b + 39b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 47c + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c - 39a*d - 29b*d - 30c*d │ │ │ │ + 2 2 2 │ │ │ │ + 47c + 21b*d + 31c*d - d , a*b - 31b*c - 18c + 44a*d + 21b*d + 6c*d + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - + 16d , a + 17b*c - 20c - 4a*d - b*d - 2c*d + 46d )} │ │ │ │ + 2 2 2 2 │ │ │ │ + 21d , a + 50b*c - 15c - 39a*d - 31b*d - 13c*d - 6d )} │ │ │ │ │ │ │ │ o18 : List │ │ │ │ It turns out that this is the ideal of 2 skew lines, just not defined over this │ │ │ │ field. │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ This routine expects the input to represent an irreducible variety │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./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 │ │ │ + -- 2.17347s elapsed │ │ │ │ │ │ o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16] │ │ │ │ │ │ o5 : GroebnerBasis │ │ │ │ │ │ i6 : elapsedTime groebnerWalk(gb I1, R2) │ │ │ - -- 2.99521s elapsed │ │ │ + -- 1.74356s 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
│ │ │ + -- 2.17347s 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
│ │ │ + -- 1.74356s 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 │ │ │ │ + -- 2.17347s 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 │ │ │ │ + -- 1.74356s 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 │ │ │ + -- .000006014s elapsed │ │ │ + -- .000006949s elapsed │ │ │ + -- .000008056s elapsed │ │ │ + -- .000008408s elapsed │ │ │ + -- .000006727s 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 │ │ │ + -- .000006233s 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 │ │ │ + -- .000004795s 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
│ │ │ + -- .000006014s elapsed
│ │ │ + -- .000006949s elapsed
│ │ │ + -- .000008056s elapsed
│ │ │ + -- .000008408s elapsed
│ │ │ + -- .000006727s 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
│ │ │ │ + -- .000006014s elapsed
│ │ │ │ + -- .000006949s elapsed
│ │ │ │ + -- .000008056s elapsed
│ │ │ │ + -- .000008408s elapsed
│ │ │ │ + -- .000006727s 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
│ │ │ + -- .000006233s 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 @@
│ │ │          
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│ │ │          
                  
    i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
    
│ │ │      		          
                  
    i5 : solveFrobeniusIdeal(I, W)
│ │ │ - -- .000004359s elapsed
│ │ │ + -- .000004795s 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
│ │ │ │ + -- .000006233s 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
│ │ │ │ + -- .000004795s 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 0.732093s (cpu); 0.398564s (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 0.701473s (cpu); 0.411557s (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 0.753639s (cpu); 0.382848s (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 0.896418s (cpu); 0.462318s (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 1.33456s (cpu); 0.669861s (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 0.857889s (cpu); 0.411963s (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 0.149116s (cpu); 0.0715503s (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 0.141148s (cpu); 0.0626748s (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 0.171566s (cpu); 0.094135s (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 0.161067s (cpu); 0.0819245s (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 0.153819s (cpu); 0.0841603s (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 0.169626s (cpu); 0.0940834s (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 0.585196s (cpu); 0.412787s (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 0.482436s (cpu); 0.3143s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -781,17 +781,17 @@
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │   [jacobian time 0 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 .131903 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.383431s (cpu); 0.299861s (thread); 0s (gc)
│ │ │ +  time .247525 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 .252948 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.593305s (cpu); 0.412836s (thread); 0s (gc)
│ │ │ +  time .336348 sec  #fractions 6]
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -844,20 +844,20 @@
│ │ │  i93 : R = S/sub(I,S)
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time 0 sec #minors 1]
│ │ │ + [jacobian time .00264973 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 .260959 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.884988s (cpu); 0.613503s (thread); 0s (gc)
│ │ │ +  time .620019 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 .00350033 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:  .0080163 seconds
│ │ │ +      idlizer2:  .0120503 seconds
│ │ │ +      minpres:   .0109818 seconds
│ │ │ +  time .129764 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) .00399843 seconds
│ │ │ +      idlizer1:  .0152292 seconds
│ │ │ +      idlizer2:  .097932 seconds
│ │ │ +      minpres:   .0160299 seconds
│ │ │ +  time .141938 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .0119982 seconds
│ │ │ -      idlizer2:  .0120027 seconds
│ │ │ -      minpres:   .0120025 seconds
│ │ │ -  time .126433 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00408557 seconds
│ │ │ +      idlizer1:  .0158982 seconds
│ │ │ +      idlizer2:  .0120713 seconds
│ │ │ +      minpres:   .0119544 seconds
│ │ │ +  time .152972 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) .00400037 seconds
│ │ │ +      idlizer1:  .0173236 seconds
│ │ │ +      idlizer2:  .0145925 seconds
│ │ │ +      minpres:   .109776 seconds
│ │ │ +  time .161486 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) .00387737 seconds
│ │ │ +      idlizer1:  .0159419 seconds
│ │ │ +      idlizer2:  .0160096 seconds
│ │ │ +      minpres:   .0161147 seconds
│ │ │ +  time .068113 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) .00392538 seconds
│ │ │ +      idlizer1:   -- used 0.769018s (cpu); 0.393999s (thread); 0s (gc)
│ │ │ +.0120284 seconds
│ │ │ +  time .110762 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 2.36635s (cpu); 1.04789s (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 1.78594s (cpu); 0.665758s (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 0.732093s (cpu); 0.398564s (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 0.701473s (cpu); 0.411557s (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 0.753639s (cpu); 0.382848s (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 0.896418s (cpu); 0.462318s (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 1.33456s (cpu); 0.669861s (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 0.857889s (cpu); 0.411963s (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 0.149116s (cpu); 0.0715503s (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 0.141148s (cpu); 0.0626748s (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 0.171566s (cpu); 0.094135s (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 0.161067s (cpu); 0.0819245s (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 0.153819s (cpu); 0.0841603s (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 0.169626s (cpu); 0.0940834s (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 0.585196s (cpu); 0.412787s (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 0.482436s (cpu); 0.3143s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    
│ │ │      		          
                  
    i83 : icFractions R
│ │ │ @@ -971,17 +971,17 @@
│ │ │  o85 : QuotientRing
    
│ │ │      		          
                  
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │   [jacobian time 0 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 .131903 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.383431s (cpu); 0.299861s (thread); 0s (gc)
│ │ │ +  time .247525 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 .252948 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.593305s (cpu); 0.412836s (thread); 0s (gc)
│ │ │ +  time .336348 sec  #fractions 6]
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    
│ │ │      		          
                  
    i91 : icFractions R
│ │ │ @@ -1049,20 +1049,20 @@
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
    
│ │ │      		          
                  
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time 0 sec #minors 1]
│ │ │ + [jacobian time .00264973 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 .260959 sec  #fractions 6]
│ │ │ + [step 1:  -- used 0.884988s (cpu); 0.613503s (thread); 0s (gc)
│ │ │ +  time .620019 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 0.732093s (cpu); 0.398564s (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 0.701473s (cpu); 0.411557s (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 0.753639s (cpu); 0.382848s (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 0.896418s (cpu); 0.462318s (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 1.33456s (cpu); 0.669861s (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 0.857889s (cpu); 0.411963s (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 0.149116s (cpu); 0.0715503s (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 0.141148s (cpu); 0.0626748s (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 0.171566s (cpu); 0.094135s (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 0.161067s (cpu); 0.0819245s (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 0.153819s (cpu); 0.0841603s (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 0.169626s (cpu); 0.0940834s (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 0.585196s (cpu); 0.412787s (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 0.482436s (cpu); 0.3143s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o82 = R'
│ │ │ │  
│ │ │ │  o82 : QuotientRing
│ │ │ │  i83 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -891,17 +891,17 @@
│ │ │ │  o85 = R
│ │ │ │  
│ │ │ │  o85 : QuotientRing
│ │ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ │   [jacobian time 0 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 .131903 sec  #fractions 6]
│ │ │ │ + [step 1:  -- used 0.383431s (cpu); 0.299861s (thread); 0s (gc)
│ │ │ │ +  time .247525 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 .252948 sec  #fractions 6]
│ │ │ │ + [step 1:  -- used 0.593305s (cpu); 0.412836s (thread); 0s (gc)
│ │ │ │ +  time .336348 sec  #fractions 6]
│ │ │ │  
│ │ │ │  o90 = R'
│ │ │ │  
│ │ │ │  o90 : QuotientRing
│ │ │ │  i91 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -949,20 +949,20 @@
│ │ │ │  i93 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o93 = R
│ │ │ │  
│ │ │ │  o93 : QuotientRing
│ │ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos,
│ │ │ │  StartWithOneMinor}, Verbosity => 1)
│ │ │ │ - [jacobian time 0 sec #minors 1]
│ │ │ │ + [jacobian time .00264973 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 .260959 sec  #fractions 6]
│ │ │ │ + [step 1:  -- used 0.884988s (cpu); 0.613503s (thread); 0s (gc)
│ │ │ │ +  time .620019 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 .00350033 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:  .0080163 seconds
│ │ │ +      idlizer2:  .0120503 seconds
│ │ │ +      minpres:   .0109818 seconds
│ │ │ +  time .129764 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) .00399843 seconds
│ │ │ +      idlizer1:  .0152292 seconds
│ │ │ +      idlizer2:  .097932 seconds
│ │ │ +      minpres:   .0160299 seconds
│ │ │ +  time .141938 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) 0 seconds
│ │ │ -      idlizer1:  .0119982 seconds
│ │ │ -      idlizer2:  .0120027 seconds
│ │ │ -      minpres:   .0120025 seconds
│ │ │ -  time .126433 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00408557 seconds
│ │ │ +      idlizer1:  .0158982 seconds
│ │ │ +      idlizer2:  .0120713 seconds
│ │ │ +      minpres:   .0119544 seconds
│ │ │ +  time .152972 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) .00400037 seconds
│ │ │ +      idlizer1:  .0173236 seconds
│ │ │ +      idlizer2:  .0145925 seconds
│ │ │ +      minpres:   .109776 seconds
│ │ │ +  time .161486 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) .00387737 seconds
│ │ │ +      idlizer1:  .0159419 seconds
│ │ │ +      idlizer2:  .0160096 seconds
│ │ │ +      minpres:   .0161147 seconds
│ │ │ +  time .068113 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) .00392538 seconds
│ │ │ +      idlizer1:   -- used 0.769018s (cpu); 0.393999s (thread); 0s (gc)
│ │ │ +.0120284 seconds
│ │ │ +  time .110762 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 .00350033 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:  .0080163 seconds
│ │ │ │ +      idlizer2:  .0120503 seconds
│ │ │ │ +      minpres:   .0109818 seconds
│ │ │ │ +  time .129764 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) .00399843 seconds
│ │ │ │ +      idlizer1:  .0152292 seconds
│ │ │ │ +      idlizer2:  .097932 seconds
│ │ │ │ +      minpres:   .0160299 seconds
│ │ │ │ +  time .141938 sec  #fractions 4]
│ │ │ │   [step 2:
│ │ │ │ -      radical (use minprimes) 0 seconds
│ │ │ │ -      idlizer1:  .0119982 seconds
│ │ │ │ -      idlizer2:  .0120027 seconds
│ │ │ │ -      minpres:   .0120025 seconds
│ │ │ │ -  time .126433 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00408557 seconds
│ │ │ │ +      idlizer1:  .0158982 seconds
│ │ │ │ +      idlizer2:  .0120713 seconds
│ │ │ │ +      minpres:   .0119544 seconds
│ │ │ │ +  time .152972 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) .00400037 seconds
│ │ │ │ +      idlizer1:  .0173236 seconds
│ │ │ │ +      idlizer2:  .0145925 seconds
│ │ │ │ +      minpres:   .109776 seconds
│ │ │ │ +  time .161486 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) .00387737 seconds
│ │ │ │ +      idlizer1:  .0159419 seconds
│ │ │ │ +      idlizer2:  .0160096 seconds
│ │ │ │ +      minpres:   .0161147 seconds
│ │ │ │ +  time .068113 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) .00392538 seconds
│ │ │ │ +      idlizer1:   -- used 0.769018s (cpu); 0.393999s (thread); 0s (gc)
│ │ │ │ +.0120284 seconds
│ │ │ │ +  time .110762 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 2.36635s (cpu); 1.04789s (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 1.78594s (cpu); 0.665758s (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 2.36635s (cpu); 1.04789s (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 1.78594s (cpu); 0.665758s (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
│ │ │ + -- .00436786s 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
│ │ │ + -- .000817662s 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 0.806587s (cpu); 0.559067s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   3       3
│ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.639755s (cpu); 0.336773s (thread); 0s (gc)
│ │ │ + -- used 0.663903s (cpu); 0.379502s (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 0.0296953s (cpu); 0.0300574s (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 2.09624s (cpu); 1.39856s (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
│ │ │ + -- .56739s 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
│ │ │ + -- .493131s 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
│ │ │ + -- .59523s 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
│ │ │ + -- .0646835s 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
│ │ │ + -- .00436786s 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
    
│ │ │ + -- .000817662s 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
│ │ │ │ + -- .00436786s 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
│ │ │ │ + -- .000817662s 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 0.806587s (cpu); 0.559067s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   3       3
│ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
    
│ │ │      		          
                  
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.639755s (cpu); 0.336773s (thread); 0s (gc)
│ │ │ + -- used 0.663903s (cpu); 0.379502s (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 0.0296953s (cpu); 0.0300574s (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 2.09624s (cpu); 1.39856s (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 0.806587s (cpu); 0.559067s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2   2    2   3       3
│ │ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.639755s (cpu); 0.336773s (thread); 0s (gc)
│ │ │ │ + -- used 0.663903s (cpu); 0.379502s (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 0.0296953s (cpu); 0.0300574s (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 2.09624s (cpu); 1.39856s (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
│ │ │ + -- .56739s 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
│ │ │ + -- .493131s 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
│ │ │ │ + -- .56739s 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
│ │ │ │ + -- .493131s 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
│ │ │ + -- .59523s 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
│ │ │ + -- .0646835s 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
│ │ │ │ + -- .59523s 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
│ │ │ │ + -- .0646835s 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/M2-244633-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-134606-0/0.json
│ │ │ +o10 = /tmp/M2-244633-0/0.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/M2-244633-0/0.json
│ │ │      		          
                  
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-134606-0/0.json
│ │ │ +o10 = /tmp/M2-244633-0/0.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/M2-244633-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-134606-0/0.json
│ │ │ │ +o10 = /tmp/M2-244633-0/0.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
│ │ │ + -- .00248058s 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
│ │ │ + -- .263183s 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
│ │ │ + -- .0098733s 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
│ │ │ + -- .00266867s 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
│ │ │ + -- .00241597s 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
│ │ │ + -- .0139935s 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
│ │ │ + -- .000770735s 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
│ │ │ │ + -- .00248058s 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
│ │ │ │ + -- .263183s 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
│ │ │ + -- .00248058s 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
│ │ │ + -- .263183s 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
│ │ │ + -- .0098733s 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
│ │ │ + -- .00266867s 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
│ │ │ + -- .00241597s 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
│ │ │ + -- .0139935s 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
│ │ │ + -- .000770735s 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
│ │ │ │ + -- .0098733s 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
│ │ │ │ + -- .00266867s 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
│ │ │ │ + -- .00241597s 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
│ │ │ │ + -- .0139935s 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
│ │ │ │ + -- .000770735s 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
│ │ │ + -- .0258072s 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
│ │ │ + -- .00241183s elapsed
│ │ │ + -- .00700044s elapsed
│ │ │ + -- .0274915s elapsed
│ │ │ + -- .0122929s elapsed
│ │ │ + -- .00443769s 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
│ │ │ + -- .00256693s elapsed
│ │ │ + -- .0066788s elapsed
│ │ │ + -- .0247459s elapsed
│ │ │ + -- .0116408s elapsed
│ │ │ + -- .00402924s elapsed
│ │ │ + -- .271548s 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
│ │ │ + -- .22815s 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
│ │ │ + -- .00480629s elapsed
│ │ │ + -- .0191203s elapsed
│ │ │ + -- .118898s elapsed
│ │ │ + -- 1.01638s elapsed
│ │ │ + -- .298946s elapsed
│ │ │ + -- .0418985s elapsed
│ │ │ + -- .00711003s elapsed
│ │ │ + -- 4.76143s 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
│ │ │ + -- .00275543s elapsed
│ │ │ + -- .00670533s elapsed
│ │ │ + -- .0243986s elapsed
│ │ │ + -- .0101463s elapsed
│ │ │ + -- .00349766s 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
│ │ │ + -- .218472s 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
│ │ │ + -- .0048812s elapsed
│ │ │ + -- .0304575s elapsed
│ │ │ + -- .11594s elapsed
│ │ │ + -- 1.05207s elapsed
│ │ │ + -- .361273s elapsed
│ │ │ + -- .0429173s elapsed
│ │ │ + -- .00721126s elapsed
│ │ │ + -- 5.00718s 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
│ │ │ + -- .00839452s elapsed
│ │ │ + -- .0141241s elapsed
│ │ │ + -- .00023261s elapsed
│ │ │ + -- .000205085s elapsed
│ │ │ + -- .000152934s elapsed
│ │ │ + -- .000147646s elapsed
│ │ │ + -- .000143406s elapsed
│ │ │ + -- .00016634s elapsed
│ │ │ + -- .000169231s elapsed
│ │ │ + -- .0002269s elapsed
│ │ │ + -- .000305454s elapsed
│ │ │ + -- .000160704s elapsed
│ │ │ + -- .000159123s elapsed
│ │ │ + -- .000147371s elapsed
│ │ │ + -- .000153972s elapsed
│ │ │ + -- .000185346s elapsed
│ │ │ + -- .000166205s elapsed
│ │ │ + -- .000141428s elapsed
│ │ │ + -- .000165669s elapsed
│ │ │ + -- .000170118s elapsed
│ │ │ + -- .000186707s elapsed
│ │ │ + -- .000191156s elapsed
│ │ │ + -- .000206161s elapsed
│ │ │ + -- .000157259s elapsed
│ │ │ + -- .000167066s elapsed
│ │ │ + -- .000155904s elapsed
│ │ │ + -- .000144249s elapsed
│ │ │ + -- .000137337s 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
│ │ │ + -- .191827s elapsed
│ │ │ + -- .1769s elapsed
│ │ │ + -- .154159s elapsed
│ │ │ + -- .210511s elapsed
│ │ │ + -- .165012s elapsed
│ │ │ + -- .227673s 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
│ │ │ + -- .0029318s elapsed
│ │ │ + -- .00639272s elapsed
│ │ │ + -- .0246709s elapsed
│ │ │ + -- .00984396s elapsed
│ │ │ + -- .00371155s 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
│ │ │ + -- .236718s 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
│ │ │ + -- .00277703s elapsed
│ │ │ + -- .00718625s elapsed
│ │ │ + -- .0259502s elapsed
│ │ │ + -- .011931s elapsed
│ │ │ + -- .00408104s 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
│ │ │ + -- .0156905s elapsed
│ │ │ + -- .000037259s elapsed
│ │ │ + -- .000092916s elapsed
│ │ │ + -- .000077264s elapsed
│ │ │ + -- .000090884s elapsed
│ │ │ + -- .000102933s elapsed
│ │ │ + -- .000098982s elapsed
│ │ │ + -- .000026561s elapsed
│ │ │ + -- .0284956s elapsed
│ │ │ + -- .000109862s elapsed
│ │ │ + -- .000100238s elapsed
│ │ │ + -- .000101321s elapsed
│ │ │ + -- .000081837s elapsed
│ │ │ + -- .000074337s elapsed
│ │ │ + -- .00006772s elapsed
│ │ │ + -- .000068955s elapsed
│ │ │ + -- .000022245s elapsed
│ │ │ + -- .000084553s elapsed
│ │ │ + -- .000021787s 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
│ │ │ + -- .0258072s 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
│ │ │ + -- .00241183s elapsed
│ │ │ + -- .00700044s elapsed
│ │ │ + -- .0274915s elapsed
│ │ │ + -- .0122929s elapsed
│ │ │ + -- .00443769s 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
│ │ │ │ + -- .0258072s 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
│ │ │ │ + -- .00241183s elapsed
│ │ │ │ + -- .00700044s elapsed
│ │ │ │ + -- .0274915s elapsed
│ │ │ │ + -- .0122929s elapsed
│ │ │ │ + -- .00443769s 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
│ │ │ + -- .00256693s elapsed
│ │ │ + -- .0066788s elapsed
│ │ │ + -- .0247459s elapsed
│ │ │ + -- .0116408s elapsed
│ │ │ + -- .00402924s elapsed
│ │ │ + -- .271548s 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
│ │ │ + -- .22815s 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
    
│ │ │ + -- .00480629s elapsed
│ │ │ + -- .0191203s elapsed
│ │ │ + -- .118898s elapsed
│ │ │ + -- 1.01638s elapsed
│ │ │ + -- .298946s elapsed
│ │ │ + -- .0418985s elapsed
│ │ │ + -- .00711003s elapsed
│ │ │ + -- 4.76143s 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
│ │ │ │ + -- .00256693s elapsed
│ │ │ │ + -- .0066788s elapsed
│ │ │ │ + -- .0247459s elapsed
│ │ │ │ + -- .0116408s elapsed
│ │ │ │ + -- .00402924s elapsed
│ │ │ │ + -- .271548s 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
│ │ │ │ + -- .22815s 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
│ │ │ │ + -- .00480629s elapsed
│ │ │ │ + -- .0191203s elapsed
│ │ │ │ + -- .118898s elapsed
│ │ │ │ + -- 1.01638s elapsed
│ │ │ │ + -- .298946s elapsed
│ │ │ │ + -- .0418985s elapsed
│ │ │ │ + -- .00711003s elapsed
│ │ │ │ + -- 4.76143s 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
│ │ │ + -- .00275543s elapsed
│ │ │ + -- .00670533s elapsed
│ │ │ + -- .0243986s elapsed
│ │ │ + -- .0101463s elapsed
│ │ │ + -- .00349766s 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
│ │ │ + -- .218472s 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
    
│ │ │ + -- .0048812s elapsed
│ │ │ + -- .0304575s elapsed
│ │ │ + -- .11594s elapsed
│ │ │ + -- 1.05207s elapsed
│ │ │ + -- .361273s elapsed
│ │ │ + -- .0429173s elapsed
│ │ │ + -- .00721126s elapsed
│ │ │ + -- 5.00718s 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
│ │ │ │ + -- .00275543s elapsed
│ │ │ │ + -- .00670533s elapsed
│ │ │ │ + -- .0243986s elapsed
│ │ │ │ + -- .0101463s elapsed
│ │ │ │ + -- .00349766s 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
│ │ │ │ + -- .218472s 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
│ │ │ │ + -- .0048812s elapsed
│ │ │ │ + -- .0304575s elapsed
│ │ │ │ + -- .11594s elapsed
│ │ │ │ + -- 1.05207s elapsed
│ │ │ │ + -- .361273s elapsed
│ │ │ │ + -- .0429173s elapsed
│ │ │ │ + -- .00721126s elapsed
│ │ │ │ + -- 5.00718s 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
│ │ │ + -- .00839452s elapsed
│ │ │ + -- .0141241s elapsed
│ │ │ + -- .00023261s elapsed
│ │ │ + -- .000205085s elapsed
│ │ │ + -- .000152934s elapsed
│ │ │ + -- .000147646s elapsed
│ │ │ + -- .000143406s elapsed
│ │ │ + -- .00016634s elapsed
│ │ │ + -- .000169231s elapsed
│ │ │ + -- .0002269s elapsed
│ │ │ + -- .000305454s elapsed
│ │ │ + -- .000160704s elapsed
│ │ │ + -- .000159123s elapsed
│ │ │ + -- .000147371s elapsed
│ │ │ + -- .000153972s elapsed
│ │ │ + -- .000185346s elapsed
│ │ │ + -- .000166205s elapsed
│ │ │ + -- .000141428s elapsed
│ │ │ + -- .000165669s elapsed
│ │ │ + -- .000170118s elapsed
│ │ │ + -- .000186707s elapsed
│ │ │ + -- .000191156s elapsed
│ │ │ + -- .000206161s elapsed
│ │ │ + -- .000157259s elapsed
│ │ │ + -- .000167066s elapsed
│ │ │ + -- .000155904s elapsed
│ │ │ + -- .000144249s elapsed
│ │ │ + -- .000137337s 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
│ │ │ │ + -- .00839452s elapsed
│ │ │ │ + -- .0141241s elapsed
│ │ │ │ + -- .00023261s elapsed
│ │ │ │ + -- .000205085s elapsed
│ │ │ │ + -- .000152934s elapsed
│ │ │ │ + -- .000147646s elapsed
│ │ │ │ + -- .000143406s elapsed
│ │ │ │ + -- .00016634s elapsed
│ │ │ │ + -- .000169231s elapsed
│ │ │ │ + -- .0002269s elapsed
│ │ │ │ + -- .000305454s elapsed
│ │ │ │ + -- .000160704s elapsed
│ │ │ │ + -- .000159123s elapsed
│ │ │ │ + -- .000147371s elapsed
│ │ │ │ + -- .000153972s elapsed
│ │ │ │ + -- .000185346s elapsed
│ │ │ │ + -- .000166205s elapsed
│ │ │ │ + -- .000141428s elapsed
│ │ │ │ + -- .000165669s elapsed
│ │ │ │ + -- .000170118s elapsed
│ │ │ │ + -- .000186707s elapsed
│ │ │ │ + -- .000191156s elapsed
│ │ │ │ + -- .000206161s elapsed
│ │ │ │ + -- .000157259s elapsed
│ │ │ │ + -- .000167066s elapsed
│ │ │ │ + -- .000155904s elapsed
│ │ │ │ + -- .000144249s elapsed
│ │ │ │ + -- .000137337s 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
│ │ │ + -- .191827s elapsed
│ │ │ + -- .1769s elapsed
│ │ │ + -- .154159s elapsed
│ │ │ + -- .210511s elapsed
│ │ │ + -- .165012s elapsed
│ │ │ + -- .227673s 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 │ │ │ │ + -- .191827s elapsed │ │ │ │ + -- .1769s elapsed │ │ │ │ + -- .154159s elapsed │ │ │ │ + -- .210511s elapsed │ │ │ │ + -- .165012s elapsed │ │ │ │ + -- .227673s 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
│ │ │ + -- .0029318s elapsed
│ │ │ + -- .00639272s elapsed
│ │ │ + -- .0246709s elapsed
│ │ │ + -- .00984396s elapsed
│ │ │ + -- .00371155s 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
│ │ │ + -- .236718s 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
│ │ │ + -- .00277703s elapsed
│ │ │ + -- .00718625s elapsed
│ │ │ + -- .0259502s elapsed
│ │ │ + -- .011931s elapsed
│ │ │ + -- .00408104s 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
│ │ │ │ + -- .0029318s elapsed
│ │ │ │ + -- .00639272s elapsed
│ │ │ │ + -- .0246709s elapsed
│ │ │ │ + -- .00984396s elapsed
│ │ │ │ + -- .00371155s 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
│ │ │ │ + -- .236718s 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
│ │ │ │ + -- .00277703s elapsed
│ │ │ │ + -- .00718625s elapsed
│ │ │ │ + -- .0259502s elapsed
│ │ │ │ + -- .011931s elapsed
│ │ │ │ + -- .00408104s 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
│ │ │ + -- .0156905s elapsed
│ │ │ + -- .000037259s elapsed
│ │ │ + -- .000092916s elapsed
│ │ │ + -- .000077264s elapsed
│ │ │ + -- .000090884s elapsed
│ │ │ + -- .000102933s elapsed
│ │ │ + -- .000098982s elapsed
│ │ │ + -- .000026561s elapsed
│ │ │ + -- .0284956s elapsed
│ │ │ + -- .000109862s elapsed
│ │ │ + -- .000100238s elapsed
│ │ │ + -- .000101321s elapsed
│ │ │ + -- .000081837s elapsed
│ │ │ + -- .000074337s elapsed
│ │ │ + -- .00006772s elapsed
│ │ │ + -- .000068955s elapsed
│ │ │ + -- .000022245s elapsed
│ │ │ + -- .000084553s elapsed
│ │ │ + -- .000021787s 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
│ │ │ │ + -- .0156905s elapsed
│ │ │ │ + -- .000037259s elapsed
│ │ │ │ + -- .000092916s elapsed
│ │ │ │ + -- .000077264s elapsed
│ │ │ │ + -- .000090884s elapsed
│ │ │ │ + -- .000102933s elapsed
│ │ │ │ + -- .000098982s elapsed
│ │ │ │ + -- .000026561s elapsed
│ │ │ │ + -- .0284956s elapsed
│ │ │ │ + -- .000109862s elapsed
│ │ │ │ + -- .000100238s elapsed
│ │ │ │ + -- .000101321s elapsed
│ │ │ │ + -- .000081837s elapsed
│ │ │ │ + -- .000074337s elapsed
│ │ │ │ + -- .00006772s elapsed
│ │ │ │ + -- .000068955s elapsed
│ │ │ │ + -- .000022245s elapsed
│ │ │ │ + -- .000084553s elapsed
│ │ │ │ + -- .000021787s 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 0.00789963s (cpu); 0.00996255s (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 0.0280493s (cpu); 0.0306357s (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 0.113255s (cpu); 0.113086s (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 0.0119406s (cpu); 0.0133077s (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 0.0619726s (cpu); 0.0635115s (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 0.0623665s (cpu); 0.0630821s (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 0.347136s (cpu); 0.347815s (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 0.153094s (cpu); 0.15372s (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 0.00789963s (cpu); 0.00996255s (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 0.0280493s (cpu); 0.0306357s (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 0.113255s (cpu); 0.113086s (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 0.0119406s (cpu); 0.0133077s (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 0.0619726s (cpu); 0.0635115s (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 0.0623665s (cpu); 0.0630821s (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 0.347136s (cpu); 0.347815s (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 0.153094s (cpu); 0.15372s (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 0.00789963s (cpu); 0.00996255s (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 0.0280493s (cpu); 0.0306357s (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 0.113255s (cpu); 0.113086s (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 0.0119406s (cpu); 0.0133077s (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 0.0619726s (cpu); 0.0635115s (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 0.0623665s (cpu); 0.0630821s (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 0.347136s (cpu); 0.347815s (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 0.153094s (cpu); 0.15372s (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 1.35501s (cpu); 0.546193s (thread); 0s (gc) │ │ │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ - -- used 0.694874s (cpu); 0.330095s (thread); 0s (gc) │ │ │ + -- used 0.915896s (cpu); 0.344222s (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 1.35501s (cpu); 0.546193s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ 
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.694874s (cpu); 0.330095s (thread); 0s (gc)
    │ │ │ + -- used 0.915896s (cpu); 0.344222s (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 1.35501s (cpu); 0.546193s (thread); 0s (gc) │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ │ - -- used 0.694874s (cpu); 0.330095s (thread); 0s (gc) │ │ │ │ + -- used 0.915896s (cpu); 0.344222s (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 │ │ │ + -- .08837s 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 │ │ │ + -- .0191562s 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 │ │ │ + -- 3.23899s elapsed │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ - -- .0225246s elapsed │ │ │ + -- .0237087s 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
│ │ │ + -- .08837s 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
│ │ │ + -- .0191562s 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 │ │ │ │ + -- .08837s 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 │ │ │ │ + -- .0191562s 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
│ │ │ + -- 3.23899s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    │ │ │ 
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0225246s elapsed
│ │ │ + -- .0237087s 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 │ │ │ │ + -- 3.23899s elapsed │ │ │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ │ - -- .0225246s elapsed │ │ │ │ + -- .0237087s 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 │ │ │ + -- .229838s 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 │ │ │ + -- .0139658s 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 │ │ │ + -- .367125s 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
│ │ │ + -- .229838s 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
│ │ │ + -- .0139658s 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
│ │ │ + -- .367125s 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 │ │ │ │ + -- .229838s 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 │ │ │ │ + -- .0139658s 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 │ │ │ │ + -- .367125s 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 │ │ │ @@ -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 │ │ │ +Sun Mar 1 17:09:22 UTC 2026 │ │ │ │ │ │ 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/M2-13189-0/0.dbm │ │ │ │ │ │ i2 : x = openDatabaseOut filename │ │ │ │ │ │ -o2 = /tmp/M2-11758-0/0.dbm │ │ │ +o2 = /tmp/M2-13189-0/0.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 │ │ │ + -- 2.34063s 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 │ │ │ + -- 1.45866s 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/M2-13423-0/0 │ │ │ │ │ │ i4 : fn << "hi there" << endl << close │ │ │ │ │ │ -o4 = /tmp/M2-11872-0/0 │ │ │ +o4 = /tmp/M2-13423-0/0 │ │ │ │ │ │ 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/M2-13423-0/0 │ │ │ │ │ │ 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/M2-13423-0/0 │ │ │ │ │ │ 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" => 15426533962 } │ │ │ "GC_free_space_divisor" => 3 │ │ │ "GC_LARGE_ALLOC_WARN_INTERVAL" => 1 │ │ │ "gcCpuTimeSecs" => 0 │ │ │ - "heapSize" => 194183168 │ │ │ - "numGCs" => 836 │ │ │ - "numGCThreads" => 6 │ │ │ + "heapSize" => 226869248 │ │ │ + "numGCs" => 824 │ │ │ + "numGCThreads" => 16 │ │ │ │ │ │ o1 : HashTable │ │ │ │ │ │ i2 : s#"heapSize" │ │ │ │ │ │ -o2 = 194183168 │ │ │ +o2 = 226869248 │ │ │ │ │ │ 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 0.0271492s (cpu); 0.0271485s (thread); 0s (gc) │ │ │ │ │ │ 1 428 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ i9 : time J1 = saturate(I); │ │ │ - -- used 0.605726s (cpu); 0.339625s (thread); 0s (gc) │ │ │ + -- used 0.670544s (cpu); 0.205632s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ │ │ i10 : time J = saturate(I, MinimalGenerators => false); │ │ │ - -- used 0.000202009s (cpu); 0.000202159s (thread); 0s (gc) │ │ │ + -- used 0.000131682s (cpu); 0.000130039s (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 0.0441748s (cpu); 0.0441739s (thread); 0s (gc) │ │ │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ - -- used 0.0514493s (cpu); 0.0514556s (thread); 0s (gc) │ │ │ + -- used 0.0465581s (cpu); 0.0465687s (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 0.0019357s (cpu); 0.00192729s (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/_benchmark.out │ │ │ @@ -1,9 +1,9 @@ │ │ │ -- -*- M2-comint -*- hash: 1330379359420 │ │ │ │ │ │ i1 : benchmark "sqrt 2p100000" │ │ │ │ │ │ -o1 = .0002909794917276939 │ │ │ +o1 = .0003225104896245069 │ │ │ │ │ │ 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 │ │ │ + -- 2.29837s 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 = 1093734 │ │ │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ o6 = 0 │ │ │ │ │ │ i7 : t │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ o7 : Task │ │ │ │ │ │ i8 : n │ │ │ │ │ │ -o8 = 1451545 │ │ │ +o8 = 2220118 │ │ │ │ │ │ i9 : isReady t │ │ │ │ │ │ o9 = false │ │ │ │ │ │ i10 : cancelTask t │ │ │ │ │ │ @@ -53,22 +53,22 @@ │ │ │ │ │ │ o12 = <> │ │ │ │ │ │ o12 : Task │ │ │ │ │ │ i13 : n │ │ │ │ │ │ -o13 = 1451960 │ │ │ +o13 = 2220294 │ │ │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ o14 = 0 │ │ │ │ │ │ i15 : n │ │ │ │ │ │ -o15 = 1451960 │ │ │ +o15 = 2220294 │ │ │ │ │ │ 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/M2-10897-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-10626-0/0 │ │ │ +o2 = /tmp/M2-10897-0/0 │ │ │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ -o3 = /tmp/M2-10626-0/0/ │ │ │ +o3 = /tmp/M2-10897-0/0/ │ │ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ -o4 = /tmp/M2-10626-0/0/ │ │ │ +o4 = /tmp/M2-10897-0/0/ │ │ │ │ │ │ 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 6.12.73+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.73-1 (2026-02-17) 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 6.12.73+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ " │ │ │ - (2025-02-07) x86_64 GNU/Linux │ │ │ + 6.12.73-1 (2026-02-17) 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 0x7fdbe68f1700 │ │ │ │ │ │ -- [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 0.223895s (cpu); 0.220451s (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 0.00376329s (cpu); 0.00313411s (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 1.19107s (cpu); 0.995389s (thread); 0s (gc) │ │ │ + -- used 0.576945s (cpu); 0.479905s (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/M2-12297-0/0/ │ │ │ │ │ │ i2 : dst = temporaryFileName() | "/" │ │ │ │ │ │ -o2 = /tmp/M2-11326-0/1/ │ │ │ +o2 = /tmp/M2-12297-0/1/ │ │ │ │ │ │ i3 : makeDirectory (src|"a/") │ │ │ │ │ │ -o3 = /tmp/M2-11326-0/0/a/ │ │ │ +o3 = /tmp/M2-12297-0/0/a/ │ │ │ │ │ │ i4 : makeDirectory (src|"b/") │ │ │ │ │ │ -o4 = /tmp/M2-11326-0/0/b/ │ │ │ +o4 = /tmp/M2-12297-0/0/b/ │ │ │ │ │ │ i5 : makeDirectory (src|"b/c/") │ │ │ │ │ │ -o5 = /tmp/M2-11326-0/0/b/c/ │ │ │ +o5 = /tmp/M2-12297-0/0/b/c/ │ │ │ │ │ │ i6 : src|"a/f" << "hi there" << close │ │ │ │ │ │ -o6 = /tmp/M2-11326-0/0/a/f │ │ │ +o6 = /tmp/M2-12297-0/0/a/f │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : src|"a/g" << "hi there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11326-0/0/a/g │ │ │ +o7 = /tmp/M2-12297-0/0/a/g │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : src|"b/c/g" << "ho there" << close │ │ │ │ │ │ -o8 = /tmp/M2-11326-0/0/b/c/g │ │ │ +o8 = /tmp/M2-12297-0/0/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/M2-12297-0/0/ │ │ │ + /tmp/M2-12297-0/0/a/ │ │ │ + /tmp/M2-12297-0/0/a/g │ │ │ + /tmp/M2-12297-0/0/a/f │ │ │ + /tmp/M2-12297-0/0/b/ │ │ │ + /tmp/M2-12297-0/0/b/c/ │ │ │ + /tmp/M2-12297-0/0/b/c/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/M2-12297-0/0/a/g -> /tmp/M2-12297-0/1/a/g │ │ │ + -- copying: /tmp/M2-12297-0/0/a/f -> /tmp/M2-12297-0/1/a/f │ │ │ + -- copying: /tmp/M2-12297-0/0/b/c/g -> /tmp/M2-12297-0/1/b/c/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/M2-12297-0/0/a/g not newer than /tmp/M2-12297-0/1/a/g │ │ │ + -- skipping: /tmp/M2-12297-0/0/a/f not newer than /tmp/M2-12297-0/1/a/f │ │ │ + -- skipping: /tmp/M2-12297-0/0/b/c/g not newer than /tmp/M2-12297-0/1/b/c/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/M2-12297-0/1/ │ │ │ + /tmp/M2-12297-0/1/a/ │ │ │ + /tmp/M2-12297-0/1/a/g │ │ │ + /tmp/M2-12297-0/1/a/f │ │ │ + /tmp/M2-12297-0/1/b/ │ │ │ + /tmp/M2-12297-0/1/b/c/ │ │ │ + /tmp/M2-12297-0/1/b/c/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/M2-11846-0/0 │ │ │ │ │ │ i2 : dst = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-11095-0/1 │ │ │ +o2 = /tmp/M2-11846-0/1 │ │ │ │ │ │ i3 : src << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-11095-0/0 │ │ │ +o3 = /tmp/M2-11846-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : copyFile(src,dst,Verbose=>true) │ │ │ - -- copying: /tmp/M2-11095-0/0 -> /tmp/M2-11095-0/1 │ │ │ + -- copying: /tmp/M2-11846-0/0 -> /tmp/M2-11846-0/1 │ │ │ │ │ │ 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/M2-11846-0/0 not newer than /tmp/M2-11846-0/1 │ │ │ │ │ │ i7 : src << "ho there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11095-0/0 │ │ │ +o7 = /tmp/M2-11846-0/0 │ │ │ │ │ │ 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/M2-11846-0/0 not newer than /tmp/M2-11846-0/1 │ │ │ │ │ │ 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 = 209.825281589 │ │ │ │ │ │ o1 : RR (of precision 53) │ │ │ │ │ │ i2 : for i from 0 to 1000000 do 223131321321*324234324324; │ │ │ │ │ │ i3 : t2 = cpuTime() │ │ │ │ │ │ -o3 = 261.1503754559999 │ │ │ +o3 = 210.676697278 │ │ │ │ │ │ o3 : RR (of precision 53) │ │ │ │ │ │ i4 : t2-t1 │ │ │ │ │ │ -o4 = 2.15877068399999 │ │ │ +o4 = .8514156889999924 │ │ │ │ │ │ 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 = 1772385019 │ │ │ │ │ │ i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 ) │ │ │ │ │ │ -o2 = 55.11132206304336 │ │ │ +o2 = 56.16464540048101 │ │ │ │ │ │ o2 : RR (of precision 53) │ │ │ │ │ │ i3 : 12 * (oo - floor oo) │ │ │ │ │ │ -o3 = 1.335864756520323 │ │ │ +o3 = 1.975744805772166 │ │ │ │ │ │ o3 : RR (of precision 53) │ │ │ │ │ │ i4 : run "date" │ │ │ -Sun Feb 9 23:55:45 UTC 2025 │ │ │ +Sun Mar 1 17:10:19 UTC 2026 │ │ │ │ │ │ o4 = 0 │ │ │ │ │ │ i5 : │ │ ├── ./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 │ │ │ + -- 1.00018 seconds │ │ │ │ │ │ o1 : Time │ │ │ │ │ │ i2 : peek oo │ │ │ │ │ │ -o2 = Time{1.00014, 0} │ │ │ +o2 = Time{1.00018, 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 0.142017s (cpu); 0.142017s (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 0.150032s (cpu); 0.150034s (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 0.0489821s (cpu); 0.0489833s (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 0.32447s (cpu); 0.172851s (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 0.00186723s (cpu); 0.00186381s (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 0.0420575s (cpu); 0.0420676s (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/M2-11092-0/0 │ │ │ │ │ │ i2 : fileExists fn │ │ │ │ │ │ o2 = false │ │ │ │ │ │ i3 : fn << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10721-0/0 │ │ │ +o3 = /tmp/M2-11092-0/0 │ │ │ │ │ │ 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/M2-14283-0/0 │ │ │ │ │ │ o1 : File │ │ │ │ │ │ i2 : fileLength f │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ i3 : close f │ │ │ │ │ │ -o3 = /tmp/M2-12302-0/0 │ │ │ +o3 = /tmp/M2-14283-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : filename = toString f │ │ │ │ │ │ -o4 = /tmp/M2-12302-0/0 │ │ │ +o4 = /tmp/M2-14283-0/0 │ │ │ │ │ │ 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/M2-12648-0/0 │ │ │ │ │ │ i2 : f = fn << "hi there" │ │ │ │ │ │ -o2 = /tmp/M2-11497-0/0 │ │ │ +o2 = /tmp/M2-12648-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : fileMode f │ │ │ │ │ │ o3 = 420 │ │ │ │ │ │ i4 : close f │ │ │ │ │ │ -o4 = /tmp/M2-11497-0/0 │ │ │ +o4 = /tmp/M2-12648-0/0 │ │ │ │ │ │ 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/M2-11885-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-11114-0/0 │ │ │ +o2 = /tmp/M2-11885-0/0 │ │ │ │ │ │ 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/M2-11649-0/0 │ │ │ │ │ │ i2 : f = fn << "hi there" │ │ │ │ │ │ -o2 = /tmp/M2-10998-0/0 │ │ │ +o2 = /tmp/M2-11649-0/0 │ │ │ │ │ │ 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/M2-11649-0/0 │ │ │ │ │ │ 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/M2-13953-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-12132-0/0 │ │ │ +o2 = /tmp/M2-13953-0/0 │ │ │ │ │ │ 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 = 34 │ │ │ │ │ │ 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 0x7fd6f88af000 │ │ │ │ │ │ -- [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 = Sun Mar 1 17:09:47 UTC 2026 │ │ │ │ │ │ │ │ │ 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__Directory.out │ │ │ @@ -2,19 +2,19 @@ │ │ │ │ │ │ i1 : isDirectory "." │ │ │ │ │ │ o1 = true │ │ │ │ │ │ i2 : fn = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-10543-0/0 │ │ │ +o2 = /tmp/M2-10734-0/0 │ │ │ │ │ │ i3 : fn << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10543-0/0 │ │ │ +o3 = /tmp/M2-10734-0/0 │ │ │ │ │ │ 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 │ │ │ + -- 5.47268s 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 │ │ │ + -- .061427s elapsed │ │ │ │ │ │ o23 = true │ │ │ │ │ │ i24 : elapsedTime isPseudoprime m3 │ │ │ - -- .000134241s elapsed │ │ │ + -- .000146131s 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/M2-14361-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-12340-0/0 │ │ │ +o2 = /tmp/M2-14361-0/0 │ │ │ │ │ │ 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/M2-11377-0/0 │ │ │ │ │ │ i2 : makeDirectory (dir|"/a/b/c") │ │ │ │ │ │ -o2 = /tmp/M2-10866-0/0/a/b/c │ │ │ +o2 = /tmp/M2-11377-0/0/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 0.841981s (cpu); 0.669293s (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 6.8357e-05s (cpu); 6.4574e-05s (thread); 0s (gc) │ │ │ │ │ │ o4 = 514229 │ │ │ │ │ │ i5 : time fib 28 │ │ │ - -- used 4.138e-06s (cpu); 3.807e-06s (thread); 0s (gc) │ │ │ + -- used 3.682e-06s (cpu); 3.175e-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) } │ │ │ + {20 => (truncate, BettiTally, ZZ, ZZ) } │ │ │ + {21 => (truncate, BettiTally, ZZ, InfiniteNumber) } │ │ │ {22 => (codim, BettiTally) } │ │ │ - {23 => (truncate, BettiTally, InfiniteNumber, ZZ) } │ │ │ - {24 => (truncate, BettiTally, ZZ, InfiniteNumber) } │ │ │ - {25 => (dual, BettiTally) } │ │ │ + {23 => (dual, BettiTally) } │ │ │ + {24 => (truncate, BettiTally, InfiniteNumber, ZZ) } │ │ │ + {25 => (truncate, BettiTally, InfiniteNumber, 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 => (+, Matrix, Matrix) } │ │ │ + {3 => (-, Matrix, Matrix) } │ │ │ + {4 => (contract, Matrix, Matrix) } │ │ │ + {5 => (diff, 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 => (intersection, Matrix, Matrix) } │ │ │ + {29 => (tensor, Matrix, Matrix) } │ │ │ + {30 => (intersection, Matrix, Matrix, Matrix, Matrix) } │ │ │ + {31 => (pullback, 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 │ │ │ + -- 2.32898s 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 │ │ │ + -- .974262s 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 │ │ │ + -- .0396837s 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 │ │ │ + -- 1.5699s 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/M2-11416-0/0/ │ │ │ │ │ │ i2 : mkdir p │ │ │ │ │ │ i3 : isDirectory p │ │ │ │ │ │ o3 = true │ │ │ │ │ │ i4 : (fn = p | "foo") << "hi there" << close │ │ │ │ │ │ -o4 = /tmp/M2-10885-0/0/foo │ │ │ +o4 = /tmp/M2-11416-0/0/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/M2-11170-0/0 │ │ │ │ │ │ i2 : dst = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-10759-0/1 │ │ │ +o2 = /tmp/M2-11170-0/1 │ │ │ │ │ │ i3 : src << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10759-0/0 │ │ │ +o3 = /tmp/M2-11170-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : moveFile(src,dst,Verbose=>true) │ │ │ ---moving: /tmp/M2-10759-0/0 -> /tmp/M2-10759-0/1 │ │ │ +--moving: /tmp/M2-11170-0/0 -> /tmp/M2-11170-0/1 │ │ │ │ │ │ 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/M2-11170-0/1.bak │ │ │ │ │ │ -o6 = /tmp/M2-10759-0/1.bak │ │ │ +o6 = /tmp/M2-11170-0/1.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 │ │ │ + -- .500131s 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 │ │ │ + -- .707402s elapsed │ │ │ │ │ │ i4 : elapsedTime parallelApply(1..100, n -> sort L); │ │ │ - -- .295654s elapsed │ │ │ + -- .18421s 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 │ │ │ + -- 2.77843s 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 │ │ │ + -- 2.59402s 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 │ │ │ + -- 2.73766s 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 │ │ │ + -- 2.67254s 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 │ │ │ + -- 2.72571s 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 │ │ │ + -- 3.9588s 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 │ │ │ + -- 9.18656s 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 │ │ │ + -- 3.58389s 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 │ │ │ + -- .973654s 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 │ │ │ + -- 1.49393s 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 0.00212013s (cpu); 1.2957e-05s (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 0x7fd784494e00 │ │ │ │ │ │ -- [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 0.0178341s (cpu); 0.0188647s (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 0x7fd784494c40 │ │ │ │ │ │ -- [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 0x7fd7844948c0 │ │ │ │ │ │ -- [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 0.00395161s (cpu); 0.00647218s (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 0x7fd784494700 │ │ │ │ │ │ -- [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 0.0480263s (cpu); 0.0471098s (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 0.00370876s (cpu); 0.00369659s (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 .0339338 seconds │ │ │ +h: 22 times, used .0677341 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 0.178306s (cpu); 0.0826095s (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 0.354295s (cpu); 0.228087s (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 3.1947s (cpu); 1.81257s (thread); 0s (gc) │ │ │ │ │ │ o15 = 58 │ │ │ │ │ │ i16 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_read__Directory.out │ │ │ @@ -1,26 +1,26 @@ │ │ │ -- -*- M2-comint -*- hash: 20910736704070514 │ │ │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-11649-0/0 │ │ │ +o1 = /tmp/M2-12960-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-11649-0/0 │ │ │ +o2 = /tmp/M2-12960-0/0 │ │ │ │ │ │ i3 : (fn = dir | "/" | "foo") << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-11649-0/0/foo │ │ │ +o3 = /tmp/M2-12960-0/0/foo │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : readDirectory dir │ │ │ │ │ │ -o4 = {., .., foo} │ │ │ +o4 = {.., ., foo} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : removeFile fn │ │ │ │ │ │ i6 : removeDirectory dir │ │ ├── ./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/M2-12123-0/0 │ │ │ │ │ │ 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/M2-12123-0/0 │ │ │ │ │ │ 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/M2-12123-0/0 │ │ │ │ │ │ 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/M2-13592-0/0 │ │ │ │ │ │ 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/M2-13631-0/0 │ │ │ │ │ │ i3 : q = temporaryFileName() │ │ │ │ │ │ -o3 = /tmp/M2-11980-0/1 │ │ │ +o3 = /tmp/M2-13631-0/1 │ │ │ │ │ │ i4 : symlinkFile(p,q) │ │ │ │ │ │ i5 : p << close │ │ │ │ │ │ -o5 = /tmp/M2-11980-0/0 │ │ │ +o5 = /tmp/M2-13631-0/0 │ │ │ │ │ │ o5 : File │ │ │ │ │ │ i6 : readlink q │ │ │ │ │ │ -o6 = /tmp/M2-11980-0/0 │ │ │ +o6 = /tmp/M2-13631-0/0 │ │ │ │ │ │ i7 : realpath q │ │ │ │ │ │ -o7 = /tmp/M2-11980-0/0 │ │ │ +o7 = /tmp/M2-13631-0/0 │ │ │ │ │ │ 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: (3)[6]: -- finalizing sequence #7 -- │ │ │ +--finalization: (4)[3]: -- finalizing sequence #4 -- │ │ │ +--finalization: (5)[1]: -- finalizing sequence #2 -- │ │ │ --finalization: (6)[5]: -- finalizing sequence #6 -- │ │ │ ---finalization: (7)[6]: -- finalizing sequence #7 -- │ │ │ +--finalization: (7)[2]: -- finalizing sequence #3 -- │ │ │ --finalization: (8)[0]: -- finalizing sequence #1 -- │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out │ │ │ @@ -1,19 +1,19 @@ │ │ │ -- -*- M2-comint -*- hash: 8532980310097060089 │ │ │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10923-0/0 │ │ │ +o1 = /tmp/M2-11494-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-10923-0/0 │ │ │ +o2 = /tmp/M2-11494-0/0 │ │ │ │ │ │ i3 : readDirectory dir │ │ │ │ │ │ -o3 = {., ..} │ │ │ +o3 = {.., .} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : removeDirectory dir │ │ │ │ │ │ i5 : │ │ ├── ./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/M2-10576-0/0 │ │ │ │ │ │ i2 : rootPath | fn │ │ │ │ │ │ -o2 = /tmp/M2-10465-0/0 │ │ │ +o2 = /tmp/M2-10576-0/0 │ │ │ │ │ │ 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/M2-12843-0/0 │ │ │ │ │ │ i2 : rootURI | fn │ │ │ │ │ │ -o2 = file:///tmp/M2-11592-0/0 │ │ │ +o2 = file:///tmp/M2-12843-0/0 │ │ │ │ │ │ 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/M2-12609-0/0 │ │ │ │ │ │ i6 : f << toString (p,m,M) << close │ │ │ │ │ │ -o6 = /tmp/M2-11478-0/0 │ │ │ +o6 = /tmp/M2-12609-0/0 │ │ │ │ │ │ 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 0.000332467s (cpu); 0.000323809s (thread); 0s (gc) │ │ │ │ │ │ i31 : time X = solve(A,B, MaximalRank=>true); │ │ │ - -- used 0.000168847s (cpu); 0.000168906s (thread); 0s (gc) │ │ │ + -- used 0.000189007s (cpu); 0.000189401s (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 0.16989s (cpu); 0.169605s (thread); 0s (gc) │ │ │ │ │ │ i39 : time X = solve(A,B, MaximalRank=>true); │ │ │ - -- used 0.238738s (cpu); 0.23868s (thread); 0s (gc) │ │ │ + -- used 0.150266s (cpu); 0.150277s (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/M2-12203-0/0/ │ │ │ │ │ │ i2 : dst = temporaryFileName() | "/" │ │ │ │ │ │ -o2 = /tmp/M2-11272-0/1/ │ │ │ +o2 = /tmp/M2-12203-0/1/ │ │ │ │ │ │ i3 : makeDirectory (src|"a/") │ │ │ │ │ │ -o3 = /tmp/M2-11272-0/0/a/ │ │ │ +o3 = /tmp/M2-12203-0/0/a/ │ │ │ │ │ │ i4 : makeDirectory (src|"b/") │ │ │ │ │ │ -o4 = /tmp/M2-11272-0/0/b/ │ │ │ +o4 = /tmp/M2-12203-0/0/b/ │ │ │ │ │ │ i5 : makeDirectory (src|"b/c/") │ │ │ │ │ │ -o5 = /tmp/M2-11272-0/0/b/c/ │ │ │ +o5 = /tmp/M2-12203-0/0/b/c/ │ │ │ │ │ │ i6 : src|"a/f" << "hi there" << close │ │ │ │ │ │ -o6 = /tmp/M2-11272-0/0/a/f │ │ │ +o6 = /tmp/M2-12203-0/0/a/f │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : src|"a/g" << "hi there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11272-0/0/a/g │ │ │ +o7 = /tmp/M2-12203-0/0/a/g │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : src|"b/c/g" << "ho there" << close │ │ │ │ │ │ -o8 = /tmp/M2-11272-0/0/b/c/g │ │ │ +o8 = /tmp/M2-12203-0/0/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/a/g -> /tmp/M2-12203-0/1/a/g │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12203-0/1/a/f │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12203-0/1/b/c/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/a/g -> /tmp/M2-12203-0/1/a/g │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12203-0/1/a/f │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12203-0/1/b/c/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/M2-12336-0/0 │ │ │ │ │ │ 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/M2-14322-0/0.tex │ │ │ │ │ │ i2 : temporaryFileName () | ".html" │ │ │ │ │ │ -o2 = /tmp/M2-12321-0/1.html │ │ │ +o2 = /tmp/M2-14322-0/1.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 2.3392e-05s (cpu); 8.943e-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 │ │ │ + -- .000014607 seconds │ │ │ │ │ │ o1 : Time │ │ │ │ │ │ i2 : peek oo │ │ │ │ │ │ -o2 = Time{.000014217, 205891132094649} │ │ │ +o2 = Time{.000014607, 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" => 6.12.73+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
│ │ │ +Sun Mar  1 17:09:22 UTC 2026
│ │ │  
│ │ │  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 │ │ │ │ +Sun Mar 1 17:09:22 UTC 2026 │ │ │ │ │ │ │ │ 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/M2-13189-0/0.dbm │ │ │ 
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11758-0/0.dbm
│ │ │ +o2 = /tmp/M2-13189-0/0.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/M2-13189-0/0.dbm │ │ │ │ i2 : x = openDatabaseOut filename │ │ │ │ │ │ │ │ -o2 = /tmp/M2-11758-0/0.dbm │ │ │ │ +o2 = /tmp/M2-13189-0/0.dbm │ │ │ │ │ │ │ │ o2 : Database │ │ │ │ i3 : x#"first" = "hi there" │ │ │ │ │ │ │ │ o3 = hi there │ │ │ │ i4 : x#"first" │ │ ├── ./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
│ │ │ + -- 2.34063s 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
│ │ │ + -- 1.45866s 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 │ │ │ │ + -- 2.34063s 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 │ │ │ │ + -- 1.45866s 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/M2-13423-0/0 │ │ │ 
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11872-0/0
│ │ │ +o4 = /tmp/M2-13423-0/0
│ │ │  
│ │ │  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/M2-13423-0/0
│ │ │  
│ │ │  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/M2-13423-0/0
│ │ │  
│ │ │  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/M2-13423-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11872-0/0
│ │ │ │ +o4 = /tmp/M2-13423-0/0
│ │ │ │  
│ │ │ │  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/M2-13423-0/0
│ │ │ │  
│ │ │ │  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/M2-13423-0/0
│ │ │ │  
│ │ │ │  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" => 15426533962        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 194183168
│ │ │ -               "numGCs" => 836
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 226869248
│ │ │ +               "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 = 226869248
│ │ │      		          
    
│ │ │          
    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" => 15426533962        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 194183168
│ │ │ │ -               "numGCs" => 836
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 226869248
│ │ │ │ +               "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 = 226869248
│ │ │ │  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/___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 0.0271492s (cpu); 0.0271485s (thread); 0s (gc)
│ │ │  
│ │ │               1      428
│ │ │  o8 : Matrix R  <-- R
    │ │ │ 
    i9 : time J1 = saturate(I);
│ │ │ - -- used 0.605726s (cpu); 0.339625s (thread); 0s (gc)
│ │ │ + -- used 0.670544s (cpu); 0.205632s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    │ │ │ 
    i10 : time J = saturate(I, MinimalGenerators => false);
│ │ │ - -- used 0.000202009s (cpu); 0.000202159s (thread); 0s (gc)
│ │ │ + -- used 0.000131682s (cpu); 0.000130039s (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 0.0271492s (cpu); 0.0271485s (thread); 0s (gc) │ │ │ │ │ │ │ │ 1 428 │ │ │ │ o8 : Matrix R <-- R │ │ │ │ i9 : time J1 = saturate(I); │ │ │ │ - -- used 0.605726s (cpu); 0.339625s (thread); 0s (gc) │ │ │ │ + -- used 0.670544s (cpu); 0.205632s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : time J = saturate(I, MinimalGenerators => false); │ │ │ │ - -- used 0.000202009s (cpu); 0.000202159s (thread); 0s (gc) │ │ │ │ + -- used 0.000131682s (cpu); 0.000130039s (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 0.0441748s (cpu); 0.0441739s (thread); 0s (gc) │ │ │ 
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0514493s (cpu); 0.0514556s (thread); 0s (gc)
    │ │ │ + -- used 0.0465581s (cpu); 0.0465687s (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 0.0441748s (cpu); 0.0441739s (thread); 0s (gc) │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ │ - -- used 0.0514493s (cpu); 0.0514556s (thread); 0s (gc) │ │ │ │ + -- used 0.0465581s (cpu); 0.0465687s (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/___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. │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -112,15 +112,15 @@ │ │ │ o7 = <<task, running>> │ │ │ │ │ │ o7 : Task │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -138,25 +138,25 @@ │ │ │ o12 = <<task, canceled>> │ │ │ │ │ │ o12 : Task │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i59 : time C = resolution M
│ │ │ - -- used 0.00187636s (cpu); 0.00186805s (thread); 0s (gc)
│ │ │ + -- used 0.0019357s (cpu); 0.00192729s (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 0.0019357s (cpu); 0.00192729s (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/_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 = .0003225104896245069
│ │ │  
│ │ │  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 = .0003225104896245069 │ │ │ │ │ │ │ │ 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
│ │ │ + -- 2.29837s 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 │ │ │ │ + -- 2.29837s 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 = 1093734 │ │ │ 
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    │ │ │ 
    i8 : n
│ │ │  
│ │ │ -o8 = 1451545
    │ │ │ +o8 = 2220118 │ │ │ 
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    │ │ │ 
    i13 : n
│ │ │  
│ │ │ -o13 = 1451960
    │ │ │ +o13 = 2220294 │ │ │ 
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    │ │ │ 
    i15 : n
│ │ │  
│ │ │ -o15 = 1451960
    │ │ │ +o15 = 2220294 │ │ │ 
    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 = 1093734 │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ │ │ o6 = 0 │ │ │ │ i7 : t │ │ │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ │ │ o7 : Task │ │ │ │ i8 : n │ │ │ │ │ │ │ │ -o8 = 1451545 │ │ │ │ +o8 = 2220118 │ │ │ │ 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 = 2220294 │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ │ │ o14 = 0 │ │ │ │ i15 : n │ │ │ │ │ │ │ │ -o15 = 1451960 │ │ │ │ +o15 = 2220294 │ │ │ │ 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/M2-10897-0/0 │ │ │ 
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10626-0/0
    │ │ │ +o2 = /tmp/M2-10897-0/0 │ │ │ 
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10626-0/0/
    │ │ │ +o3 = /tmp/M2-10897-0/0/ │ │ │ 
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10626-0/0/
    │ │ │ +o4 = /tmp/M2-10897-0/0/ │ │ │
    │ │ │
    │ │ │

    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/M2-10897-0/0 │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ │ │ -o2 = /tmp/M2-10626-0/0 │ │ │ │ +o2 = /tmp/M2-10897-0/0 │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ │ │ -o3 = /tmp/M2-10626-0/0/ │ │ │ │ +o3 = /tmp/M2-10897-0/0/ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ │ │ -o4 = /tmp/M2-10626-0/0/ │ │ │ │ +o4 = /tmp/M2-10897-0/0/ │ │ │ │ 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 6.12.73+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.73-1 (2026-02-17) 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 6.12.73+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │       "
│ │ │ -     (2025-02-07) x86_64 GNU/Linux
    │ │ │ + 6.12.73-1 (2026-02-17) 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 6.12.73+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.73-1
│ │ │ │ +(2026-02-17) 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 6.12.73+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │       "
│ │ │ │ -     (2025-02-07) x86_64 GNU/Linux
│ │ │ │ +     6.12.73-1 (2026-02-17) 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 0x7fdbe68f1700
│ │ │  
│ │ │     -- [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 0.223895s (cpu); 0.220451s (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 0.00376329s (cpu); 0.00313411s (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 0x7fdbe68f1700
│ │ │ │  
│ │ │ │     -- [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 0.223895s (cpu); 0.220451s (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 0.00376329s (cpu); 0.00313411s (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 1.19107s (cpu); 0.995389s (thread); 0s (gc)
│ │ │ + -- used 0.576945s (cpu); 0.479905s (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 1.19107s (cpu); 0.995389s (thread); 0s (gc)
│ │ │ │ + -- used 0.576945s (cpu); 0.479905s (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/M2-12297-0/0/
│ │ │      		          
                  
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11326-0/1/
    
│ │ │ +o2 = /tmp/M2-12297-0/1/
│ │ │      		          
                  
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11326-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12297-0/0/a/
│ │ │      		          
                  
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11326-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12297-0/0/b/
│ │ │      		          
                  
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11326-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12297-0/0/b/c/
│ │ │      		          
                  
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11326-0/0/a/f
│ │ │ +o6 = /tmp/M2-12297-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │      		          
                  
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11326-0/0/a/g
│ │ │ +o7 = /tmp/M2-12297-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │      		          
                  
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11326-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12297-0/0/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/M2-12297-0/0/
│ │ │ +     /tmp/M2-12297-0/0/a/
│ │ │ +     /tmp/M2-12297-0/0/a/g
│ │ │ +     /tmp/M2-12297-0/0/a/f
│ │ │ +     /tmp/M2-12297-0/0/b/
│ │ │ +     /tmp/M2-12297-0/0/b/c/
│ │ │ +     /tmp/M2-12297-0/0/b/c/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/M2-12297-0/0/a/g -> /tmp/M2-12297-0/1/a/g
│ │ │ + -- copying: /tmp/M2-12297-0/0/a/f -> /tmp/M2-12297-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12297-0/0/b/c/g -> /tmp/M2-12297-0/1/b/c/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/M2-12297-0/0/a/g not newer than /tmp/M2-12297-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-12297-0/0/a/f not newer than /tmp/M2-12297-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12297-0/0/b/c/g not newer than /tmp/M2-12297-0/1/b/c/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/M2-12297-0/1/
│ │ │ +      /tmp/M2-12297-0/1/a/
│ │ │ +      /tmp/M2-12297-0/1/a/g
│ │ │ +      /tmp/M2-12297-0/1/a/f
│ │ │ +      /tmp/M2-12297-0/1/b/
│ │ │ +      /tmp/M2-12297-0/1/b/c/
│ │ │ +      /tmp/M2-12297-0/1/b/c/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/M2-12297-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11326-0/1/
│ │ │ │ +o2 = /tmp/M2-12297-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11326-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12297-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11326-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12297-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11326-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12297-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11326-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12297-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11326-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12297-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11326-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12297-0/0/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/M2-12297-0/0/
│ │ │ │ +     /tmp/M2-12297-0/0/a/
│ │ │ │ +     /tmp/M2-12297-0/0/a/g
│ │ │ │ +     /tmp/M2-12297-0/0/a/f
│ │ │ │ +     /tmp/M2-12297-0/0/b/
│ │ │ │ +     /tmp/M2-12297-0/0/b/c/
│ │ │ │ +     /tmp/M2-12297-0/0/b/c/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/M2-12297-0/0/a/g -> /tmp/M2-12297-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-12297-0/0/a/f -> /tmp/M2-12297-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12297-0/0/b/c/g -> /tmp/M2-12297-0/1/b/c/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/M2-12297-0/0/a/g not newer than /tmp/M2-12297-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-12297-0/0/a/f not newer than /tmp/M2-12297-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12297-0/0/b/c/g not newer than /tmp/M2-12297-0/1/b/c/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/M2-12297-0/1/
│ │ │ │ +      /tmp/M2-12297-0/1/a/
│ │ │ │ +      /tmp/M2-12297-0/1/a/g
│ │ │ │ +      /tmp/M2-12297-0/1/a/f
│ │ │ │ +      /tmp/M2-12297-0/1/b/
│ │ │ │ +      /tmp/M2-12297-0/1/b/c/
│ │ │ │ +      /tmp/M2-12297-0/1/b/c/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/M2-11846-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11095-0/1
│ │ │ │ +o2 = /tmp/M2-11846-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11095-0/0
│ │ │ │ +o3 = /tmp/M2-11846-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11095-0/0 -> /tmp/M2-11095-0/1
│ │ │ │ + -- copying: /tmp/M2-11846-0/0 -> /tmp/M2-11846-0/1
│ │ │ │  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/M2-11846-0/0 not newer than /tmp/M2-11846-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11095-0/0
│ │ │ │ +o7 = /tmp/M2-11846-0/0
│ │ │ │  
│ │ │ │  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/M2-11846-0/0 not newer than /tmp/M2-11846-0/1
│ │ │ │  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/M2-11846-0/0
│ │ │      		          
                  
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11095-0/1
    
│ │ │ +o2 = /tmp/M2-11846-0/1
│ │ │      		          
                  
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11095-0/0
│ │ │ +o3 = /tmp/M2-11846-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11095-0/0 -> /tmp/M2-11095-0/1
    
│ │ │ + -- copying: /tmp/M2-11846-0/0 -> /tmp/M2-11846-0/1
│ │ │      		          
                  
    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/M2-11846-0/0 not newer than /tmp/M2-11846-0/1
│ │ │      		          
                  
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11095-0/0
│ │ │ +o7 = /tmp/M2-11846-0/0
│ │ │  
│ │ │  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/M2-11846-0/0 not newer than /tmp/M2-11846-0/1
│ │ │      		          
                  
    i9 : get dst
│ │ │  
│ │ │  o9 = hi there
    
│ │ │      		          
    
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 258.991604772
│ │ │ +o1 = 209.825281589
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │      		          
                  
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │      		          
                  
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 261.1503754559999
│ │ │ +o3 = 210.676697278
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │      		          
                  
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 2.15877068399999
│ │ │ +o4 = .8514156889999924
│ │ │  
│ │ │  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 = 209.825281589
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 261.1503754559999
│ │ │ │ +o3 = 210.676697278
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 2.15877068399999
│ │ │ │ +o4 = .8514156889999924
│ │ │ │  
│ │ │ │  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 = 1772385019
│ │ │      		          
    
│ │ │          
    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 = 56.16464540048101
│ │ │  
│ │ │  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 = 1.975744805772166
│ │ │  
│ │ │  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
│ │ │ +Sun Mar  1 17:10:19 UTC 2026
│ │ │  
│ │ │  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 = 1772385019
│ │ │ │  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 = 56.16464540048101
│ │ │ │  
│ │ │ │  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 = 1.975744805772166
│ │ │ │  
│ │ │ │  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
│ │ │ │ +Sun Mar  1 17:10:19 UTC 2026
│ │ │ │  
│ │ │ │  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__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
│ │ │ +     -- 1.00018 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │      		          
                  
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00014, 0}
    
│ │ │ +o2 = Time{1.00018, 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
│ │ │ │ +     -- 1.00018 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{1.00014, 0}
│ │ │ │ +o2 = Time{1.00018, 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 0.142017s (cpu); 0.142017s (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 0.150032s (cpu); 0.150034s (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 0.0489821s (cpu); 0.0489833s (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 0.32447s (cpu); 0.172851s (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 0.00186723s (cpu); 0.00186381s (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 0.0420575s (cpu); 0.0420676s (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 0.142017s (cpu); 0.142017s (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 0.150032s (cpu); 0.150034s (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 0.0489821s (cpu); 0.0489833s (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 0.32447s (cpu); 0.172851s (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 0.00186723s (cpu); 0.00186381s (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 0.0420575s (cpu); 0.0420676s (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/M2-11092-0/0
│ │ │      		          
                  
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │      		          
                  
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10721-0/0
│ │ │ +o3 = /tmp/M2-11092-0/0
│ │ │  
│ │ │  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/M2-11092-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10721-0/0
│ │ │ │ +o3 = /tmp/M2-11092-0/0
│ │ │ │  
│ │ │ │  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/M2-14283-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12302-0/0
│ │ │ │ +o3 = /tmp/M2-14283-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12302-0/0
│ │ │ │ +o4 = /tmp/M2-14283-0/0
│ │ │ │  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/M2-14283-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │      		          
                  
    i2 : fileLength f
│ │ │  
│ │ │  o2 = 8
    
│ │ │      		          
                  
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12302-0/0
│ │ │ +o3 = /tmp/M2-14283-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12302-0/0
    
│ │ │ +o4 = /tmp/M2-14283-0/0
│ │ │      		          
                  
    i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
    
│ │ │      		          
    
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11497-0/0
    
│ │ │ +o1 = /tmp/M2-12648-0/0
│ │ │      		          
                  
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11497-0/0
│ │ │ +o2 = /tmp/M2-12648-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │      		          
                  
    i3 : fileMode f
│ │ │  
│ │ │  o3 = 420
    
│ │ │      		          
                  
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11497-0/0
│ │ │ +o4 = /tmp/M2-12648-0/0
│ │ │  
│ │ │  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/M2-12648-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11497-0/0
│ │ │ │ +o2 = /tmp/M2-12648-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11497-0/0
│ │ │ │ +o4 = /tmp/M2-12648-0/0
│ │ │ │  
│ │ │ │  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/M2-11885-0/0
│ │ │      		          
                  
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11114-0/0
│ │ │ +o2 = /tmp/M2-11885-0/0
│ │ │  
│ │ │  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/M2-11885-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11114-0/0
│ │ │ │ +o2 = /tmp/M2-11885-0/0
│ │ │ │  
│ │ │ │  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/M2-11649-0/0
│ │ │      		          
                  
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-10998-0/0
│ │ │ +o2 = /tmp/M2-11649-0/0
│ │ │  
│ │ │  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/M2-11649-0/0
│ │ │  
│ │ │  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/M2-11649-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10998-0/0
│ │ │ │ +o2 = /tmp/M2-11649-0/0
│ │ │ │  
│ │ │ │  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/M2-11649-0/0
│ │ │ │  
│ │ │ │  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/M2-13953-0/0
│ │ │      		          
                  
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12132-0/0
│ │ │ +o2 = /tmp/M2-13953-0/0
│ │ │  
│ │ │  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/M2-13953-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12132-0/0
│ │ │ │ +o2 = /tmp/M2-13953-0/0
│ │ │ │  
│ │ │ │  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 = 34
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = 34
│ │ │ │  ********** 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 0x7fd6f88af000
│ │ │  
│ │ │     -- [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 0x7fd6f88af000
│ │ │ │  
│ │ │ │     -- [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 = Sun Mar  1 17:09:47 UTC 2026
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = Sun Mar  1 17:09:47 UTC 2026
│ │ │ │  ********** 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__Directory.html
│ │ │ @@ -72,20 +72,20 @@
│ │ │  
    		              
    i1 : isDirectory "."
│ │ │  
│ │ │  o1 = true
    
│ │ │      		          
                  
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10543-0/0
    
│ │ │ +o2 = /tmp/M2-10734-0/0
│ │ │      		          
                  
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10543-0/0
│ │ │ +o3 = /tmp/M2-10734-0/0
│ │ │  
│ │ │  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/M2-10734-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10543-0/0
│ │ │ │ +o3 = /tmp/M2-10734-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ + -- 5.47268s 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
│ │ │ + -- .061427s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │      		          
                  
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000134241s elapsed
│ │ │ + -- .000146131s 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
│ │ │ │ + -- 5.47268s 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
│ │ │ │ + -- .061427s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000134241s elapsed
│ │ │ │ + -- .000146131s 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/M2-14361-0/0
│ │ │      		          
                  
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12340-0/0
│ │ │ +o2 = /tmp/M2-14361-0/0
│ │ │  
│ │ │  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/M2-14361-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12340-0/0
│ │ │ │ +o2 = /tmp/M2-14361-0/0
│ │ │ │  
│ │ │ │  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/M2-11377-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10866-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-11377-0/0/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/M2-11377-0/0
│ │ │      		          
                  
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10866-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-11377-0/0/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 0.841981s (cpu); 0.669293s (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 6.8357e-05s (cpu); 6.4574e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │      		          
                  
    i5 : time fib 28
│ │ │ - -- used 4.138e-06s (cpu); 3.807e-06s (thread); 0s (gc)
│ │ │ + -- used 3.682e-06s (cpu); 3.175e-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 0.841981s (cpu); 0.669293s (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 6.8357e-05s (cpu); 6.4574e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 4.138e-06s (cpu); 3.807e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.682e-06s (cpu); 3.175e-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)                        }
│ │ │ +     {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │       {22 => (codim, BettiTally)                                   }
│ │ │ -     {23 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {25 => (dual, BettiTally)                                    }
│ │ │ +     {23 => (dual, BettiTally)                                    }
│ │ │ +     {24 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ +     {25 => (truncate, BettiTally, InfiniteNumber, 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 => (+, Matrix, Matrix)                                   }
│ │ │ +     {3 => (-, Matrix, Matrix)                                   }
│ │ │ +     {4 => (contract, Matrix, Matrix)                            }
│ │ │ +     {5 => (diff, 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 => (intersection, Matrix, Matrix)                       }
│ │ │ +     {29 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {30 => (intersection, Matrix, Matrix, Matrix, Matrix)       }
│ │ │ +     {31 => (pullback, 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)                        }
│ │ │ │ +     {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │       {22 => (codim, BettiTally)                                   }
│ │ │ │ -     {23 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ -     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {25 => (dual, BettiTally)                                    }
│ │ │ │ +     {23 => (dual, BettiTally)                                    }
│ │ │ │ +     {24 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ +     {25 => (truncate, BettiTally, InfiniteNumber, 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 => (+, Matrix, Matrix)                                   }
│ │ │ │ +     {3 => (-, Matrix, Matrix)                                   }
│ │ │ │ +     {4 => (contract, Matrix, Matrix)                            }
│ │ │ │ +     {5 => (diff, 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 => (intersection, Matrix, Matrix)                       }
│ │ │ │ +     {29 => (tensor, Matrix, Matrix)                             }
│ │ │ │ +     {30 => (intersection, Matrix, Matrix, Matrix, Matrix)       }
│ │ │ │ +     {31 => (pullback, 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
│ │ │ + -- 2.32898s 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
│ │ │ + -- .974262s 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
│ │ │ + -- .0396837s 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
│ │ │ + -- 1.5699s 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
│ │ │ │ + -- 2.32898s 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
│ │ │ │ + -- .974262s 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
│ │ │ │ + -- .0396837s 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
│ │ │ │ + -- 1.5699s 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/M2-11416-0/0/
│ │ │      		          
                  
    i2 : mkdir p
    
│ │ │      		          
                  
    i3 : isDirectory p
│ │ │  
│ │ │  o3 = true
    
│ │ │      		          
                  
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10885-0/0/foo
│ │ │ +o4 = /tmp/M2-11416-0/0/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/M2-11416-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10885-0/0/foo
│ │ │ │ +o4 = /tmp/M2-11416-0/0/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/M2-11170-0/0
│ │ │      		          
                  
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10759-0/1
    
│ │ │ +o2 = /tmp/M2-11170-0/1
│ │ │      		          
                  
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10759-0/0
│ │ │ +o3 = /tmp/M2-11170-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10759-0/0 -> /tmp/M2-10759-0/1
    
│ │ │ +--moving: /tmp/M2-11170-0/0 -> /tmp/M2-11170-0/1
│ │ │      		          
                  
    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/M2-11170-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10759-0/1.bak
    
│ │ │ +o6 = /tmp/M2-11170-0/1.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/M2-11170-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10759-0/1
│ │ │ │ +o2 = /tmp/M2-11170-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10759-0/0
│ │ │ │ +o3 = /tmp/M2-11170-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-10759-0/0 -> /tmp/M2-10759-0/1
│ │ │ │ +--moving: /tmp/M2-11170-0/0 -> /tmp/M2-11170-0/1
│ │ │ │  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/M2-11170-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-10759-0/1.bak
│ │ │ │ +o6 = /tmp/M2-11170-0/1.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
│ │ │ + -- .500131s 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
│ │ │ │ + -- .500131s 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
    
│ │ │ + -- .707402s elapsed
│ │ │      		          
                  
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .295654s elapsed
    
│ │ │ + -- .18421s 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
│ │ │ │ + -- .707402s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .295654s elapsed
│ │ │ │ + -- .18421s 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
│ │ │ + -- 2.77843s 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
│ │ │ + -- 2.59402s 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
│ │ │ + -- 2.73766s 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
│ │ │ + -- 2.67254s 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
│ │ │ + -- 2.72571s 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
│ │ │ + -- 3.9588s 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
│ │ │ + -- 9.18656s 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
│ │ │ + -- 3.58389s 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
│ │ │ + -- .973654s 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
│ │ │ + -- 1.49393s 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
│ │ │ │ + -- 2.77843s 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
│ │ │ │ + -- 2.59402s 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
│ │ │ │ + -- 2.73766s 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
│ │ │ │ + -- 2.67254s 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
│ │ │ │ + -- 2.72571s 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
│ │ │ │ + -- 3.9588s 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
│ │ │ │ + -- 9.18656s 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
│ │ │ │ + -- 3.58389s 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
│ │ │ │ + -- .973654s 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
│ │ │ │ + -- 1.49393s 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 0.00212013s (cpu); 1.2957e-05s (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 0x7fd784494e00
│ │ │  
│ │ │     -- [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 0.0178341s (cpu); 0.0188647s (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 0x7fd784494c40
│ │ │  
│ │ │     -- [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 0x7fd7844948c0
│ │ │  
│ │ │     -- [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 0.00395161s (cpu); 0.00647218s (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 0x7fd784494700
│ │ │  
│ │ │     -- [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 0.0480263s (cpu); 0.0471098s (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 0.00212013s (cpu); 1.2957e-05s (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 0x7fd784494e00
│ │ │ │  
│ │ │ │     -- [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 0.0178341s (cpu); 0.0188647s (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 0x7fd784494c40
│ │ │ │  
│ │ │ │     -- [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 0x7fd7844948c0
│ │ │ │  
│ │ │ │     -- [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 0.00395161s (cpu); 0.00647218s (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 0x7fd784494700
│ │ │ │  
│ │ │ │     -- [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 0.0480263s (cpu); 0.0471098s (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 0.00370876s (cpu); 0.00369659s (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 .0339338 seconds
│ │ │ +h: 22 times, used .0677341 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 0.00370876s (cpu); 0.00369659s (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 .0339338 seconds
│ │ │ │ +h: 22 times, used .0677341 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 0.178306s (cpu); 0.0826095s (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 0.354295s (cpu); 0.228087s (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 3.1947s (cpu); 1.81257s (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 0.178306s (cpu); 0.0826095s (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 0.354295s (cpu); 0.228087s (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 3.1947s (cpu); 1.81257s (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
│ │ │ @@ -67,32 +67,32 @@
│ │ │        
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,26 +11,26 @@
│ │ │ │      * 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/M2-12960-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11649-0/0
│ │ │ │ +o2 = /tmp/M2-12960-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11649-0/0/foo
│ │ │ │ +o3 = /tmp/M2-12960-0/0/foo
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : readDirectory dir
│ │ │ │  
│ │ │ │ -o4 = {., .., foo}
│ │ │ │ +o4 = {.., ., foo}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : removeFile fn
│ │ │ │  i6 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_m_o_v_e_D_i_r_e_c_t_o_r_y -- remove a directory
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ ├── ./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 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11649-0/0
    
│ │ │ +o1 = /tmp/M2-12960-0/0
│ │ │      		          
                  
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11649-0/0
    
│ │ │ +o2 = /tmp/M2-12960-0/0
│ │ │      		          
                  
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11649-0/0/foo
│ │ │ +o3 = /tmp/M2-12960-0/0/foo
│ │ │  
│ │ │  o3 : File
    
│ │ │      		          
                  
    i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
    
│ │ │      		          
                  
    i5 : removeFile fn
    
│ │ │      		          
    
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11232-0/0
    
│ │ │ +o1 = /tmp/M2-12123-0/0
│ │ │      		          
                  
    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/M2-12123-0/0
│ │ │  
│ │ │  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/M2-12123-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │      		          
    
│ │ │  Now verify that it contains the desired text with get.        
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │ ├── 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
│ │ │ │ + -- .118102s 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
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    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/M2-12123-0/0
│ │ │ │  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/M2-12123-0/0
│ │ │ │  
│ │ │ │  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/M2-12123-0/0
│ │ │ │  
│ │ │ │  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/M2-13592-0/0
│ │ │      		          
                  
    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/M2-13592-0/0
│ │ │ │  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/M2-13631-0/0
│ │ │      		          
                  
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11980-0/1
    
│ │ │ +o3 = /tmp/M2-13631-0/1
│ │ │      		          
                  
    i4 : symlinkFile(p,q)
    
│ │ │      		          
                  
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-11980-0/0
│ │ │ +o5 = /tmp/M2-13631-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │      		          
                  
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-11980-0/0
    
│ │ │ +o6 = /tmp/M2-13631-0/0
│ │ │      		          
                  
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-11980-0/0
    
│ │ │ +o7 = /tmp/M2-13631-0/0
│ │ │      		          
                  
    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/M2-13631-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11980-0/1
│ │ │ │ +o3 = /tmp/M2-13631-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11980-0/0
│ │ │ │ +o5 = /tmp/M2-13631-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11980-0/0
│ │ │ │ +o6 = /tmp/M2-13631-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11980-0/0
│ │ │ │ +o7 = /tmp/M2-13631-0/0
│ │ │ │  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
│ │ │ @@ -75,19 +75,19 @@
│ │ │            
                  
    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: (3)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (4)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (5)[1]: -- finalizing sequence #2 --
│ │ │  --finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │  --finalization: (8)[0]: -- finalizing sequence #1 --
    
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 {}
│ │ │ │ @@ -17,19 +17,19 @@
│ │ │ │              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: (3)[6]: -- finalizing sequence #7 --
│ │ │ │ +--finalization: (4)[3]: -- finalizing sequence #4 --
│ │ │ │ +--finalization: (5)[1]: -- finalizing sequence #2 --
│ │ │ │  --finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │ │ +--finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ │  --finalization: (8)[0]: -- finalizing sequence #1 --
│ │ │ │  ********** 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.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_remove__Directory.html
│ │ │ @@ -69,25 +69,25 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                  
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10923-0/0
    
│ │ │ +o1 = /tmp/M2-11494-0/0
│ │ │      		          
                  
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10923-0/0
    
│ │ │ +o2 = /tmp/M2-11494-0/0
│ │ │      		          
                  
    i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
    
│ │ │      		          
                  
    i4 : removeDirectory dir
    
│ │ │      		          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,21 +11,21 @@
│ │ │ │      * 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/M2-11494-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10923-0/0
│ │ │ │ +o2 = /tmp/M2-11494-0/0
│ │ │ │  i3 : readDirectory dir
│ │ │ │  
│ │ │ │ -o3 = {., ..}
│ │ │ │ +o3 = {.., .}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d_D_i_r_e_c_t_o_r_y -- read the contents of a directory
│ │ │ │      * _m_a_k_e_D_i_r_e_c_t_o_r_y -- make a directory
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./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/M2-10576-0/0
│ │ │      		          
                  
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10465-0/0
    
│ │ │ +o2 = /tmp/M2-10576-0/0
│ │ │      		          
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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/M2-10576-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10465-0/0
│ │ │ │ +o2 = /tmp/M2-10576-0/0
│ │ │ │  ********** 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/M2-12843-0/0
│ │ │        		          
                    
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11592-0/0
      
│ │ │ +o2 = file:///tmp/M2-12843-0/0
│ │ │        		          
      
│ │ │        
      
│ │ │        
      
│ │ │          
      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/M2-12843-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-11592-0/0
│ │ │ │ +o2 = file:///tmp/M2-12843-0/0
│ │ │ │  ********** 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/M2-12609-0/0
│ │ │      		          
                  
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11478-0/0
│ │ │ +o6 = /tmp/M2-12609-0/0
│ │ │  
│ │ │  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/M2-12609-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11478-0/0
│ │ │ │ +o6 = /tmp/M2-12609-0/0
│ │ │ │  
│ │ │ │  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 0.000332467s (cpu); 0.000323809s (thread); 0s (gc)
│ │ │      		          
                  
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000168847s (cpu); 0.000168906s (thread); 0s (gc)
    
│ │ │ + -- used 0.000189007s (cpu); 0.000189401s (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 0.16989s (cpu); 0.169605s (thread); 0s (gc)
│ │ │      		          
                  
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.238738s (cpu); 0.23868s (thread); 0s (gc)
    
│ │ │ + -- used 0.150266s (cpu); 0.150277s (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 0.000332467s (cpu); 0.000323809s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000168847s (cpu); 0.000168906s (thread); 0s (gc)
│ │ │ │ + -- used 0.000189007s (cpu); 0.000189401s (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 0.16989s (cpu); 0.169605s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.238738s (cpu); 0.23868s (thread); 0s (gc)
│ │ │ │ + -- used 0.150266s (cpu); 0.150277s (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
│ │ │ @@ -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 {}
│ │ │ │ @@ -22,20 +22,20 @@
│ │ │ │      * _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/M2-12203-0/0/
│ │ │      		          
                  
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11272-0/1/
    
│ │ │ +o2 = /tmp/M2-12203-0/1/
│ │ │      		          
                  
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11272-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12203-0/0/a/
│ │ │      		          
                  
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11272-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12203-0/0/b/
│ │ │      		          
                  
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11272-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12203-0/0/b/c/
│ │ │      		          
                  
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11272-0/0/a/f
│ │ │ +o6 = /tmp/M2-12203-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │      		          
                  
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11272-0/0/a/g
│ │ │ +o7 = /tmp/M2-12203-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │      		          
                  
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11272-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12203-0/0/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/a/g -> /tmp/M2-12203-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12203-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12203-0/1/b/c/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/a/g -> /tmp/M2-12203-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12203-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12203-0/1/b/c/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/M2-12203-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11272-0/1/
│ │ │ │ +o2 = /tmp/M2-12203-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11272-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12203-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11272-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12203-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11272-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12203-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11272-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12203-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11272-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12203-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11272-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12203-0/0/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/a/g -> /tmp/M2-12203-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12203-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12203-0/1/b/c/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/a/g -> /tmp/M2-12203-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12203-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12203-0/1/b/c/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/M2-12336-0/0
│ │ │      		          
                  
    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/M2-12336-0/0
│ │ │ │  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/M2-14322-0/0.tex
│ │ │      		          
                  
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12321-0/1.html
    
│ │ │ +o2 = /tmp/M2-14322-0/1.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/M2-14322-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12321-0/1.html
│ │ │ │ +o2 = /tmp/M2-14322-0/1.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 2.3392e-05s (cpu); 8.943e-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 2.3392e-05s (cpu); 8.943e-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
│ │ │ +     -- .000014607 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │      		          
                  
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000014217, 205891132094649}
    
│ │ │ +o2 = Time{.000014607, 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
│ │ │ │ +     -- .000014607 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000014217, 205891132094649}
│ │ │ │ +o2 = Time{.000014607, 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" => 6.12.73+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" => 6.12.73+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
│ │ │ @@ -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 {}
│ │ │ │ @@ -2063,20 +2063,20 @@
│ │ │ │            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 3.04124s (cpu); 1.54357s (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 3.04124s (cpu); 1.54357s (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 3.04124s (cpu); 1.54357s (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 0.098596s (cpu); 0.0342409s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.010001s (cpu); 0.0137971s (thread); 0s (gc)
│ │ │ + -- used 0.014822s (cpu); 0.0165059s (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 0.000967019s (cpu); 0.000286834s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 5
│ │ │  
│ │ │  i16 : time regularity comodule I
│ │ │ - -- used 0.0148788s (cpu); 0.0151641s (thread); 0s (gc)
│ │ │ + -- used 0.0147461s (cpu); 0.0168093s (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 0.00403464s (cpu); 0.00454323s (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 0.00191332s (cpu); 0.00206766s (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 0.098596s (cpu); 0.0342409s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
    
│ │ │      		          
                  
    i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.010001s (cpu); 0.0137971s (thread); 0s (gc)
│ │ │ + -- used 0.014822s (cpu); 0.0165059s (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 0.098596s (cpu); 0.0342409s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.010001s (cpu); 0.0137971s (thread); 0s (gc)
│ │ │ │ + -- used 0.014822s (cpu); 0.0165059s (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 0.000967019s (cpu); 0.000286834s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 5
    
│ │ │      		          
                  
    i16 : time regularity comodule I
│ │ │ - -- used 0.0148788s (cpu); 0.0151641s (thread); 0s (gc)
│ │ │ + -- used 0.0147461s (cpu); 0.0168093s (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 0.000967019s (cpu); 0.000286834s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 5
│ │ │ │  i16 : time regularity comodule I
│ │ │ │ - -- used 0.0148788s (cpu); 0.0151641s (thread); 0s (gc)
│ │ │ │ + -- used 0.0147461s (cpu); 0.0168093s (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 0.00403464s (cpu); 0.00454323s (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 0.00191332s (cpu); 0.00206766s (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 0.00403464s (cpu); 0.00454323s (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 0.00191332s (cpu); 0.00206766s (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 0.165298s (cpu); 0.0769713s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.145769s (cpu); 0.0633631s (thread); 0s (gc)
│ │ │ + -- used 0.138252s (cpu); 0.0614281s (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 0.00043359s (cpu); 0.000226777s (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
│ │ │ + -- .118102s 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 0.256728s (cpu); 0.0824651s (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 1.81496s (cpu); 1.19966s (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 0.016001s (cpu); 0.0147268s (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 5.33865s (cpu); 2.80056s (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 0.00169562s (cpu); 0.000488947s (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
│ │ │ + -- .0426517s 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 0.165298s (cpu); 0.0769713s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │      		          
                  
    i11 : time fVector R10
│ │ │ - -- used 0.145769s (cpu); 0.0633631s (thread); 0s (gc)
│ │ │ + -- used 0.138252s (cpu); 0.0614281s (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 0.00043359s (cpu); 0.000226777s (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 0.165298s (cpu); 0.0769713s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.145769s (cpu); 0.0633631s (thread); 0s (gc)
│ │ │ │ + -- used 0.138252s (cpu); 0.0614281s (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 0.00043359s (cpu); 0.000226777s (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
    
│ │ │ + -- .118102s elapsed
│ │ │      		          
                  
    i4 : #L
│ │ │  
│ │ │  o4 = 64
    
│ │ │      		          
              
                  
    i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.228153s (cpu); 0.0609033s (thread); 0s (gc)
    
│ │ │ + -- used 0.256728s (cpu); 0.0824651s (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 0.256728s (cpu); 0.0824651s (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 1.81496s (cpu); 1.19966s (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 1.81496s (cpu); 1.19966s (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 0.016001s (cpu); 0.0147268s (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 5.33865s (cpu); 2.80056s (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 0.016001s (cpu); 0.0147268s (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 5.33865s (cpu); 2.80056s (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 0.00169562s (cpu); 0.000488947s (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 0.00169562s (cpu); 0.000488947s (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
│ │ │ + -- .0426517s 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
│ │ │ │ + -- .0426517s 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
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  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: Birational        (time .0398604)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .00398451)  #primes = 0 #prunedViaCodim = 0
│ │ │    Strategy: DecomposeMonomials -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : 0
│ │ │ - -- .0346098s elapsed
│ │ │ + --  Time taken : .000679851
│ │ │ + -- .0303344s 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
│ │ │ + -- .00054039s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0108096s elapsed
│ │ │ + -- .0150629s 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
│ │ │ + -- .0868819s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00150213s elapsed
│ │ │ + -- .00176033s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00113311s elapsed
│ │ │ + -- .00162421s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -59,19 +59,19 @@
│ │ │                
    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: Birational        (time .0398604)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .00398451)  #primes = 0 #prunedViaCodim = 0
│ │ │    Strategy: DecomposeMonomials -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : 0
│ │ │ - -- .0346098s elapsed
│ │ │ + --  Time taken : .000679851
│ │ │ + -- .0303344s elapsed
│ │ │  (time 0)         #primes = 1 #prunedViaCodim = 0
    
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -12,19 +12,19 @@ │ │ │ │ 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: Birational (time .0398604) #primes = 0 #prunedViaCodim = 0 │ │ │ │ + Strategy: Factorization (time .00398451) #primes = 0 #prunedViaCodim = 0 │ │ │ │ Strategy: DecomposeMonomials -- Converting annotated ideals to ideals and │ │ │ │ selecting minimal primes... │ │ │ │ - -- Time taken : 0 │ │ │ │ - -- .0346098s elapsed │ │ │ │ + -- Time taken : .000679851 │ │ │ │ + -- .0303344s 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
│ │ │ + -- .00054039s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
      │ │ │ │ │ │ │ │ │ 
      i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0108096s elapsed
│ │ │ + -- .0150629s 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 │ │ │ │ + -- .00054039s elapsed │ │ │ │ │ │ │ │ o6 = ideal (a, b, c) │ │ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ i7 : elapsedTime radical(ideal I_*, Unmixed => true) │ │ │ │ - -- .0108096s elapsed │ │ │ │ + -- .0150629s 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
│ │ │ + -- .0868819s elapsed
│ │ │  
│ │ │  o7 = true
      │ │ │ │ │ │ │ │ │ 
      i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00150213s elapsed
│ │ │ + -- .00176033s elapsed
│ │ │  
│ │ │  o8 = true
      │ │ │ │ │ │ │ │ │ 
      i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00113311s elapsed
│ │ │ + -- .00162421s 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 │ │ │ │ + -- .0868819s elapsed │ │ │ │ │ │ │ │ o7 = true │ │ │ │ i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar") │ │ │ │ - -- .00150213s elapsed │ │ │ │ + -- .00176033s elapsed │ │ │ │ │ │ │ │ o8 = true │ │ │ │ i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar") │ │ │ │ - -- .00113311s elapsed │ │ │ │ + -- .00162421s 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 0.197496s (cpu); 0.0876222s (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 0.023584s (cpu); 0.0127982s (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 0.197496s (cpu); 0.0876222s (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 0.023584s (cpu); 0.0127982s (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 0.197496s (cpu); 0.0876222s (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 0.023584s (cpu); 0.0127982s (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 0.463557s (cpu); 0.336743s (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 0.661254s (cpu); 0.545474s (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 0.463557s (cpu); 0.336743s (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 0.661254s (cpu); 0.545474s (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 0.463557s (cpu); 0.336743s (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 0.661254s (cpu); 0.545474s (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
│ │ │ + -- .00358926s elapsed
│ │ │ + -- .00352023s elapsed
│ │ │ + -- .000396871s elapsed
│ │ │ + -- .00330888s elapsed
│ │ │ + -- .0353415s elapsed
│ │ │ + -- .000377405s elapsed
│ │ │ + -- .00345157s elapsed
│ │ │ + -- .00335566s elapsed
│ │ │ + -- .000326485s elapsed
│ │ │ + -- .00319682s elapsed
│ │ │ + -- .00333625s elapsed
│ │ │ + -- .000320728s elapsed
│ │ │ +--backup directory created: /tmp/M2-152176-0/1
│ │ │    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
│ │ │ + -- .00358926s elapsed
│ │ │ + -- .00352023s elapsed
│ │ │ + -- .000396871s elapsed
│ │ │ + -- .00330888s elapsed
│ │ │ + -- .0353415s elapsed
│ │ │ + -- .000377405s elapsed
│ │ │ + -- .00345157s elapsed
│ │ │ + -- .00335566s elapsed
│ │ │ + -- .000326485s elapsed
│ │ │ + -- .00319682s elapsed
│ │ │ + -- .00333625s elapsed
│ │ │ + -- .000320728s elapsed
│ │ │ +--backup directory created: /tmp/M2-152176-0/1
│ │ │    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
│ │ │ │ + -- .00358926s elapsed
│ │ │ │ + -- .00352023s elapsed
│ │ │ │ + -- .000396871s elapsed
│ │ │ │ + -- .00330888s elapsed
│ │ │ │ + -- .0353415s elapsed
│ │ │ │ + -- .000377405s elapsed
│ │ │ │ + -- .00345157s elapsed
│ │ │ │ + -- .00335566s elapsed
│ │ │ │ + -- .000326485s elapsed
│ │ │ │ + -- .00319682s elapsed
│ │ │ │ + -- .00333625s elapsed
│ │ │ │ + -- .000320728s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-152176-0/1
│ │ │ │    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/M2-264701-0/0-in.ms -o /tmp/M2-264701-0/0-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 0.4% │ │ │ +update 0.00 sec 77.1% │ │ │ +convert 0.00 sec 2.2% │ │ │ +linear algebra 0.00 sec 1.6% │ │ │ reduce gb 0.00 sec 0.0% │ │ │ ----------------------------------------- │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ size of basis 3 │ │ │ #terms in basis 3 │ │ │ #pairs reduced 0 │ │ │ @@ -89,30 +89,30 @@ │ │ │ ----------------------------------------- │ │ │ │ │ │ multi-modular steps │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ {1}{2}<100.00%> │ │ │ │ │ │ ------------------------------------------------------------------------------------ │ │ │ -msolve overall time 0.31 sec (elapsed) / 0.91 sec (cpu) │ │ │ ------------------------------------------------------------------------------------- │ │ │ │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ Max coeff. bitsize 1 │ │ │ #primes 3 │ │ │ #bad primes 0 │ │ │ ----------------------------------------- │ │ │ │ │ │ ---------------- TIMINGS ---------------- │ │ │ CRT (elapsed) 0.00 sec │ │ │ ratrecon(elapsed) 0.00 sec │ │ │ ----------------------------------------- │ │ │ +msolve overall time 0.03 sec (elapsed) / 0.07 sec (cpu) │ │ │ +------------------------------------------------------------------------------------ │ │ │ │ │ │ o3 = | z y x | │ │ │ │ │ │ 1 3 │ │ │ o3 : Matrix R <-- R │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/_msolve__Real__Solutions.out │ │ │ @@ -11,61 +11,61 @@ │ │ │ 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 9603838835 4801919417 │ │ │ +o3 = {{{----------, ----------}, {- ----------, - ----------}}, {{- │ │ │ + 4294967296 4294967296 4294967296 2147483648 │ │ │ + ------------------------------------------------------------------------ │ │ │ + 1 1 9603838835 4801919417 4294967295 │ │ │ + ----------, ----------}, {- ----------, - ----------}}, {{----------, │ │ │ + 4294967296 4294967296 4294967296 2147483648 4294967296 │ │ │ + ------------------------------------------------------------------------ │ │ │ + 4294967297 4801919417 9603838835 1 1 │ │ │ + ----------}, {----------, ----------}}, {{- ----------, ----------}, │ │ │ + 4294967296 2147483648 4294967296 4294967296 4294967296 │ │ │ + ------------------------------------------------------------------------ │ │ │ + 4801919417 9603838835 │ │ │ + {----------, ----------}}} │ │ │ + 2147483648 4294967296 │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : rationalApproxSols = msolveRealSolutions(I, QQ) │ │ │ │ │ │ - 19207677669 19207677669 19207677669 │ │ │ -o4 = {{0, -----------}, {1, -----------}, {0, - -----------}, {1, - │ │ │ - 8589934592 8589934592 8589934592 │ │ │ + 19207677669 19207677669 19207677669 │ │ │ +o4 = {{1, - -----------}, {0, - -----------}, {1, -----------}, {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]}, {[-2.32831e-10,2.32831e-10], │ │ │ ------------------------------------------------------------------------ │ │ │ - [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], [-2.23633,-2.23535]}, {[-.000976562,.000976562], │ │ │ ------------------------------------------------------------------------ │ │ │ - [-2.23633,-2.23535]}, {[-.000976562,.000976562], [2.23535,2.23633]}, │ │ │ + [-2.23633,-2.23535]}, {[.999023,1.00098], [2.23535,2.23633]}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {[.999023,1.00098], [2.23535,2.23633]}} │ │ │ + {[-.000976562,.000976562], [2.23535,2.23633]}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : floatApproxSols = msolveRealSolutions(I, RR) │ │ │ │ │ │ o7 = {{1, -2.23607}, {0, -2.23607}, {1, 2.23607}, {0, 2.23607}} │ │ │ │ │ │ @@ -82,16 +82,16 @@ │ │ │ 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]}, {[-2.32831e-10,2.32831e-10], │ │ │ ----------------------------------------------------------------------- │ │ │ - [-2.23607,-2.23607]}, {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}, │ │ │ + [-2.23607,-2.23607]}, {[1,1], [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 │ │ │ @@ -99,64 +99,64 @@ │ │ │ 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      9603838835    4801919417       
│ │ │ +o3 = {{{----------, ----------}, {- ----------, - ----------}}, {{-
│ │ │ +        4294967296  4294967296      4294967296    2147483648       
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +          1           1          9603838835    4801919417      4294967295 
│ │ │ +     ----------, ----------}, {- ----------, - ----------}}, {{----------,
│ │ │ +     4294967296  4294967296      4294967296    2147483648      4294967296 
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     4294967297    4801919417  9603838835             1           1      
│ │ │ +     ----------}, {----------, ----------}}, {{- ----------, ----------},
│ │ │ +     4294967296    2147483648  4294967296        4294967296  4294967296  
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +      4801919417  9603838835
│ │ │ +     {----------, ----------}}}
│ │ │ +      2147483648  4294967296
│ │ │  
│ │ │  o3 : List
      │ │ │ │ │ │ │ │ │ 
      i4 : rationalApproxSols = msolveRealSolutions(I, QQ)
│ │ │  
│ │ │ -          19207677669       19207677669         19207677669        
│ │ │ -o4 = {{0, -----------}, {1, -----------}, {0, - -----------}, {1, -
│ │ │ -           8589934592        8589934592          8589934592        
│ │ │ +            19207677669         19207677669       19207677669      
│ │ │ +o4 = {{1, - -----------}, {0, - -----------}, {1, -----------}, {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]}, {[-2.32831e-10,2.32831e-10],
│ │ │       ------------------------------------------------------------------------
│ │ │ -     [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], [-2.23633,-2.23535]}, {[-.000976562,.000976562],
│ │ │       ------------------------------------------------------------------------
│ │ │ -     [-2.23633,-2.23535]}, {[-.000976562,.000976562], [2.23535,2.23633]},
│ │ │ +     [-2.23633,-2.23535]}, {[.999023,1.00098], [2.23535,2.23633]},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {[.999023,1.00098], [2.23535,2.23633]}}
│ │ │ +     {[-.000976562,.000976562], [2.23535,2.23633]}}
│ │ │  
│ │ │  o6 : List
      │ │ │ │ │ │ │ │ │ 
      i7 : floatApproxSols = msolveRealSolutions(I, RR)
│ │ │  
│ │ │  o7 = {{1, -2.23607}, {0, -2.23607}, {1, 2.23607}, {0, 2.23607}}
│ │ │ @@ -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]}, {[-2.32831e-10,2.32831e-10],
│ │ │        -----------------------------------------------------------------------
│ │ │ -      [-2.23607,-2.23607]}, {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]},
│ │ │ +      [-2.23607,-2.23607]}, {[1,1], [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 {} │ │ │ │ @@ -44,58 +44,58 @@ │ │ │ │ │ │ │ │ 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 9603838835 4801919417 │ │ │ │ +o3 = {{{----------, ----------}, {- ----------, - ----------}}, {{- │ │ │ │ + 4294967296 4294967296 4294967296 2147483648 │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + 1 1 9603838835 4801919417 4294967295 │ │ │ │ + ----------, ----------}, {- ----------, - ----------}}, {{----------, │ │ │ │ + 4294967296 4294967296 4294967296 2147483648 4294967296 │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + 4294967297 4801919417 9603838835 1 1 │ │ │ │ + ----------}, {----------, ----------}}, {{- ----------, ----------}, │ │ │ │ + 4294967296 2147483648 4294967296 4294967296 4294967296 │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + 4801919417 9603838835 │ │ │ │ + {----------, ----------}}} │ │ │ │ + 2147483648 4294967296 │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : rationalApproxSols = msolveRealSolutions(I, QQ) │ │ │ │ │ │ │ │ - 19207677669 19207677669 19207677669 │ │ │ │ -o4 = {{0, -----------}, {1, -----------}, {0, - -----------}, {1, - │ │ │ │ - 8589934592 8589934592 8589934592 │ │ │ │ + 19207677669 19207677669 19207677669 │ │ │ │ +o4 = {{1, - -----------}, {0, - -----------}, {1, -----------}, {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]}, {[-2.32831e-10,2.32831e-10], │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - [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], [-2.23633,-2.23535]}, {[-.000976562,.000976562], │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - [-2.23633,-2.23535]}, {[-.000976562,.000976562], [2.23535,2.23633]}, │ │ │ │ + [-2.23633,-2.23535]}, {[.999023,1.00098], [2.23535,2.23633]}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {[.999023,1.00098], [2.23535,2.23633]}} │ │ │ │ + {[-.000976562,.000976562], [2.23535,2.23633]}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : floatApproxSols = msolveRealSolutions(I, RR) │ │ │ │ │ │ │ │ o7 = {{1, -2.23607}, {0, -2.23607}, {1, 2.23607}, {0, 2.23607}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ @@ -111,19 +111,19 @@ │ │ │ │ │ │ │ │ 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]}, {[-2.32831e-10,2.32831e-10], │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - [-2.23607,-2.23607]}, {[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}, │ │ │ │ + [-2.23607,-2.23607]}, {[1,1], [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/M2-264701-0/0-in.ms -o /tmp/M2-264701-0/0-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   0.4%
│ │ │ +update                  0.00 sec  77.1%
│ │ │ +convert                 0.00 sec   2.2%
│ │ │ +linear algebra          0.00 sec   1.6%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -146,30 +146,30 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%>
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.31 sec (elapsed) /  0.91 sec (cpu)
│ │ │ -------------------------------------------------------------------------------------
│ │ │   
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  Max coeff. bitsize                1
│ │ │  #primes                           3
│ │ │  #bad primes                       0
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │ +msolve overall time           0.03 sec (elapsed) /  0.07 sec (cpu)
│ │ │ +------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
      │ │ │ │ │ │ │ │ │ ├── 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/M2-264701-0/0-in.ms -o / │ │ │ │ +tmp/M2-264701-0/0-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 0.4% │ │ │ │ +update 0.00 sec 77.1% │ │ │ │ +convert 0.00 sec 2.2% │ │ │ │ +linear algebra 0.00 sec 1.6% │ │ │ │ reduce gb 0.00 sec 0.0% │ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ │ size of basis 3 │ │ │ │ #terms in basis 3 │ │ │ │ #pairs reduced 0 │ │ │ │ @@ -119,17 +119,14 @@ │ │ │ │ multi-modular steps │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----------------------- │ │ │ │ {1}{2}<100.00%> │ │ │ │ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----- │ │ │ │ -msolve overall time 0.31 sec (elapsed) / 0.91 sec (cpu) │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ ------ │ │ │ │ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----------------------- │ │ │ │ │ │ │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ │ Max coeff. bitsize 1 │ │ │ │ @@ -137,14 +134,17 @@ │ │ │ │ #bad primes 0 │ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ ---------------- TIMINGS ---------------- │ │ │ │ CRT (elapsed) 0.00 sec │ │ │ │ ratrecon(elapsed) 0.00 sec │ │ │ │ ----------------------------------------- │ │ │ │ +msolve overall time 0.03 sec (elapsed) / 0.07 sec (cpu) │ │ │ │ +------------------------------------------------------------------------------- │ │ │ │ +----- │ │ │ │ │ │ │ │ o3 = | z y x | │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o3 : Matrix R <-- R │ │ │ │ ********** RReeffeerreenncceess ********** │ │ │ │ [1] The msolve library: _h_t_t_p_s_:_/_/_m_s_o_l_v_e_._l_i_p_6_._f_r; │ │ ├── ./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 │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkZWdyZWVPZk1hcCxJZGVhbCksImRlZ3JlZU9mTWFw │ │ ├── ./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 │ │ │ ZSBkaWZmZXJlbnRpYWwiLCAibGluZW51bSIgPT4gNDYzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkQi │ │ ├── ./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 │ │ │ + -- .0288176s elapsed │ │ │ │ │ │ o3 = 2 │ │ │ │ │ │ i4 : elapsedTime monjMult I │ │ │ - -- .24679s elapsed │ │ │ + -- .186169s elapsed │ │ │ │ │ │ o4 = 2 │ │ │ │ │ │ i5 : elapsedTime multiplicitySequence I │ │ │ - -- .155598s elapsed │ │ │ + -- .16433s 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 │ │ │ + -- .132461s 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 │ │ │ + -- .203252s elapsed │ │ │ │ │ │ o3 = 192 │ │ │ │ │ │ i4 : elapsedTime jMult I │ │ │ - -- 1.04785s elapsed │ │ │ + -- 1.10258s 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
│ │ │ + -- .0288176s elapsed
│ │ │  
│ │ │  o3 = 2
      │ │ │ │ │ │ │ │ │ 
      i4 : elapsedTime monjMult I
│ │ │ - -- .24679s elapsed
│ │ │ + -- .186169s elapsed
│ │ │  
│ │ │  o4 = 2
      │ │ │ │ │ │ │ │ │ 
      i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .155598s elapsed
│ │ │ + -- .16433s 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 │ │ │ │ + -- .0288176s elapsed │ │ │ │ │ │ │ │ o3 = 2 │ │ │ │ i4 : elapsedTime monjMult I │ │ │ │ - -- .24679s elapsed │ │ │ │ + -- .186169s elapsed │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : elapsedTime multiplicitySequence I │ │ │ │ - -- .155598s elapsed │ │ │ │ + -- .16433s 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
│ │ │ + -- .132461s 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 │ │ │ │ + -- .132461s 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
│ │ │ + -- .203252s elapsed
│ │ │  
│ │ │  o3 = 192
      │ │ │ │ │ │ │ │ │ 
      i4 : elapsedTime jMult I
│ │ │ - -- 1.04785s elapsed
│ │ │ + -- 1.10258s 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 │ │ │ │ + -- .203252s elapsed │ │ │ │ │ │ │ │ o3 = 192 │ │ │ │ i4 : elapsedTime jMult I │ │ │ │ - -- 1.04785s elapsed │ │ │ │ + -- 1.10258s 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 3.39802s (cpu); 1.91804s (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 4.01058s (cpu); 2.08527s (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 6.69987s (cpu); 4.70276s (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 3.96052s (cpu); 3.36606s (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 0.17974s (cpu); 0.12472s (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 2.07741s (cpu); 1.37801s (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 4.37218s (cpu); 2.63371s (thread); 0s (gc) │ │ │ │ │ │ o4 = 2 │ │ │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ - -- used 0.312606s (cpu); 0.236089s (thread); 0s (gc) │ │ │ + -- used 0.380604s (cpu); 0.309147s (thread); 0s (gc) │ │ │ │ │ │ o5 = 2 │ │ │ │ │ │ i6 : time degree Phi │ │ │ - -- used 0.326294s (cpu); 0.24353s (thread); 0s (gc) │ │ │ + -- used 0.236808s (cpu); 0.240456s (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 0.547122s (cpu); 0.394752s (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 0.00303895s (cpu); 0.000166371s (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 0.00289442s (cpu); 0.000242602s (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 1.24358s (cpu); 1.03589s (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 0.000127446s (cpu); 0.000440516s (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 0.0466847s (cpu); 0.0275186s (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 0.0515877s (cpu); 0.0399714s (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 0.288531s (cpu); 0.161622s (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 0.14103s (cpu); 0.128861s (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 9.58541s (cpu); 2.67502s (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 0.343194s (cpu); 0.326242s (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 1.28553s (cpu); 1.07185s (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 0.0969241s (cpu); 0.0593903s (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 0.244495s (cpu); 0.10515s (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 0.525217s (cpu); 0.327461s (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 0.00170041s (cpu); 1.0957e-05s (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 0.488569s (cpu); 0.190837s (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 1.61573s (cpu); 0.808866s (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 0.274375s (cpu); 0.200762s (thread); 0s (gc) │ │ │ │ │ │ o3 = false │ │ │ │ │ │ i4 : time Psi = first graph Phi; │ │ │ - -- used 0.158078s (cpu); 0.0838548s (thread); 0s (gc) │ │ │ + -- used 0.181944s (cpu); 0.102368s (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 3.9016s (cpu); 3.33247s (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 0.0119348s (cpu); 0.0103492s (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 0.568381s (cpu); 0.493607s (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 0.597206s (cpu); 0.388841s (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 0.00197666s (cpu); 0.00165213s (thread); 0s (gc) │ │ │ + -- used 0.214606s (cpu); 0.146251s (thread); 0s (gc) │ │ │ + -- used 0.244017s (cpu); 0.168326s (thread); 0s (gc) │ │ │ + -- used 0.213684s (cpu); 0.132164s (thread); 0s (gc) │ │ │ + -- used 0.181247s (cpu); 0.114521s (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 0.129091s (cpu); 0.0685029s (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 0.0357822s (cpu); 0.0248371s (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 1.54303s (cpu); 1.0301s (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 1.45135s (cpu); 0.706416s (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 0.218812s (cpu); 0.146875s (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 3.39802s (cpu); 1.91804s (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 4.01058s (cpu); 2.08527s (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 6.69987s (cpu); 4.70276s (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 3.39802s (cpu); 1.91804s (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 4.01058s (cpu); 2.08527s (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 6.69987s (cpu); 4.70276s (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 3.96052s (cpu); 3.36606s (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 3.96052s (cpu); 3.36606s (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 0.17974s (cpu); 0.12472s (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 2.07741s (cpu); 1.37801s (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 0.17974s (cpu); 0.12472s (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 2.07741s (cpu); 1.37801s (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 4.37218s (cpu); 2.63371s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
      
│ │ │            
│ │ │            
│ │ │                
      i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.312606s (cpu); 0.236089s (thread); 0s (gc)
│ │ │ + -- used 0.380604s (cpu); 0.309147s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
      
│ │ │            
│ │ │            
│ │ │                
      i6 : time degree Phi
│ │ │ - -- used 0.326294s (cpu); 0.24353s (thread); 0s (gc)
│ │ │ + -- used 0.236808s (cpu); 0.240456s (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 4.37218s (cpu); 2.63371s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ │ - -- used 0.312606s (cpu); 0.236089s (thread); 0s (gc) │ │ │ │ + -- used 0.380604s (cpu); 0.309147s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 2 │ │ │ │ i6 : time degree Phi │ │ │ │ - -- used 0.326294s (cpu); 0.24353s (thread); 0s (gc) │ │ │ │ + -- used 0.236808s (cpu); 0.240456s (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 0.547122s (cpu); 0.394752s (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 0.547122s (cpu); 0.394752s (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 0.00303895s (cpu); 0.000166371s (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 0.00289442s (cpu); 0.000242602s (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 1.24358s (cpu); 1.03589s (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 0.000127446s (cpu); 0.000440516s (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 0.00303895s (cpu); 0.000166371s (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 0.00289442s (cpu); 0.000242602s (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 1.24358s (cpu); 1.03589s (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 0.000127446s (cpu); 0.000440516s (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 0.0466847s (cpu); 0.0275186s (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 0.0515877s (cpu); 0.0399714s (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 0.288531s (cpu); 0.161622s (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 0.0466847s (cpu); 0.0275186s (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 0.0515877s (cpu); 0.0399714s (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 0.288531s (cpu); 0.161622s (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 0.14103s (cpu); 0.128861s (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 9.58541s (cpu); 2.67502s (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 0.14103s (cpu); 0.128861s (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 9.58541s (cpu); 2.67502s (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 0.343194s (cpu); 0.326242s (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 1.28553s (cpu); 1.07185s (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 0.343194s (cpu); 0.326242s (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 1.28553s (cpu); 1.07185s (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 0.0969241s (cpu); 0.0593903s (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 0.244495s (cpu); 0.10515s (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 0.525217s (cpu); 0.327461s (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 0.0969241s (cpu); 0.0593903s (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 0.244495s (cpu); 0.10515s (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 0.525217s (cpu); 0.327461s (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 0.00170041s (cpu); 1.0957e-05s (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 0.488569s (cpu); 0.190837s (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 1.61573s (cpu); 0.808866s (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 0.00170041s (cpu); 1.0957e-05s (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 0.488569s (cpu); 0.190837s (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 1.61573s (cpu); 0.808866s (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 0.274375s (cpu); 0.200762s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
      
│ │ │            
│ │ │            
│ │ │                
      i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.158078s (cpu); 0.0838548s (thread); 0s (gc)
│ │ │ + -- used 0.181944s (cpu); 0.102368s (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 3.9016s (cpu); 3.33247s (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 0.274375s (cpu); 0.200762s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.158078s (cpu); 0.0838548s (thread); 0s (gc)
│ │ │ │ + -- used 0.181944s (cpu); 0.102368s (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 3.9016s (cpu); 3.33247s (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 0.0119348s (cpu); 0.0103492s (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 0.568381s (cpu); 0.493607s (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 0.0119348s (cpu); 0.0103492s (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 0.568381s (cpu); 0.493607s (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 0.597206s (cpu); 0.388841s (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 0.597206s (cpu); 0.388841s (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 0.00197666s (cpu); 0.00165213s (thread); 0s (gc)
│ │ │ + -- used 0.214606s (cpu); 0.146251s (thread); 0s (gc)
│ │ │ + -- used 0.244017s (cpu); 0.168326s (thread); 0s (gc)
│ │ │ + -- used 0.213684s (cpu); 0.132164s (thread); 0s (gc)
│ │ │ + -- used 0.181247s (cpu); 0.114521s (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 0.129091s (cpu); 0.0685029s (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 0.00197666s (cpu); 0.00165213s (thread); 0s (gc) │ │ │ │ + -- used 0.214606s (cpu); 0.146251s (thread); 0s (gc) │ │ │ │ + -- used 0.244017s (cpu); 0.168326s (thread); 0s (gc) │ │ │ │ + -- used 0.213684s (cpu); 0.132164s (thread); 0s (gc) │ │ │ │ + -- used 0.181247s (cpu); 0.114521s (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 0.129091s (cpu); 0.0685029s (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 0.0357822s (cpu); 0.0248371s (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 1.54303s (cpu); 1.0301s (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 0.0357822s (cpu); 0.0248371s (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 1.54303s (cpu); 1.0301s (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 1.45135s (cpu); 0.706416s (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 1.45135s (cpu); 0.706416s (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 0.218812s (cpu); 0.146875s (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 0.218812s (cpu); 0.146875s (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, 80, 88, 90, 93, 93, 94, 93, 94, 97, 97, 97, 97, 99, 97, 98, 97, 96,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     94, 99, 97, 99, 99, 98, 98, 99, 98, 99, 99, 98)
│ │ │ +     97, 98, 97, 100, 96, 97, 99, 99, 100, 99, 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 = (16, 16, 11, 3, 3, 2, 2, 1, 1, 2, 1, 0, 1, 2, 2, 0, 2, 0, 0, 1, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 1, 1, 1, 0)
│ │ │ +      0, 0, 0, 0, 0, 0, 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 = {DFW, D[c, DdS, DE{, D`s}
│ │ │  
│ │ │  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
│ │ │ @@ -1,18 +1,18 @@
│ │ │  -- -*- 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, b}, {a, d}, {c, d}, {b, e}, {c, 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}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │       Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ ├── ./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 0.000648575s (cpu); 0.000524126s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0467584s (cpu); 0.0451558s (thread); 0s (gc)
│ │ │ + -- used 0.0624064s (cpu); 0.0599056s (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, 80, 88, 90, 93, 93, 94, 93, 94, 97, 97, 97, 97, 99, 97, 98, 97, 96,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     94, 99, 97, 99, 99, 98, 98, 99, 98, 99, 99, 98)
│ │ │ +     97, 98, 97, 100, 96, 97, 99, 99, 100, 99, 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 = (16, 16, 11, 3, 3, 2, 2, 1, 1, 2, 1, 0, 1, 2, 2, 0, 2, 0, 0, 1, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 1, 1, 1, 0)
│ │ │ +      0, 0, 0, 0, 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 = (69, 80, 81, 89, 94, 95, 96, 94, 100, 97, 96, 97, 96, 98, 99, 97, 99, │ │ │ │ +o9 = (70, 80, 88, 90, 93, 93, 94, 93, 94, 97, 97, 97, 97, 99, 97, 98, 97, 96, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 94, 99, 97, 99, 99, 98, 98, 99, 98, 99, 99, 98) │ │ │ │ + 97, 98, 97, 100, 96, 97, 99, 99, 100, 99, 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 = (16, 16, 11, 3, 3, 2, 2, 1, 1, 2, 1, 0, 1, 2, 2, 0, 2, 0, 0, 1, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 0, 0, 1, 1, 1, 0) │ │ │ │ + 0, 0, 0, 0, 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/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 = {DFW, D[c, DdS, DE{, D`s}
│ │ │  
│ │ │  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 = {DFW, D[c, DdS, DE{, D`s}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -89,19 +89,19 @@
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │          
                    
      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, b}, {a, d}, {c, d}, {b, e}, {c, 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}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │       Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,19 +25,19 @@
│ │ │ │  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, b}, {a, d}, {c, d}, {b, e}, {c, 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}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}}}
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ ├── ./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 0.000648575s (cpu); 0.000524126s (thread); 0s (gc)
│ │ │        		          
                    
      i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0467584s (cpu); 0.0451558s (thread); 0s (gc)
      
│ │ │ + -- used 0.0624064s (cpu); 0.0599056s (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 0.000648575s (cpu); 0.000524126s (thread); 0s (gc)
│ │ │ │  i6 : time (graphComplement \ G);
│ │ │ │ - -- used 0.0467584s (cpu); 0.0451558s (thread); 0s (gc)
│ │ │ │ + -- used 0.0624064s (cpu); 0.0599056s (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 = (66, 83, 93, 91, 92, 89, 99, 98, 95, 96, 95, 98, 99, 98, 98, 98, 97, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 98, 97, 99, 100, 99, 98, 100, 99, 97, 97, 99)
│ │ │ +     98, 97, 96, 98, 97, 98, 99, 98, 99, 99, 98)
│ │ │  
│ │ │  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 = (14, 10, 11, 6, 1, 3, 2, 0, 1, 2, 0, 3, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 0, 0, 0, 1)
│ │ │ +      0, 1, 2, 0, 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 = {Dt?, DtS, DUO, DVO, 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 = {D[S, DdW, DYc}
│ │ │  
│ │ │  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 0.000590194s (cpu); 0.000513549s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : time (graphComplement \ G);
│ │ │ - -- used 0.132423s (cpu); 0.0571132s (thread); 0s (gc)
│ │ │ + -- used 0.152001s (cpu); 0.0690716s (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 = (66, 83, 93, 91, 92, 89, 99, 98, 95, 96, 95, 98, 99, 98, 98, 98, 97, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 98, 97, 99, 100, 99, 98, 100, 99, 97, 97, 99)
│ │ │ +     98, 97, 96, 98, 97, 98, 99, 98, 99, 99, 98)
│ │ │  
│ │ │  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 = (14, 10, 11, 6, 1, 3, 2, 0, 1, 2, 0, 3, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 0, 0, 0, 1)
│ │ │ +      0, 1, 2, 0, 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 = (66, 83, 93, 91, 92, 89, 99, 98, 95, 96, 95, 98, 99, 98, 98, 98, 97, 97, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 97, 98, 97, 99, 100, 99, 98, 100, 99, 97, 97, 99) │ │ │ │ + 98, 97, 96, 98, 97, 98, 99, 98, 99, 99, 98) │ │ │ │ │ │ │ │ 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 = (14, 10, 11, 6, 1, 3, 2, 0, 1, 2, 0, 3, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 0, 0, 0, 0, 0, 1) │ │ │ │ + 0, 1, 2, 0, 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 = {Dt?, DtS, DUO, DVO, 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 = {Dt?, DtS, DUO, DVO, 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 = {D[S, DdW, DYc}
│ │ │  
│ │ │  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 = {D[S, DdW, DYc} │ │ │ │ │ │ │ │ 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 0.000590194s (cpu); 0.000513549s (thread); 0s (gc) │ │ │ 
      i5 : time (graphComplement \ G);
│ │ │ - -- used 0.132423s (cpu); 0.0571132s (thread); 0s (gc)
      │ │ │ + -- used 0.152001s (cpu); 0.0690716s (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 0.000590194s (cpu); 0.000513549s (thread); 0s (gc) │ │ │ │ i5 : time (graphComplement \ G); │ │ │ │ - -- used 0.132423s (cpu); 0.0571132s (thread); 0s (gc) │ │ │ │ + -- used 0.152001s (cpu); 0.0690716s (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 │ │ │ + -- .0671594s 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
│ │ │ + -- .0671594s 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 │ │ │ │ + -- .0671594s 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 0.251852s (cpu); 0.254268s (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 0.00390117s (cpu); 0.0014076s (thread); 0s (gc) │ │ │ + -- used 3.1333e-05s (cpu); 0.000107057s (thread); 0s (gc) │ │ │ + -- used 1.0367e-05s (cpu); 7.748e-05s (thread); 0s (gc) │ │ │ + -- used 9.272e-06s (cpu); 7.8412e-05s (thread); 0s (gc) │ │ │ + -- used 9.212e-06s (cpu); 7.1396e-05s (thread); 0s (gc) │ │ │ + -- used 1.3926e-05s (cpu); 9.9014e-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 = {1, 2, 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, 4, 5, 6, 8} │ │ │ │ │ │ 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 │ │ │ + -- .0319344s 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 │ │ │ + -- .00135472s 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 │ │ │ + -- .0973978s 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 │ │ │ + -- .00166105s 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 │ │ │ + -- 28.3694s elapsed │ │ │ │ │ │ o11 = 7909 │ │ │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3)) │ │ │ - -- .0257955s elapsed │ │ │ + -- .0325329s 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 0.000546371s (cpu); 2.8673e-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 0.238262s (cpu); 0.0864925s (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 0.122286s (cpu); 0.0440022s (thread); 0s (gc) │ │ │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ - -- used 0.0044979s (cpu); 0.00448742s (thread); 0s (gc) │ │ │ + -- used 0.000554321s (cpu); 0.00285448s (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 0.251852s (cpu); 0.254268s (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 0.00390117s (cpu); 0.0014076s (thread); 0s (gc)
│ │ │ + -- used 3.1333e-05s (cpu); 0.000107057s (thread); 0s (gc)
│ │ │ + -- used 1.0367e-05s (cpu); 7.748e-05s (thread); 0s (gc)
│ │ │ + -- used 9.272e-06s (cpu); 7.8412e-05s (thread); 0s (gc)
│ │ │ + -- used 9.212e-06s (cpu); 7.1396e-05s (thread); 0s (gc)
│ │ │ + -- used 1.3926e-05s (cpu); 9.9014e-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 0.251852s (cpu); 0.254268s (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 0.00390117s (cpu); 0.0014076s (thread); 0s (gc) │ │ │ │ + -- used 3.1333e-05s (cpu); 0.000107057s (thread); 0s (gc) │ │ │ │ + -- used 1.0367e-05s (cpu); 7.748e-05s (thread); 0s (gc) │ │ │ │ + -- used 9.272e-06s (cpu); 7.8412e-05s (thread); 0s (gc) │ │ │ │ + -- used 9.212e-06s (cpu); 7.1396e-05s (thread); 0s (gc) │ │ │ │ + -- used 1.3926e-05s (cpu); 9.9014e-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 = {1, 2, 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, 4, 5, 6, 8}
│ │ │  
│ │ │  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 = {1, 2, 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, 4, 5, 6, 8} │ │ │ │ │ │ │ │ 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
│ │ │ + -- .0319344s 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
    │ │ │ + -- .00135472s 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
│ │ │ + -- .0973978s 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
    │ │ │ + -- .00166105s 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
│ │ │ + -- 28.3694s elapsed
│ │ │  
│ │ │  o11 = 7909
    │ │ │ 
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0257955s elapsed
    │ │ │ + -- .0325329s 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 │ │ │ │ + -- .0319344s 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 │ │ │ │ + -- .00135472s 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 │ │ │ │ + -- .0973978s 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 │ │ │ │ + -- .00166105s 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 │ │ │ │ + -- 28.3694s elapsed │ │ │ │ │ │ │ │ o11 = 7909 │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety │ │ │ │ D3)) │ │ │ │ - -- .0257955s elapsed │ │ │ │ + -- .0325329s 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 0.000546371s (cpu); 2.8673e-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 0.238262s (cpu); 0.0864925s (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 0.000546371s (cpu); 2.8673e-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 0.238262s (cpu); 0.0864925s (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 0.122286s (cpu); 0.0440022s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.0044979s (cpu); 0.00448742s (thread); 0s (gc)
    │ │ │ + -- used 0.000554321s (cpu); 0.00285448s (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 0.122286s (cpu); 0.0440022s (thread); 0s (gc) │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ │ - -- used 0.0044979s (cpu); 0.00448742s (thread); 0s (gc) │ │ │ │ + -- used 0.000554321s (cpu); 0.00285448s (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 .00798908 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .00399574 seconds │ │ │ + -- used .00798829 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used 0 seconds │ │ │ + -- used .00404815 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 .00394571 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0022852 seconds │ │ │ + -- used .00411183 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00168203 seconds │ │ │ + -- used 0 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 .0120593 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0102348 seconds │ │ │ + -- used .0122885 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00876388 seconds │ │ │ + -- used .00738438 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 .00402804 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00697058 seconds │ │ │ + -- used .00799025 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .00394506 seconds │ │ │ + -- used 0 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 0.0676672s (cpu); 0.0677077s (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 │ │ │ + -- .526979s 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 .00798908 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00399574 seconds
│ │ │ +     -- used .00798829 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used 0 seconds
│ │ │ +     -- used .00404815 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 .00798908 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .00399574 seconds │ │ │ │ + -- used .00798829 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used 0 seconds │ │ │ │ + -- used .00404815 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 .00394571 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0022852 seconds
│ │ │ +     -- used .00411183 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00168203 seconds
│ │ │ +     -- used 0 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 .00394571 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .0022852 seconds
│ │ │ │ +     -- used .00411183 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00168203 seconds
│ │ │ │ +     -- used 0 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 .0120593 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0102348 seconds
│ │ │ +     -- used .0122885 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00876388 seconds
│ │ │ +     -- used .00738438 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 .00402804 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00697058 seconds
│ │ │ +     -- used .00799025 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .00394506 seconds
│ │ │ +     -- used 0 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 .0120593 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0102348 seconds │ │ │ │ + -- used .0122885 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00876388 seconds │ │ │ │ + -- used .00738438 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 .00402804 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00697058 seconds │ │ │ │ + -- used .00799025 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .00394506 seconds │ │ │ │ + -- used 0 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 0.0676672s (cpu); 0.0677077s (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 0.0676672s (cpu); 0.0677077s (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
│ │ │ + -- .526979s 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
│ │ │ │ + -- .526979s 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
│ │ │ + -- .102594s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .140684s elapsed
│ │ │ + -- .0877377s elapsed
│ │ │  next gb
│ │ │ - -- .00124739s elapsed
│ │ │ + -- .00078065s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .131949s elapsed
│ │ │ + -- .0906957s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .108241s elapsed
│ │ │ + -- .0934511s elapsed
│ │ │  next gb
│ │ │ - -- .000704851s elapsed
│ │ │ + -- .000688215s elapsed
│ │ │  true
│ │ │ - -- 1.55972s elapsed
│ │ │ + -- 1.07992s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.14586s elapsed
│ │ │ + -- .981828s 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
│ │ │ + -- .171282s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .222111s elapsed
│ │ │ + -- .114606s elapsed
│ │ │  next gb
│ │ │ - -- .00146195s elapsed
│ │ │ + -- .000978626s elapsed
│ │ │  true
│ │ │ - -- .851703s elapsed
│ │ │ -(5, 8,  all semigroups are smoothable) -- .915236s elapsed
│ │ │ + -- .469819s elapsed
│ │ │ +(5, 8,  all semigroups are smoothable) -- .512511s 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
│ │ │ + -- .0570459s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .446926s elapsed
│ │ │ + -- .299163s 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
│ │ │ + -- 2.50267s 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
│ │ │ + -- .737372s elapsed
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.57876s elapsed
│ │ │ + -- 3.2277s 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
│ │ │ + -- 3.30603s 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) -- 1.12339s 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
│ │ │ + -- .331576s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .178994s elapsed
│ │ │ + -- .13593s elapsed
│ │ │  next gb
│ │ │ - -- .00179905s elapsed
│ │ │ + -- .00187252s elapsed
│ │ │  true
│ │ │ - -- .959568s elapsed
│ │ │ + -- .806821s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .402129s elapsed
│ │ │ + -- .32493s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .190718s elapsed
│ │ │ + -- .141738s elapsed
│ │ │  next gb
│ │ │ - -- .00255459s elapsed
│ │ │ + -- .00259788s elapsed
│ │ │  decompose
│ │ │ - -- .183642s elapsed
│ │ │ + -- .153445s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 2.51269s elapsed
│ │ │ + -- 2.25267s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .0737587s elapsed
│ │ │ + -- .107521s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .0919318s elapsed
│ │ │ + -- .0899849s elapsed
│ │ │  next gb
│ │ │ - -- .000445154s elapsed
│ │ │ + -- .000450413s elapsed
│ │ │  true
│ │ │ - -- .609331s elapsed
│ │ │ + -- .591702s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .475404s elapsed
│ │ │ + -- .436282s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .292145s elapsed
│ │ │ + -- .178299s elapsed
│ │ │  next gb
│ │ │ - -- .00644164s elapsed
│ │ │ + -- .00452152s elapsed
│ │ │  decompose
│ │ │ - -- 1.13216s elapsed
│ │ │ + -- .742778s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 3.15341s elapsed
│ │ │ - -- 7.23516s elapsed
│ │ │ + -- 2.2446s elapsed
│ │ │ + -- 5.89595s elapsed
│ │ │  0
│ │ │  
│ │ │ - -- .0000054s elapsed
│ │ │ - -- 7.26994s elapsed
│ │ │ + -- .000005882s elapsed
│ │ │ + -- 5.93411s 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
│ │ │ + -- .102594s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .140684s elapsed
│ │ │ + -- .0877377s elapsed
│ │ │  next gb
│ │ │ - -- .00124739s elapsed
│ │ │ + -- .00078065s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .131949s elapsed
│ │ │ + -- .0906957s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .108241s elapsed
│ │ │ + -- .0934511s elapsed
│ │ │  next gb
│ │ │ - -- .000704851s elapsed
│ │ │ + -- .000688215s elapsed
│ │ │  true
│ │ │ - -- 1.55972s elapsed
│ │ │ + -- 1.07992s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
    
│ │ │            
│ │ │            
│ │ │                
    i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.14586s elapsed
│ │ │ + -- .981828s 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
│ │ │ + -- .171282s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .222111s elapsed
│ │ │ + -- .114606s elapsed
│ │ │  next gb
│ │ │ - -- .00146195s elapsed
│ │ │ + -- .000978626s elapsed
│ │ │  true
│ │ │ - -- .851703s elapsed
│ │ │ -(5, 8,  all semigroups are smoothable) -- .915236s elapsed
│ │ │ + -- .469819s elapsed
│ │ │ +(5, 8,  all semigroups are smoothable) -- .512511s elapsed
│ │ │  
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
    
│ │ │            
│ │ │          
│ │ │ @@ -200,22 +200,22 @@
│ │ │            
│ │ │                
    i13 : genus L
│ │ │  
│ │ │  o13 = 8
    
│ │ │            
│ │ │            
│ │ │                
    i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .0789936s elapsed
│ │ │ + -- .0570459s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .446926s elapsed
│ │ │ + -- .299163s 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
│ │ │ │ + -- .102594s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .140684s elapsed
│ │ │ │ + -- .0877377s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00124739s elapsed
│ │ │ │ + -- .00078065s elapsed
│ │ │ │  true
│ │ │ │  unfolding
│ │ │ │ - -- .131949s elapsed
│ │ │ │ + -- .0906957s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .108241s elapsed
│ │ │ │ + -- .0934511s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000704851s elapsed
│ │ │ │ + -- .000688215s elapsed
│ │ │ │  true
│ │ │ │ - -- 1.55972s elapsed
│ │ │ │ + -- 1.07992s elapsed
│ │ │ │  
│ │ │ │  o6 = {}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ │ - -- 1.14586s elapsed
│ │ │ │ + -- .981828s 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
│ │ │ │ + -- .171282s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .222111s elapsed
│ │ │ │ + -- .114606s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00146195s elapsed
│ │ │ │ + -- .000978626s elapsed
│ │ │ │  true
│ │ │ │ - -- .851703s elapsed
│ │ │ │ -(5, 8,  all semigroups are smoothable) -- .915236s elapsed
│ │ │ │ + -- .469819s elapsed
│ │ │ │ +(5, 8,  all semigroups are smoothable) -- .512511s 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
│ │ │ │ + -- .0570459s elapsed
│ │ │ │  6
│ │ │ │  false
│ │ │ │  5
│ │ │ │  false
│ │ │ │  4
│ │ │ │  decompose
│ │ │ │ - -- .446926s elapsed
│ │ │ │ + -- .299163s 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
│ │ │ + -- 2.50267s 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
│ │ │ │ + -- 2.50267s 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
│ │ │ + -- .737372s elapsed
│ │ │  
│ │ │  o3 = false
    
│ │ │            
│ │ │            
│ │ │                
    i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.57876s elapsed
│ │ │ + -- 3.2277s 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 │ │ │ │ + -- .737372s elapsed │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : elapsedTime isSmoothableSemigroup(L,0.14,0) │ │ │ │ - -- 4.57876s elapsed │ │ │ │ + -- 3.2277s 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
│ │ │ + -- 3.30603s 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 │ │ │ │ + -- 3.30603s 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) -- 1.12339s elapsed
│ │ │  
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
    │ │ │ 
    i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .380066s elapsed
│ │ │ + -- .331576s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .178994s elapsed
│ │ │ + -- .13593s elapsed
│ │ │  next gb
│ │ │ - -- .00179905s elapsed
│ │ │ + -- .00187252s elapsed
│ │ │  true
│ │ │ - -- .959568s elapsed
│ │ │ + -- .806821s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .402129s elapsed
│ │ │ + -- .32493s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .190718s elapsed
│ │ │ + -- .141738s elapsed
│ │ │  next gb
│ │ │ - -- .00255459s elapsed
│ │ │ + -- .00259788s elapsed
│ │ │  decompose
│ │ │ - -- .183642s elapsed
│ │ │ + -- .153445s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 2.51269s elapsed
│ │ │ + -- 2.25267s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .0737587s elapsed
│ │ │ + -- .107521s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .0919318s elapsed
│ │ │ + -- .0899849s elapsed
│ │ │  next gb
│ │ │ - -- .000445154s elapsed
│ │ │ + -- .000450413s elapsed
│ │ │  true
│ │ │ - -- .609331s elapsed
│ │ │ + -- .591702s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .475404s elapsed
│ │ │ + -- .436282s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .292145s elapsed
│ │ │ + -- .178299s elapsed
│ │ │  next gb
│ │ │ - -- .00644164s elapsed
│ │ │ + -- .00452152s elapsed
│ │ │  decompose
│ │ │ - -- 1.13216s elapsed
│ │ │ + -- .742778s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 3.15341s elapsed
│ │ │ - -- 7.23516s elapsed
│ │ │ + -- 2.2446s elapsed
│ │ │ + -- 5.89595s elapsed
│ │ │  0
│ │ │  
│ │ │ - -- .0000054s elapsed
│ │ │ - -- 7.26994s elapsed
│ │ │ + -- .000005882s elapsed
│ │ │ + -- 5.93411s 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) -- 1.12339s 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 │ │ │ │ + -- .331576s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .178994s elapsed │ │ │ │ + -- .13593s elapsed │ │ │ │ next gb │ │ │ │ - -- .00179905s elapsed │ │ │ │ + -- .00187252s elapsed │ │ │ │ true │ │ │ │ - -- .959568s elapsed │ │ │ │ + -- .806821s elapsed │ │ │ │ {6, 7, 9, 17} │ │ │ │ unfolding │ │ │ │ - -- .402129s elapsed │ │ │ │ + -- .32493s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .190718s elapsed │ │ │ │ + -- .141738s elapsed │ │ │ │ next gb │ │ │ │ - -- .00255459s elapsed │ │ │ │ + -- .00259788s elapsed │ │ │ │ decompose │ │ │ │ - -- .183642s elapsed │ │ │ │ + -- .153445s elapsed │ │ │ │ number of components: 2 │ │ │ │ support c, codim c: {(2, 2), (5, 2)} │ │ │ │ {0, -1} │ │ │ │ - -- 2.51269s elapsed │ │ │ │ + -- 2.25267s elapsed │ │ │ │ {6, 8, 9, 10} │ │ │ │ unfolding │ │ │ │ - -- .0737587s elapsed │ │ │ │ + -- .107521s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .0919318s elapsed │ │ │ │ + -- .0899849s elapsed │ │ │ │ next gb │ │ │ │ - -- .000445154s elapsed │ │ │ │ + -- .000450413s elapsed │ │ │ │ true │ │ │ │ - -- .609331s elapsed │ │ │ │ + -- .591702s elapsed │ │ │ │ {6, 8, 10, 11, 13} │ │ │ │ unfolding │ │ │ │ - -- .475404s elapsed │ │ │ │ + -- .436282s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .292145s elapsed │ │ │ │ + -- .178299s elapsed │ │ │ │ next gb │ │ │ │ - -- .00644164s elapsed │ │ │ │ + -- .00452152s elapsed │ │ │ │ decompose │ │ │ │ - -- 1.13216s elapsed │ │ │ │ + -- .742778s elapsed │ │ │ │ number of components: 1 │ │ │ │ support c, codim c: {(5, 1)} │ │ │ │ {-1} │ │ │ │ - -- 3.15341s elapsed │ │ │ │ - -- 7.23516s elapsed │ │ │ │ + -- 2.2446s elapsed │ │ │ │ + -- 5.89595s elapsed │ │ │ │ 0 │ │ │ │ │ │ │ │ - -- .0000054s elapsed │ │ │ │ - -- 7.26994s elapsed │ │ │ │ + -- .000005882s elapsed │ │ │ │ + -- 5.93411s 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 0.174754s (cpu); 0.118369s (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 0.200049s (cpu); 0.137105s (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 0.359695s (cpu); 0.262202s (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 0.145488s (cpu); 0.0998033s (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) │ │ ├── ./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 0.156541s (cpu); 0.0987291s (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 0.384712s (cpu); 0.261475s (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 0.0279982s (cpu); 0.0245139s (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 0.0317659s (cpu); 0.0297236s (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 0.161714s (cpu); 0.0994519s (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 0.0320012s (cpu); 0.0290698s (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 0.340403s (cpu); 0.249586s (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 0.180491s (cpu); 0.133101s (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 0.174754s (cpu); 0.118369s (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 0.174754s (cpu); 0.118369s (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 0.200049s (cpu); 0.137105s (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 0.200049s (cpu); 0.137105s (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 0.359695s (cpu); 0.262202s (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 0.359695s (cpu); 0.262202s (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 0.145488s (cpu); 0.0998033s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ 
    i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │                  Basis element widths: {2}
│ │ │                  Degree shifts: {-2}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  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 0.145488s (cpu); 0.0998033s (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
│ │ ├── ./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 0.156541s (cpu); 0.0987291s (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 0.156541s (cpu); 0.0987291s (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 0.384712s (cpu); 0.261475s (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 0.384712s (cpu); 0.261475s (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 0.0279982s (cpu); 0.0245139s (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 0.0279982s (cpu); 0.0245139s (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 0.0317659s (cpu); 0.0297236s (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 0.0317659s (cpu); 0.0297236s (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 0.161714s (cpu); 0.0994519s (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 0.161714s (cpu); 0.0994519s (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 0.0320012s (cpu); 0.0290698s (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 0.0320012s (cpu); 0.0290698s (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 0.340403s (cpu); 0.249586s (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 0.340403s (cpu); 0.249586s (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 0.180491s (cpu); 0.133101s (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 0.180491s (cpu); 0.133101s (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
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│ │ ├── ./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
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│ │ ├── ./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
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│ │ ├── ./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==
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│ │ │  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=
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│ │ ├── ./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
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│ │ ├── ./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
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│ │ │  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
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│ │ ├── ./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
│ │ │ + -- .942021s 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
│ │ │ + -- .387016s 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
│ │ │ + -- 14.4614s 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
│ │ │ + -- .593194s 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
│ │ │ + -- 1.83906s 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
    
│ │ │ + -- .942021s 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     │ │ │ + -- .387016s 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
    │ │ │ + -- 14.4614s 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 │ │ │ │ + -- .942021s 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 │ │ │ │ + -- .387016s 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 │ │ │ │ + -- 14.4614s 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
    │ │ │ + -- .593194s 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
    
│ │ │ + -- 1.83906s 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
│ │ │ │ + -- .593194s 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
│ │ │ │ + -- 1.83906s 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
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│ │ │  #:format=standard
│ │ │  # End of header
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│ │ ├── ./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
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│ │ ├── ./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
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│ │ ├── ./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
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│ │ ├── ./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
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│ │ │  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
│ │ │ + -- 2.01647s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.0879s elapsed
│ │ │ + -- 3.54673s 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
    │ │ │ + -- 2.01647s elapsed │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.0879s elapsed
    │ │ │ + -- 3.54673s 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 │ │ │ │ + -- 2.01647s elapsed │ │ │ │ i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R); │ │ │ │ - -- 4.0879s elapsed │ │ │ │ + -- 3.54673s 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/_polyo__Ideal.out │ │ │ @@ -18,31 +18,31 @@ │ │ │ 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 1,3 4,3 1,4 3,4 2,3 3,3 2,4 4,2 2,1 4,1 2,2 │ │ │ ------------------------------------------------------------------------ │ │ │ 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 1,2 2,2 1,3 2,4 1,1 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 │ │ │ ------------------------------------------------------------------------ │ │ │ - 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 │ │ │ + 4,2 3,3 3,4 1,3 3,3 1,4 3,2 2,1 3,1 2,2 4,2 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 │ │ │ + 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 │ │ │ ------------------------------------------------------------------------ │ │ │ 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 │ │ │ + 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 │ │ │ ------------------------------------------------------------------------ │ │ │ - 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 │ │ │ + 2,1 1,2 4,2 3,1 4,1 3,2 2,3 1,1 2,1 1,3 4,3 3,1 │ │ │ ------------------------------------------------------------------------ │ │ │ x x ) │ │ │ - 4,2 3,4 │ │ │ + 4,1 3,3 │ │ │ │ │ │ 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/_polyo__Ideal.html │ │ │ @@ -102,33 +102,33 @@ │ │ │ │ │ │ 
    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 1,3    4,3 1,4   3,4 2,3    3,3 2,4   4,2 2,1    4,1 2,2 
│ │ │       ------------------------------------------------------------------------
│ │ │       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 1,2    2,2 1,3   2,4 1,1    2,1 1,4   4,4 3,1    4,1 3,4   4,3 3,2
│ │ │       ------------------------------------------------------------------------
│ │ │       - 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  
│ │ │ +        4,2 3,3   3,4 1,3    3,3 1,4   3,2 2,1    3,1 2,2   4,2 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 
│ │ │ +      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 
│ │ │       ------------------------------------------------------------------------
│ │ │       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
│ │ │ +      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
│ │ │       ------------------------------------------------------------------------
│ │ │       - 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  
│ │ │ +        2,1 1,2   4,2 3,1    4,1 3,2   2,3 1,1    2,1 1,3   4,3 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   )
│ │ │ -      4,2 3,4
│ │ │ +      4,1 3,3
│ │ │  
│ │ │  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 {} │ │ │ │ @@ -46,33 +46,33 @@ │ │ │ │ │ │ │ │ │ │ │ │ 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 1,3 4,3 1,4 3,4 2,3 3,3 2,4 4,2 2,1 4,1 2,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 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 1,2 2,2 1,3 2,4 1,1 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 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 │ │ │ │ + 4,2 3,3 3,4 1,3 3,3 1,4 3,2 2,1 3,1 2,2 4,2 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 │ │ │ │ + 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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 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 │ │ │ │ + 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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 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 │ │ │ │ + 2,1 1,2 4,2 3,1 4,1 3,2 2,3 1,1 2,1 1,3 4,3 3,1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ - 4,2 3,4 │ │ │ │ + 4,1 3,3 │ │ │ │ │ │ │ │ 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 0.00254053s (cpu); 6.537e-06s (thread); 0s (gc) │ │ │ │ │ │ o7 = true │ │ │ │ │ │ i8 : time isDistributive P │ │ │ - -- used 6.1046s (cpu); 4.14657s (thread); 0s (gc) │ │ │ + -- used 6.14146s (cpu); 4.06569s (thread); 0s (gc) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : C' = dual C; │ │ │ │ │ │ i10 : time isDistributive C' │ │ │ - -- used 0.000291406s (cpu); 6.492e-06s (thread); 0s (gc) │ │ │ + -- used 0.000337251s (cpu); 9.577e-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 0.377449s (cpu); 0.215621s (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 0.000198378s (cpu); 1.1864e-05s (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 0.00254053s (cpu); 6.537e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    │ │ │ │ │ │ │ │ │ 
    i8 : time isDistributive P
│ │ │ - -- used 6.1046s (cpu); 4.14657s (thread); 0s (gc)
│ │ │ + -- used 6.14146s (cpu); 4.06569s (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 0.000337251s (cpu); 9.577e-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 0.00254053s (cpu); 6.537e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.1046s (cpu); 4.14657s (thread); 0s (gc)
│ │ │ │ + -- used 6.14146s (cpu); 4.06569s (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 0.000337251s (cpu); 9.577e-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 0.377449s (cpu); 0.215621s (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 0.000198378s (cpu); 1.1864e-05s (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 0.377449s (cpu); 0.215621s (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 0.000198378s (cpu); 1.1864e-05s (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
│ │ │ + -- .196735s 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
│ │ │ + -- .0759532s 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
│ │ │ + -- .0230758s 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
│ │ │ + -- .0598273s 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
│ │ │ + -- .00886634s 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
│ │ │ + -- .196735s 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
│ │ │ + -- .0759532s 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
│ │ │ + -- .0230758s 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 │ │ │ │ + -- .196735s 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 │ │ │ │ + -- .0759532s 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 │ │ │ │ + -- .0230758s 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
│ │ │ + -- .0598273s 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
│ │ │ + -- .00886634s 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 │ │ │ │ + -- .0598273s 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 │ │ │ │ + -- .00886634s 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 0x7f3678dbf390>
│ │ │  
│ │ │  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 0x7f139b68b2a0>
│ │ │  
│ │ │  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 0x7f0ea6d4e890>
│ │ │  
│ │ │  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
│ │ │ + -- 2.4304s 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
│ │ │ + -- 1.84969s 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
│ │ │ + -- 2.4304s 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
    │ │ │ + -- 1.84969s 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 │ │ │ │ + -- 2.4304s 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 │ │ │ │ + -- 1.84969s 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 6.93103s (cpu); 5.79721s (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 6.93103s (cpu); 5.79721s (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 6.93103s (cpu); 5.79721s (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}, .00800395, .0440583) │ │ │ │ │ │ o4 : Sequence │ │ │ │ │ │ i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2}) │ │ │ │ │ │ -o5 = {{.00415423, .0129743}, {.00772311, .0045526}, {.00327175, .00800128}, │ │ │ +o5 = {{.00803125, .0240436}, {.0120115, .00797395}, {.00797544, .0120223}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00449345, .012301}, {.00573879, .0131784}, {.00740736, .0112207}, │ │ │ + {.00796986, .00800719}, {.00399967, .015999}, {.0040009, .0159992}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00400166, .00799792}, {.00798013, .00399967}, {.00800154, .00399867}, │ │ │ + {.00400035, .0119943}, {.00400355, .00799583}, {.00400321, .00799734}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00800417, .00799329}} │ │ │ + {.00800112, .0079959}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : 1/10*sum(L,t->t_0) │ │ │ │ │ │ -o6 = .006077619700000004 │ │ │ +o6 = .006399680800000018 │ │ │ │ │ │ o6 : RR (of precision 53) │ │ │ │ │ │ i7 : 1/10*sum(L,t->t_1) │ │ │ │ │ │ -o7 = .008621774699999963 │ │ │ +o7 = .01200286699999991 │ │ │ │ │ │ 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}, .00800395, .0440583)
│ │ │  
│ │ │  o4 : Sequence
    │ │ │ 
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00415423, .0129743}, {.00772311, .0045526}, {.00327175, .00800128},
│ │ │ +o5 = {{.00803125, .0240436}, {.0120115, .00797395}, {.00797544, .0120223},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00449345, .012301}, {.00573879, .0131784}, {.00740736, .0112207},
│ │ │ +     {.00796986, .00800719}, {.00399967, .015999}, {.0040009, .0159992},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00400166, .00799792}, {.00798013, .00399967}, {.00800154, .00399867},
│ │ │ +     {.00400035, .0119943}, {.00400355, .00799583}, {.00400321, .00799734},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00800417, .00799329}}
│ │ │ +     {.00800112, .0079959}}
│ │ │  
│ │ │  o5 : List
    │ │ │ 
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .006077619700000004
│ │ │ +o6 = .006399680800000018
│ │ │  
│ │ │  o6 : RR (of precision 53)
    │ │ │ 
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008621774699999963
│ │ │ +o7 = .01200286699999991
│ │ │  
│ │ │  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}, .00800395, .0440583) │ │ │ │ │ │ │ │ o4 : Sequence │ │ │ │ i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2}) │ │ │ │ │ │ │ │ -o5 = {{.00415423, .0129743}, {.00772311, .0045526}, {.00327175, .00800128}, │ │ │ │ +o5 = {{.00803125, .0240436}, {.0120115, .00797395}, {.00797544, .0120223}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00449345, .012301}, {.00573879, .0131784}, {.00740736, .0112207}, │ │ │ │ + {.00796986, .00800719}, {.00399967, .015999}, {.0040009, .0159992}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00400166, .00799792}, {.00798013, .00399967}, {.00800154, .00399867}, │ │ │ │ + {.00400035, .0119943}, {.00400355, .00799583}, {.00400321, .00799734}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00800417, .00799329}} │ │ │ │ + {.00800112, .0079959}} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : 1/10*sum(L,t->t_0) │ │ │ │ │ │ │ │ -o6 = .006077619700000004 │ │ │ │ +o6 = .006399680800000018 │ │ │ │ │ │ │ │ o6 : RR (of precision 53) │ │ │ │ i7 : 1/10*sum(L,t->t_1) │ │ │ │ │ │ │ │ -o7 = .008621774699999963 │ │ │ │ +o7 = .01200286699999991 │ │ │ │ │ │ │ │ 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 1.27653s (cpu); 1.1535s (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 = 1772386465 │ │ │ │ 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, c*d), {a*b, a*d, c*d}, {b*d, b*c, 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 5.10391s (cpu); 3.39458s (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 │ │ │ @@ -47,32 +47,33 @@ │ │ │
    │ │ │

    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 1.27653s (cpu); 1.1535s (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 1.27653s (cpu); 1.1535s (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 1.66736s (cpu); 1.47553s (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 1.66736s (cpu); 1.47553s (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 1.66736s (cpu); 1.47553s (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 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147038
│ │ │ +o1 = 1772386393
│ │ │  
│ │ │  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 1.87556s (cpu); 1.37582s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 32 }
│ │ │ -           5 => 219
│ │ │ -           6 => 166
│ │ │ -           7 => 68
│ │ │ -           8 => 14
│ │ │ -           9 => 1
│ │ │ +o4 = Tally{3 => 1  }
│ │ │ +           4 => 45
│ │ │ +           5 => 207
│ │ │ +           6 => 173
│ │ │ +           7 => 61
│ │ │ +           8 => 11
│ │ │ +           9 => 2
│ │ │  
│ │ │  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 = 1772386429
│ │ │  
│ │ │  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*c
│ │ │  
│ │ │  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 = 1772386478
│ │ │  
│ │ │  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*d)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*c, d*e, b*d, b*c, b*e)
│ │ │ +o6 = ideal (b*c, c*e, a*d, b*d, 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 = 1772386465
│ │ │  
│ │ │  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, c*d), {a*b, a*d, c*d}, {b*d, b*c, 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 5.10391s (cpu); 3.39458s (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 = 1772386429
│ │ │      		          
                  
    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*c
│ │ │  
│ │ │  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 = 1772386429
│ │ │ │  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*c
│ │ │ │  
│ │ │ │  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 = 1772386478
│ │ │      		          
                  
    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*d)
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │      		          
                  
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*c, d*e, b*d, b*c, b*e)
│ │ │ +o6 = ideal (b*c, c*e, a*d, b*d, 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 = 1772386478
│ │ │ │  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*d)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (a*c, d*e, b*d, b*c, b*e)
│ │ │ │ +o6 = ideal (b*c, c*e, a*d, b*d, 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 = 1772386465
│ │ │      		          
                  
    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, c*d), {a*b, a*d, c*d}, {b*d, b*c, 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 5.10391s (cpu); 3.39458s (thread); 0s (gc) │ │ │ 
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i1 : setRandomSeed(currentTime())
│ │ │  
│ │ │ -o1 = 1739147038
    │ │ │ +o1 = 1772386393 │ │ │ 
    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 1.87556s (cpu); 1.37582s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 32 }
│ │ │ -           5 => 219
│ │ │ -           6 => 166
│ │ │ -           7 => 68
│ │ │ -           8 => 14
│ │ │ -           9 => 1
│ │ │ +o4 = Tally{3 => 1  }
│ │ │ +           4 => 45
│ │ │ +           5 => 207
│ │ │ +           6 => 173
│ │ │ +           7 => 61
│ │ │ +           8 => 11
│ │ │ +           9 => 2
│ │ │  
│ │ │  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 {} │ │ │ │ @@ -10,26 +10,27 @@ │ │ │ │ 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 = 1772386393 │ │ │ │ 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 1.87556s (cpu); 1.37582s (thread); 0s (gc) │ │ │ │ │ │ │ │ -o4 = Tally{4 => 32 } │ │ │ │ - 5 => 219 │ │ │ │ - 6 => 166 │ │ │ │ - 7 => 68 │ │ │ │ - 8 => 14 │ │ │ │ - 9 => 1 │ │ │ │ +o4 = Tally{3 => 1 } │ │ │ │ + 4 => 45 │ │ │ │ + 5 => 207 │ │ │ │ + 6 => 173 │ │ │ │ + 7 => 61 │ │ │ │ + 8 => 11 │ │ │ │ + 9 => 2 │ │ │ │ │ │ │ │ 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. │ │ ├── ./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 │ │ │ + -- 1.38971s elapsed │ │ │ │ │ │ o4 = 4 │ │ │ │ │ │ i5 : elapsedTime dim I │ │ │ - -- 2.94822s elapsed │ │ │ + -- 3.08335s 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 │ │ │ + -- 1.63006s 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 │ │ │ + -- 2.76987s 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 │ │ │ + -- 2.05744s 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
│ │ │ + -- 1.38971s elapsed
│ │ │  
│ │ │  o4 = 4
    │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime dim I
│ │ │ - -- 2.94822s elapsed
│ │ │ + -- 3.08335s 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 │ │ │ │ + -- 1.38971s elapsed │ │ │ │ │ │ │ │ o4 = 4 │ │ │ │ i5 : elapsedTime dim I │ │ │ │ - -- 2.94822s elapsed │ │ │ │ + -- 3.08335s 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
    │ │ │ + -- 1.63006s 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 │ │ │ │ + -- 1.63006s 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
│ │ │ + -- 2.76987s 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
│ │ │ + -- 2.05744s 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 │ │ │ │ + -- 2.76987s 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 │ │ │ │ + -- 2.05744s 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 0.872226s (cpu); 0.440995s (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 0.872226s (cpu); 0.440995s (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 0.872226s (cpu); 0.440995s (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 0.00402941s (cpu); 0.00389351s (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 1.07758s (cpu); 0.861679s (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 0.307127s (cpu); 0.226667s (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 0.00402941s (cpu); 0.00389351s (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 1.07758s (cpu); 0.861679s (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 0.307127s (cpu); 0.226667s (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 0.00402941s (cpu); 0.00389351s (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 1.07758s (cpu); 0.861679s (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 0.307127s (cpu); 0.226667s (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 0.898667s (cpu); 0.816926s (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 1.01618s (cpu); 0.784392s (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 1.68204s (cpu); 1.36807s (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 1.69903s (cpu); 1.42127s (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 0.0962184s (cpu); 0.035128s (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 0.173074s (cpu); 0.15497s (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 0.0935788s (cpu); 0.0256217s (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 0.0392464s (cpu); 0.0130909s (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 0.898667s (cpu); 0.816926s (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 0.898667s (cpu); 0.816926s (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 1.01618s (cpu); 0.784392s (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 1.68204s (cpu); 1.36807s (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 1.69903s (cpu); 1.42127s (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 1.01618s (cpu); 0.784392s (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 1.68204s (cpu); 1.36807s (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 1.69903s (cpu); 1.42127s (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 0.0962184s (cpu); 0.035128s (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 0.173074s (cpu); 0.15497s (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 0.0935788s (cpu); 0.0256217s (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 0.0392464s (cpu); 0.0130909s (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 0.0962184s (cpu); 0.035128s (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 0.173074s (cpu); 0.15497s (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 0.0935788s (cpu); 0.0256217s (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 0.0392464s (cpu); 0.0130909s (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 = .2515452450909091
│ │ │  
│ │ │  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 = .0718294853650794
│ │ │  
│ │ │  o11 : RR (of precision 53)
│ │ │  
│ │ │  i12 : time regularity I2  
│ │ │ - -- used 0.00416868s (cpu); 0.00256224s (thread); 0s (gc)
│ │ │ + -- used 0.00119659s (cpu); 0.00240278s (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 = .2515452450909091
│ │ │  
│ │ │  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 = .0718294853650794
│ │ │  
│ │ │  o11 : RR (of precision 53)
    │ │ │ 
    i12 : time regularity I2  
│ │ │ - -- used 0.00416868s (cpu); 0.00256224s (thread); 0s (gc)
│ │ │ + -- used 0.00119659s (cpu); 0.00240278s (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 = .2515452450909091 │ │ │ │ │ │ │ │ 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 = .0718294853650794 │ │ │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ i12 : time regularity I2 │ │ │ │ - -- used 0.00416868s (cpu); 0.00256224s (thread); 0s (gc) │ │ │ │ + -- used 0.00119659s (cpu); 0.00240278s (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 0.120089s (cpu); 0.0675517s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ - -- used 0.05218s (cpu); 0.0522233s (thread); 0s (gc) │ │ │ + -- used 0.0590641s (cpu); 0.0591862s (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 0.15133s (cpu); 0.0934306s (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 0.155528s (cpu); 0.100207s (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 0.123168s (cpu); 0.0563825s (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 0.0439701s (cpu); 0.0418194s (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 0.163642s (cpu); 0.102523s (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 0.140164s (cpu); 0.0832001s (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 5.47757s (cpu); 5.14268s (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 1.17086s (cpu); 1.10823s (thread); 0s (gc) │ │ │ │ │ │ i5 : -- X-resultant of V │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ - -- used 0.218349s (cpu); 0.168032s (thread); 0s (gc) │ │ │ + -- used 0.281694s (cpu); 0.225164s (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 0.270705s (cpu); 0.204162s (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 0.0119784s (cpu); 0.0109673s (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 0.0863703s (cpu); 0.0260779s (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 0.418765s (cpu); 0.418269s (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 0.209643s (cpu); 0.144597s (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 0.219108s (cpu); 0.156772s (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 0.075987s (cpu); 0.0752172s (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 0.841738s (cpu); 0.780641s (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 0.11487s (cpu); 0.0592711s (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 0.0480065s (cpu); 0.0462558s (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 0.100436s (cpu); 0.04215s (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 0.0200009s (cpu); 0.021033s (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 0.116061s (cpu); 0.057382s (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 0.0120281s (cpu); 0.0119106s (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 0.008084s (cpu); 0.0103908s (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 0.0240137s (cpu); 0.0230333s (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 0.00403292s (cpu); 0.00342789s (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 0.000346408s (cpu); 0.00250617s (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 0.00800021s (cpu); 0.00542838s (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 0.108553s (cpu); 0.0469091s (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 0.0400119s (cpu); 0.0395872s (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 0.191413s (cpu); 0.194569s (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 0.31447s (cpu); 0.187981s (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 0.152885s (cpu); 0.0909933s (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 0.325467s (cpu); 0.328044s (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 0.731535s (cpu); 0.671966s (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 0.0279954s (cpu); 0.029536s (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 2.56126s (cpu); 2.05766s (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 0.476701s (cpu); 0.407729s (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 0.110306s (cpu); 0.0468759s (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 0.063985s (cpu); 0.0638555s (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 0.0476613s (cpu); 0.0454011s (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 0.123651s (cpu); 0.061426s (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 0.17717s (cpu); 0.112914s (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 0.120089s (cpu); 0.0675517s (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 0.0590641s (cpu); 0.0591862s (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 0.15133s (cpu); 0.0934306s (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 0.155528s (cpu); 0.100207s (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 0.120089s (cpu); 0.0675517s (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 0.0590641s (cpu); 0.0591862s (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 0.15133s (cpu); 0.0934306s (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 0.155528s (cpu); 0.100207s (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 0.123168s (cpu); 0.0563825s (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 0.0439701s (cpu); 0.0418194s (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 0.163642s (cpu); 0.102523s (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 0.140164s (cpu); 0.0832001s (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 0.123168s (cpu); 0.0563825s (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 0.0439701s (cpu); 0.0418194s (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 0.163642s (cpu); 0.102523s (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 0.140164s (cpu); 0.0832001s (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 5.47757s (cpu); 5.14268s (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 1.17086s (cpu); 1.10823s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ 
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.218349s (cpu); 0.168032s (thread); 0s (gc)
    │ │ │ + -- used 0.281694s (cpu); 0.225164s (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 0.270705s (cpu); 0.204162s (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 5.47757s (cpu); 5.14268s (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 1.17086s (cpu); 1.10823s (thread); 0s (gc) │ │ │ │ i5 : -- X-resultant of V │ │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ │ - -- used 0.218349s (cpu); 0.168032s (thread); 0s (gc) │ │ │ │ + -- used 0.281694s (cpu); 0.225164s (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 0.270705s (cpu); 0.204162s (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 0.0119784s (cpu); 0.0109673s (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 0.0863703s (cpu); 0.0260779s (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 0.418765s (cpu); 0.418269s (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 0.0119784s (cpu); 0.0109673s (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 0.0863703s (cpu); 0.0260779s (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 0.418765s (cpu); 0.418269s (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 0.209643s (cpu); 0.144597s (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 0.219108s (cpu); 0.156772s (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 0.075987s (cpu); 0.0752172s (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 0.841738s (cpu); 0.780641s (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 0.209643s (cpu); 0.144597s (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 0.219108s (cpu); 0.156772s (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 0.075987s (cpu); 0.0752172s (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 0.841738s (cpu); 0.780641s (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 0.11487s (cpu); 0.0592711s (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 0.0480065s (cpu); 0.0462558s (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 0.100436s (cpu); 0.04215s (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 0.0200009s (cpu); 0.021033s (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 0.11487s (cpu); 0.0592711s (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 0.0480065s (cpu); 0.0462558s (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 0.100436s (cpu); 0.04215s (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 0.0200009s (cpu); 0.021033s (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 0.116061s (cpu); 0.057382s (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 0.116061s (cpu); 0.057382s (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 0.0120281s (cpu); 0.0119106s (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 0.008084s (cpu); 0.0103908s (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 0.0120281s (cpu); 0.0119106s (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 0.008084s (cpu); 0.0103908s (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 0.0240137s (cpu); 0.0230333s (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 0.0240137s (cpu); 0.0230333s (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 0.00403292s (cpu); 0.00342789s (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 0.000346408s (cpu); 0.00250617s (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 0.00403292s (cpu); 0.00342789s (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 0.000346408s (cpu); 0.00250617s (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 0.00800021s (cpu); 0.00542838s (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 0.108553s (cpu); 0.0469091s (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 0.0400119s (cpu); 0.0395872s (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 0.191413s (cpu); 0.194569s (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 0.00800021s (cpu); 0.00542838s (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 0.108553s (cpu); 0.0469091s (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 0.0400119s (cpu); 0.0395872s (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 0.191413s (cpu); 0.194569s (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 0.31447s (cpu); 0.187981s (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 0.152885s (cpu); 0.0909933s (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 0.325467s (cpu); 0.328044s (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 0.731535s (cpu); 0.671966s (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 0.31447s (cpu); 0.187981s (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 0.152885s (cpu); 0.0909933s (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 0.325467s (cpu); 0.328044s (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 0.731535s (cpu); 0.671966s (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 0.0279954s (cpu); 0.029536s (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 2.56126s (cpu); 2.05766s (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 0.476701s (cpu); 0.407729s (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 0.0279954s (cpu); 0.029536s (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 2.56126s (cpu); 2.05766s (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 0.476701s (cpu); 0.407729s (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 0.110306s (cpu); 0.0468759s (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 0.063985s (cpu); 0.0638555s (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 0.0476613s (cpu); 0.0454011s (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 0.123651s (cpu); 0.061426s (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 0.17717s (cpu); 0.112914s (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 0.110306s (cpu); 0.0468759s (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 0.063985s (cpu); 0.0638555s (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 0.0476613s (cpu); 0.0454011s (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 0.123651s (cpu); 0.061426s (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 0.17717s (cpu); 0.112914s (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 63520 │ │ │ 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/M2-133700-0/0.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/M2-133700-0/1.m2" >"/tmp/M2-133700-0/1.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i7 : h │ │ │ │ │ │ o7 = HashTable{"answer file" => null} │ │ │ "exit code" => 0 │ │ │ "output file" => null │ │ │ @@ -33,23 +33,23 @@ │ │ │ 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/M2-133700-0/2.m2" >"/tmp/M2-133700-0/2.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/M2-133700-0/2.ans} │ │ │ "exit code" => 27 │ │ │ - "output file" => /tmp/M2-76770-0/2.out │ │ │ + "output file" => /tmp/M2-133700-0/2.out │ │ │ "return code" => 6912 │ │ │ "statistics" => null │ │ │ "time used" => 1 │ │ │ value => null │ │ │ │ │ │ o11 : HashTable │ │ │ │ │ │ @@ -58,80 +58,80 @@ │ │ │ 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/M2-133700-0/3.m2" >"/tmp/M2-133700-0/3.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/M2-133700-0/3.ans} │ │ │ "exit code" => 0 │ │ │ - "output file" => /tmp/M2-76770-0/3.out │ │ │ + "output file" => /tmp/M2-133700-0/3.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/M2-133700-0/3.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-76770-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-133700-0/0.m2"; │ │ │ │ │ │ - i3 : runExternalM2ReturnAnswer("/tmp/M2-76770-0/3.ans",spin (10)); │ │ │ + i3 : runExternalM2ReturnAnswer("/tmp/M2-133700-0/3.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/M2-133700-0/4.m2" >"/tmp/M2-133700-0/4.out" 2>&1') >"/tmp/M2-133700-0/4.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/M2-133700-0/4.m2" >"/tmp/M2-133700-0/4.out" 2>&1" │ │ │ + User time (seconds): 4.11 │ │ │ + System time (seconds): 0.24 │ │ │ + Percent of CPU this job got: 115% │ │ │ + Elapsed (wall clock) time (h:mm:ss or m:ss): 0:03.78 │ │ │ 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): 335772 │ │ │ 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: 10936 │ │ │ + Voluntary context switches: 7985 │ │ │ + Involuntary context switches: 879 │ │ │ Swaps: 0 │ │ │ File system inputs: 0 │ │ │ - File system outputs: 0 │ │ │ + File system outputs: 24 │ │ │ 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/M2-133700-0/6.m2" >"/tmp/M2-133700-0/6.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/M2-133700-0/7.m2" >"/tmp/M2-133700-0/7.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/M2-133700-0/7.ans} │ │ │ "exit code" => 1 │ │ │ - "output file" => /tmp/M2-76770-0/7.out │ │ │ + "output file" => /tmp/M2-133700-0/7.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/M2-133700-0/7.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-76770-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-133700-0/0.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/M2-133700-0/7.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/M2-133700-0/8.m2" >"/tmp/M2-133700-0/8.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/M2-133700-0/9.m2" >"/tmp/M2-133700-0/9.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              63520
│ │ │  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 63520 │ │ │ │ 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/M2-133700-0/0.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/M2-133700-0/1.m2" >"/tmp/M2-133700-0/1.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │ 
    i7 : h
│ │ │  
│ │ │  o7 = HashTable{"answer file" => null}
│ │ │                 "exit code" => 0
│ │ │ @@ -146,24 +146,24 @@
│ │ │          
    
│ │ │            
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │  
│ │ │            
│ │ │  
│ │ │ @@ -180,79 +180,79 @@
│ │ │          
                  
    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/M2-133700-0/2.m2" >"/tmp/M2-133700-0/2.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/M2-133700-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-76770-0/2.out
│ │ │ +                "output file" => /tmp/M2-133700-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │                  "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
    
│ │ │      		          
    
│ │ │          
    
│ │ │            
    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/M2-133700-0/3.m2" >"/tmp/M2-133700-0/3.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/M2-133700-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-76770-0/3.out
│ │ │ +                "output file" => /tmp/M2-133700-0/3.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/M2-133700-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-76770-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-133700-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-76770-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-133700-0/3.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/M2-133700-0/4.m2" >"/tmp/M2-133700-0/4.out" 2>&1') >"/tmp/M2-133700-0/4.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/M2-133700-0/4.m2" >"/tmp/M2-133700-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 4.11
│ │ │ +              System time (seconds): 0.24
│ │ │ +              Percent of CPU this job got: 115%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:03.78
│ │ │                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): 335772
│ │ │                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: 10936
│ │ │ +              Voluntary context switches: 7985
│ │ │ +              Involuntary context switches: 879
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 24
│ │ │                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/M2-133700-0/6.m2" >"/tmp/M2-133700-0/6.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/M2-133700-0/7.m2" >"/tmp/M2-133700-0/7.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/M2-133700-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-76770-0/7.out
│ │ │ +                "output file" => /tmp/M2-133700-0/7.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/M2-133700-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-76770-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-133700-0/0.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/M2-133700-0/7.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/M2-133700-0/8.m2" >"/tmp/M2-133700-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │ @@ -345,15 +345,15 @@ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── 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 │ │ │ │ + -- .00745254s 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 │ │ │ │ + -- .00244858s 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 │ │ │ │ + -- .0049554s 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 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    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/M2-133700-0/9.m2" >"/tmp/M2-133700-0/9.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/M2-133700-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-76770-0/1.out" 2>&1 ))
│ │ │ │ +M2-133700-0/1.m2" >"/tmp/M2-133700-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │ @@ -80,22 +80,22 @@
│ │ │ │  
│ │ │ │  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 ))
│ │ │ │ +M2-133700-0/2.m2" >"/tmp/M2-133700-0/2.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/M2-133700-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-76770-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-133700-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  i12 : fileExists(h#"output file")
│ │ │ │ @@ -104,143 +104,143 @@
│ │ │ │  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/M2-133700-0/3.m2" >"/tmp/M2-133700-0/3.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/M2-133700-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-76770-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-133700-0/3.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/M2-133700-0/3.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-76770-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-133700-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-76770-0/3.ans",spin (10));
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-133700-0/3.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/M2-133700-0/4.m2" >"/tmp/M2-133700-0/4.out"
│ │ │ │ +2>&1') >"/tmp/M2-133700-0/4.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/M2-133700-0/4.m2" >"/tmp/M2-133700-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 4.11
│ │ │ │ +              System time (seconds): 0.24
│ │ │ │ +              Percent of CPU this job got: 115%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:03.78
│ │ │ │                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): 335772
│ │ │ │                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: 10936
│ │ │ │ +              Voluntary context switches: 7985
│ │ │ │ +              Involuntary context switches: 879
│ │ │ │                Swaps: 0
│ │ │ │                File system inputs: 0
│ │ │ │ -              File system outputs: 0
│ │ │ │ +              File system outputs: 24
│ │ │ │                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 ))
│ │ │ │ +M2-133700-0/6.m2" >"/tmp/M2-133700-0/6.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 ))
│ │ │ │ +M2-133700-0/7.m2" >"/tmp/M2-133700-0/7.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/M2-133700-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-76770-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-133700-0/7.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/M2-133700-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-76770-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-133700-0/0.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/M2-133700-0/7.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 ))
│ │ │ │ +M2-133700-0/8.m2" >"/tmp/M2-133700-0/8.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 ))
│ │ │ │ +M2-133700-0/9.m2" >"/tmp/M2-133700-0/9.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 0.00280645s (cpu); 0.000241721s (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 4.31752s (cpu); 3.22511s (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 0.00280645s (cpu); 0.000241721s (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 4.31752s (cpu); 3.22511s (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 0.00280645s (cpu); 0.000241721s (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 4.31752s (cpu); 3.22511s (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/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
│ │ │ + -- .00745254s 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
│ │ │ + -- .00244858s 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
│ │ │ + -- .0049554s 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
│ │ │ + -- .00314005s 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
│ │ │ + -- .000777619s 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
│ │ │ + -- .00148808s 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
│ │ │ + -- .00412364s 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
│ │ │ + -- .00314184s 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
│ │ │ + -- .00308668s 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
│ │ │ + -- .00745254s 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
    │ │ │ + -- .00244858s 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
    │ │ │ + -- .0049554s elapsed │ │ │ 
    i14 : hL === h
│ │ │  
│ │ │  o14 = true
    │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00438766s elapsed
│ │ │ + -- .00314005s 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
│ │ │ + -- .000777619s 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
│ │ │ + -- .00148808s 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 │ │ │ │ + -- .00314005s 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 │ │ │ │ + -- .000777619s 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 │ │ │ │ + -- .00148808s 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
│ │ │ + -- .00412364s 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 │ │ │ │ + -- .00412364s 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
│ │ │ + -- .00314184s 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 │ │ │ │ + -- .00314184s 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
│ │ │ + -- .00308668s 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 │ │ │ │ + -- .00308668s 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 0.351988s (cpu); 0.353624s (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 0.681813s (cpu); 0.605175s (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 0.0220288s (cpu); 0.0240888s (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 0.0084224s (cpu); 0.00894973s (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 0.351988s (cpu); 0.353624s (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 0.681813s (cpu); 0.605175s (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 0.0220288s (cpu); 0.0240888s (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 0.0084224s (cpu); 0.00894973s (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 0.351988s (cpu); 0.353624s (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 0.681813s (cpu); 0.605175s (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 0.0220288s (cpu); 0.0240888s (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 0.0084224s (cpu); 0.00894973s (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 0.00800135s (cpu); 0.00515618s (thread); 0s (gc) │ │ │ + -- used 0.00615376s (cpu); 0.00794971s (thread); 0s (gc) │ │ │ + -- used 0.00958423s (cpu); 0.0121729s (thread); 0s (gc) │ │ │ + -- used 0.0172877s (cpu); 0.019031s (thread); 0s (gc) │ │ │ + -- used 0.034148s (cpu); 0.0372794s (thread); 0s (gc) │ │ │ + -- used 0.0605682s (cpu); 0.0630337s (thread); 0s (gc) │ │ │ + -- used 0.111349s (cpu); 0.113803s (thread); 0s (gc) │ │ │ + -- used 0.175416s (cpu); 0.175807s (thread); 0s (gc) │ │ │ + -- used 0.372784s (cpu); 0.305455s (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 0.00800135s (cpu); 0.00515618s (thread); 0s (gc)
│ │ │ + -- used 0.00615376s (cpu); 0.00794971s (thread); 0s (gc)
│ │ │ + -- used 0.00958423s (cpu); 0.0121729s (thread); 0s (gc)
│ │ │ + -- used 0.0172877s (cpu); 0.019031s (thread); 0s (gc)
│ │ │ + -- used 0.034148s (cpu); 0.0372794s (thread); 0s (gc)
│ │ │ + -- used 0.0605682s (cpu); 0.0630337s (thread); 0s (gc)
│ │ │ + -- used 0.111349s (cpu); 0.113803s (thread); 0s (gc)
│ │ │ + -- used 0.175416s (cpu); 0.175807s (thread); 0s (gc)
│ │ │ + -- used 0.372784s (cpu); 0.305455s (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 0.00800135s (cpu); 0.00515618s (thread); 0s (gc) │ │ │ │ + -- used 0.00615376s (cpu); 0.00794971s (thread); 0s (gc) │ │ │ │ + -- used 0.00958423s (cpu); 0.0121729s (thread); 0s (gc) │ │ │ │ + -- used 0.0172877s (cpu); 0.019031s (thread); 0s (gc) │ │ │ │ + -- used 0.034148s (cpu); 0.0372794s (thread); 0s (gc) │ │ │ │ + -- used 0.0605682s (cpu); 0.0630337s (thread); 0s (gc) │ │ │ │ + -- used 0.111349s (cpu); 0.113803s (thread); 0s (gc) │ │ │ │ + -- used 0.175416s (cpu); 0.175807s (thread); 0s (gc) │ │ │ │ + -- used 0.372784s (cpu); 0.305455s (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 8.06579s (cpu); 4.07492s (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 62.9315s (cpu); 58.5598s (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 0.666738s (cpu); 0.200353s (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 0.351504s (cpu); 0.127487s (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 8.06579s (cpu); 4.07492s (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 62.9315s (cpu); 58.5598s (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 8.06579s (cpu); 4.07492s (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 62.9315s (cpu); 58.5598s (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 0.666738s (cpu); 0.200353s (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 0.351504s (cpu); 0.127487s (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 0.666738s (cpu); 0.200353s (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 0.351504s (cpu); 0.127487s (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") -- .0618978s 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") -- .0618978s 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") -- .0618978s 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/SlackIdeals/example-output/_rehomogenize__Polynomial.out │ │ │ @@ -9,14 +9,14 @@ │ │ │ │ │ │ i3 : (Y, T) = setOnesForest X; │ │ │ │ │ │ i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)}; │ │ │ │ │ │ i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1) │ │ │ │ │ │ - 2 2 2 2 2 2 2 2 2 2 │ │ │ -o5 = x x x x x x + x x x x x x x x - x x x x x x │ │ │ - 1 4 6 7 10 11 1 2 3 4 5 8 10 11 2 3 6 7 9 12 │ │ │ + 2 2 2 2 2 2 2 2 2 2 │ │ │ +o5 = x x x x x x - x x x x x x + x x x x x x x x │ │ │ + 1 4 6 7 10 11 2 3 5 8 10 11 1 2 3 4 6 7 9 12 │ │ │ │ │ │ o5 : R │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Polynomial.html │ │ │ @@ -93,17 +93,17 @@ │ │ │ │ │ │ │ │ │ 
      i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)};
      │ │ │ │ │ │ │ │ │ 
      i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1)
│ │ │  
│ │ │ -      2 2     2  2                 2  2     2 2     2 2
│ │ │ -o5 = x x x x x  x   + x x x x x x x  x   - x x x x x x
│ │ │ -      1 4 6 7 10 11    1 2 3 4 5 8 10 11    2 3 6 7 9 12
│ │ │ +      2 2 2 2          2 2 2 2                  2 2
│ │ │ +o5 = x x x x x  x   - x x x x x  x   + x x x x x x x x
│ │ │ +      1 4 6 7 10 11    2 3 5 8 10 11    1 2 3 4 6 7 9 12
│ │ │  
│ │ │  o5 : R
      │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │ ├── html2text {} │ │ │ │ @@ -32,17 +32,17 @@ │ │ │ │ │ │ │ │ 6 5 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : (Y, T) = setOnesForest X; │ │ │ │ i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)}; │ │ │ │ i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1) │ │ │ │ │ │ │ │ - 2 2 2 2 2 2 2 2 2 2 │ │ │ │ -o5 = x x x x x x + x x x x x x x x - x x x x x x │ │ │ │ - 1 4 6 7 10 11 1 2 3 4 5 8 10 11 2 3 6 7 9 12 │ │ │ │ + 2 2 2 2 2 2 2 2 2 2 │ │ │ │ +o5 = x x x x x x - x x x x x x + x x x x x x x x │ │ │ │ + 1 4 6 7 10 11 2 3 5 8 10 11 1 2 3 4 6 7 9 12 │ │ │ │ │ │ │ │ o5 : R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_e_t_O_n_e_s_F_o_r_e_s_t -- sets to 1 variables in a symbolic slack matrix which │ │ │ │ corresponding to edges of a spanning forest │ │ │ │ * _s_l_a_c_k_I_d_e_a_l -- computes the slack ideal │ │ │ │ * _s_y_m_b_o_l_i_c_S_l_a_c_k_M_a_t_r_i_x -- computes the symbolic slack matrix │ │ ├── ./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 0.00360834s (cpu); 7.8412e-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 0.307007s (cpu); 0.123558s (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 0.362471s (cpu); 0.256241s (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 0.101523s (cpu); 0.104631s (thread); 0s (gc) │ │ │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula │ │ │ - -- used 0.289099s (cpu); 0.231488s (thread); 0s (gc) │ │ │ + -- used 0.373093s (cpu); 0.308601s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ - -- used 0.366071s (cpu); 0.319698s (thread); 0s (gc) │ │ │ + -- used 0.345105s (cpu); 0.331408s (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 0.233572s (cpu); 0.112439s (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 0.4986s (cpu); 0.444307s (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 2.57797s (cpu); 2.33692s (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 0.55827s (cpu); 0.5411s (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 0.103571s (cpu); 0.047193s (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 0.0876111s (cpu); 0.0424191s (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 0.00400136s (cpu); 0.00404202s (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 0.616291s (cpu); 0.499763s (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 0.00360834s (cpu); 7.8412e-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 0.307007s (cpu); 0.123558s (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 0.00360834s (cpu); 7.8412e-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 0.307007s (cpu); 0.123558s (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 0.362471s (cpu); 0.256241s (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 0.362471s (cpu); 0.256241s (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 0.101523s (cpu); 0.104631s (thread); 0s (gc) │ │ │ 
    i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.289099s (cpu); 0.231488s (thread); 0s (gc)
    │ │ │ + -- used 0.373093s (cpu); 0.308601s (thread); 0s (gc) │ │ │ 
    i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.366071s (cpu); 0.319698s (thread); 0s (gc)
    │ │ │ + -- used 0.345105s (cpu); 0.331408s (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 0.101523s (cpu); 0.104631s (thread); 0s (gc) │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay │ │ │ │ formula │ │ │ │ - -- used 0.289099s (cpu); 0.231488s (thread); 0s (gc) │ │ │ │ + -- used 0.373093s (cpu); 0.308601s (thread); 0s (gc) │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ │ - -- used 0.366071s (cpu); 0.319698s (thread); 0s (gc) │ │ │ │ + -- used 0.345105s (cpu); 0.331408s (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 0.233572s (cpu); 0.112439s (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 0.4986s (cpu); 0.444307s (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 0.233572s (cpu); 0.112439s (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 0.4986s (cpu); 0.444307s (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 2.57797s (cpu); 2.33692s (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 2.57797s (cpu); 2.33692s (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 0.55827s (cpu); 0.5411s (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 0.103571s (cpu); 0.047193s (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 0.0876111s (cpu); 0.0424191s (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 0.00400136s (cpu); 0.00404202s (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 0.616291s (cpu); 0.499763s (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 0.55827s (cpu); 0.5411s (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 0.103571s (cpu); 0.047193s (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 0.0876111s (cpu); 0.0424191s (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 0.00400136s (cpu); 0.00404202s (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 0.616291s (cpu); 0.499763s (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 0.00401347s (cpu); 0.00134179s (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 0.00120188s (cpu); 0.00132726s (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 0.00102551s (cpu); 0.00258454s (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 0.28537s (cpu); 0.226872s (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 0.675013s (cpu); 0.512287s (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 0.00401347s (cpu); 0.00134179s (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 0.00120188s (cpu); 0.00132726s (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 0.00102551s (cpu); 0.00258454s (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 0.00401347s (cpu); 0.00134179s (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 0.00120188s (cpu); 0.00132726s (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 0.00102551s (cpu); 0.00258454s (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 2.72105s (cpu); 1.10102s (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 2.97157s (cpu); 1.12949s (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 8.40911s (cpu); 4.65975s (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 0.28537s (cpu); 0.226872s (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 0.28537s (cpu); 0.226872s (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 0.675013s (cpu); 0.512287s (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 0.675013s (cpu); 0.512287s (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 2.72105s (cpu); 1.10102s (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 2.97157s (cpu); 1.12949s (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 8.40911s (cpu); 4.65975s (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 3.93395s (cpu); 2.11317s (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 0.459645s (cpu); 0.311982s (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 25.4423s (cpu); 8.18609s (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 0.443729s (cpu); 0.280252s (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 0.405323s (cpu); 0.144426s (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 1.10554s (cpu); 0.478733s (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 0.313203s (cpu); 0.224817s (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 0.522405s (cpu); 0.359082s (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 3.65605s (cpu); 2.73s (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 1.78245s (cpu); 0.81436s (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 0.00802813s (cpu); 0.00948857s (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 0.798328s (cpu); 0.204162s (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 1.76891s (cpu); 1.46315s (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 6.26604s (cpu); 3.29159s (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 5.5331s (cpu); 2.90206s (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 5.52583s (cpu); 2.83597s (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 1.2785s (cpu); 0.689822s (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 2.72105s (cpu); 1.10102s (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 2.97157s (cpu); 1.12949s (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 8.40911s (cpu); 4.65975s (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 3.93395s (cpu); 2.11317s (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 0.459645s (cpu); 0.311982s (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 3.93395s (cpu); 2.11317s (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 0.459645s (cpu); 0.311982s (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 25.4423s (cpu); 8.18609s (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 0.443729s (cpu); 0.280252s (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 25.4423s (cpu); 8.18609s (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 0.443729s (cpu); 0.280252s (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 0.405323s (cpu); 0.144426s (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 0.405323s (cpu); 0.144426s (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 1.10554s (cpu); 0.478733s (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 1.10554s (cpu); 0.478733s (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 0.313203s (cpu); 0.224817s (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 0.313203s (cpu); 0.224817s (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 0.522405s (cpu); 0.359082s (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 0.522405s (cpu); 0.359082s (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 3.65605s (cpu); 2.73s (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 3.65605s (cpu); 2.73s (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 1.78245s (cpu); 0.81436s (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 1.78245s (cpu); 0.81436s (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 0.00802813s (cpu); 0.00948857s (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 0.798328s (cpu); 0.204162s (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 0.00802813s (cpu); 0.00948857s (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 0.798328s (cpu); 0.204162s (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 1.76891s (cpu); 1.46315s (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 6.26604s (cpu); 3.29159s (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 1.76891s (cpu); 1.46315s (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 6.26604s (cpu); 3.29159s (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 5.5331s (cpu); 2.90206s (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 5.5331s (cpu); 2.90206s (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 5.52583s (cpu); 2.83597s (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 5.52583s (cpu); 2.83597s (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 1.2785s (cpu); 0.689822s (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 1.2785s (cpu); 0.689822s (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/_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/_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 0.253338s (cpu); 0.19718s (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 0.110227s (cpu); 0.047241s (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 0.253338s (cpu); 0.19718s (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 0.110227s (cpu); 0.047241s (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 0.253338s (cpu); 0.19718s (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 0.110227s (cpu); 0.047241s (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 0.439016s (cpu); 0.188244s (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 0.439016s (cpu); 0.188244s (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 0.439016s (cpu); 0.188244s (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 0.869385s (cpu); 0.68154s (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 2.57018s (cpu); 2.16715s (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 0.00397158s (cpu); 0.0020571s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00365471s (cpu); 0.00378473s (thread); 0s (gc)
│ │ │ + -- used 0.00198068s (cpu); 0.00483596s (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 0.028017s (cpu); 0.0288697s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.15989s (cpu); 1.16923s (thread); 0s (gc)
│ │ │ + -- used 2.35279s (cpu); 1.32899s (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 0.163091s (cpu); 0.0971248s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.0700561s (cpu); 0.0715188s (thread); 0s (gc)
│ │ │ + -- used 0.0869725s (cpu); 0.0874147s (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 1.00687s (cpu); 0.808492s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.352773s (cpu); 0.235218s (thread); 0s (gc)
│ │ │ + -- used 0.384247s (cpu); 0.261362s (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 0.126893s (cpu); 0.073038s (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 0.230775s (cpu); 0.170198s (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 0.313728s (cpu); 0.184837s (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 0.566972s (cpu); 0.371041s (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 0.869385s (cpu); 0.68154s (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 2.57018s (cpu); 2.16715s (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 0.869385s (cpu); 0.68154s (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 2.57018s (cpu); 2.16715s (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 0.00397158s (cpu); 0.0020571s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    │ │ │ │ │ │ │ │ │ 
    i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00365471s (cpu); 0.00378473s (thread); 0s (gc)
│ │ │ + -- used 0.00198068s (cpu); 0.00483596s (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 0.00397158s (cpu); 0.0020571s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : time isCohenMacaulay(R, AtOrigin => true) │ │ │ │ - -- used 0.00365471s (cpu); 0.00378473s (thread); 0s (gc) │ │ │ │ + -- used 0.00198068s (cpu); 0.00483596s (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 0.028017s (cpu); 0.0288697s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
    │ │ │ 
    i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.15989s (cpu); 1.16923s (thread); 0s (gc)
│ │ │ + -- used 2.35279s (cpu); 1.32899s (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 0.163091s (cpu); 0.0971248s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
    │ │ │ 
    i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.0700561s (cpu); 0.0715188s (thread); 0s (gc)
│ │ │ + -- used 0.0869725s (cpu); 0.0874147s (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 1.00687s (cpu); 0.808492s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
    │ │ │ │ │ │ │ │ │ 
    i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.352773s (cpu); 0.235218s (thread); 0s (gc)
│ │ │ + -- used 0.384247s (cpu); 0.261362s (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 0.028017s (cpu); 0.0288697s (thread); 0s (gc) │ │ │ │ │ │ │ │ o20 = true │ │ │ │ i21 : time isFInjective(R, CanonicalStrategy => null) │ │ │ │ - -- used 2.15989s (cpu); 1.16923s (thread); 0s (gc) │ │ │ │ + -- used 2.35279s (cpu); 1.32899s (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 0.163091s (cpu); 0.0971248s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = false │ │ │ │ i24 : time isFInjective(R, AtOrigin => true) │ │ │ │ - -- used 0.0700561s (cpu); 0.0715188s (thread); 0s (gc) │ │ │ │ + -- used 0.0869725s (cpu); 0.0874147s (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 1.00687s (cpu); 0.808492s (thread); 0s (gc) │ │ │ │ │ │ │ │ o29 = false │ │ │ │ i30 : time isFInjective(R, AssumeCM => true) │ │ │ │ - -- used 0.352773s (cpu); 0.235218s (thread); 0s (gc) │ │ │ │ + -- used 0.384247s (cpu); 0.261362s (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 0.126893s (cpu); 0.073038s (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 0.230775s (cpu); 0.170198s (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 0.126893s (cpu); 0.073038s (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 0.230775s (cpu); 0.170198s (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 0.313728s (cpu); 0.184837s (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 0.566972s (cpu); 0.371041s (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 0.313728s (cpu); 0.184837s (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 0.566972s (cpu); 0.371041s (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/___Threaded__G__B.out │ │ │ @@ -22,46 +22,31 @@ │ │ │ │ │ │ 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 │ │ │ +o4 = LineageTable{((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 │ │ │ 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 +97,23 @@ │ │ │ 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{((1, 2), 1) => 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 +135,23 @@ │ │ │ 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{((1, 2), 1) => 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 │ │ │ @@ -231,18 +206,18 @@ │ │ │ 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) │ │ │ +Adding the following remainder to GB: Adding the following remainder to GB: -1 from lineage Scheduling task for lineage (1,2) │ │ │ +((0,2),0) │ │ │ Scheduling task for lineage ((0,2),1) │ │ │ +-a^3*b-a*b^2*d+c^2 from lineage (0,1) │ │ │ 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) │ │ │ Found a unit in the Groebner basis; reducing now. │ │ │ │ │ │ o14 = LineageTable{(0, 1) => null} │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ @@ -6,16 +6,14 @@ │ │ │ │ │ │ i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2" │ │ │ │ │ │ 3 │ │ │ o3 = LineageTable{((0, 2), 0) => -c } │ │ │ 2 │ │ │ ((1, 2), 0) => -c │ │ │ - 2 │ │ │ - (0, 1) => a c │ │ │ 2 │ │ │ (0, 2) => b*c │ │ │ (1, 2) => -a*c │ │ │ 2 │ │ │ 0 => a*b*c + c │ │ │ 3 2 │ │ │ 1 => - b c + a*b + a*c │ │ │ @@ -25,15 +23,14 @@ │ │ │ o3 : LineageTable │ │ │ │ │ │ i4 : reduce T │ │ │ │ │ │ o4 = LineageTable{((0, 2), 0) => null} │ │ │ 2 │ │ │ ((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/_tgb.out │ │ │ @@ -6,46 +6,34 @@ │ │ │ │ │ │ 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 │ │ │ - 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 │ │ │ + 4 4 2 6 │ │ │ +o4 = LineageTable{((0, 1), 2) => 9y z + 23y z } │ │ │ + 2 6 2 5 │ │ │ + ((0, 1), 3) => 23y z + 48y z │ │ │ + 2 4 │ │ │ + ((0, 3), (0, 2)) => -4y z │ │ │ + 2 4 │ │ │ + ((0, 3), 2) => 5y 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 │ │ │ - 3 4 2 4 │ │ │ - (2, 3) => 7y z - 9y z │ │ │ + 3 5 2 4 │ │ │ + (0, 2) => - 24y z + 9y z │ │ │ + 5 3 4 │ │ │ + (0, 3) => 28y z - 24y z │ │ │ + 3 4 2 7 │ │ │ + (1, 2) => 21y z - 14y z │ │ │ + 2 6 2 5 │ │ │ + (1, 3) => - 14y z - 38y z │ │ │ + 2 5 2 4 │ │ │ + (2, 3) => 44y z - 9y z │ │ │ 2 │ │ │ 0 => 2x + 10y z │ │ │ 2 3 │ │ │ 1 => 8x y + 10x*y*z │ │ │ 3 2 3 │ │ │ 2 => 5x*y z + 9x*z │ │ │ 3 3 │ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ @@ -81,16 +81,14 @@ │ │ │ │ │ │ 
    i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │                                     3
│ │ │  o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │ -                             2
│ │ │ -                  (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │                            3       2
│ │ │                    1 => - b c + a*b  + a*c
│ │ │ @@ -101,15 +99,14 @@
│ │ │            
│ │ │            
│ │ │                
    i4 : reduce T
│ │ │  
│ │ │  o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │ -                  (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,16 +25,14 @@
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │                                     3
│ │ │ │  o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │ -                             2
│ │ │ │ -                  (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │                                  2
│ │ │ │                    0 => a*b*c + c
│ │ │ │                            3       2
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │ @@ -43,15 +41,14 @@
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : reduce T
│ │ │ │  
│ │ │ │  o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │                    ((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/_tgb.html
│ │ │ @@ -92,46 +92,34 @@
│ │ │            
│ │ │            
│ │ │                
    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
│ │ │ -                                   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
│ │ │ +                                   4 4      2 6
│ │ │ +o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z  }
│ │ │ +                                    2 6      2 5
│ │ │ +                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ +                                         2 4
│ │ │ +                  ((0, 3), (0, 2)) => -4y z
│ │ │ +                                   2 4
│ │ │ +                  ((0, 3), 2) => 5y 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
│ │ │ -                              3 4     2 4
│ │ │ -                  (2, 3) => 7y z  - 9y z
│ │ │ +                                 3 5     2 4
│ │ │ +                  (0, 2) => - 24y z  + 9y z
│ │ │ +                               5       3 4
│ │ │ +                  (0, 3) => 28y z - 24y z
│ │ │ +                               3 4      2 7
│ │ │ +                  (1, 2) => 21y z  - 14y z
│ │ │ +                                 2 6      2 5
│ │ │ +                  (1, 3) => - 14y z  - 38y z
│ │ │ +                               2 5     2 4
│ │ │ +                  (2, 3) => 44y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │                    2 => 5x*y z  + 9x*z
│ │ │                             3         3
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,46 +27,34 @@
│ │ │ │  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
│ │ │ │ -                                   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
│ │ │ │ +                                   4 4      2 6
│ │ │ │ +o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z  }
│ │ │ │ +                                    2 6      2 5
│ │ │ │ +                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ │ +                                         2 4
│ │ │ │ +                  ((0, 3), (0, 2)) => -4y z
│ │ │ │ +                                   2 4
│ │ │ │ +                  ((0, 3), 2) => 5y 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
│ │ │ │ -                              3 4     2 4
│ │ │ │ -                  (2, 3) => 7y z  - 9y z
│ │ │ │ +                                 3 5     2 4
│ │ │ │ +                  (0, 2) => - 24y z  + 9y z
│ │ │ │ +                               5       3 4
│ │ │ │ +                  (0, 3) => 28y z - 24y z
│ │ │ │ +                               3 4      2 7
│ │ │ │ +                  (1, 2) => 21y z  - 14y z
│ │ │ │ +                                 2 6      2 5
│ │ │ │ +                  (1, 3) => - 14y z  - 38y z
│ │ │ │ +                               2 5     2 4
│ │ │ │ +                  (2, 3) => 44y z  - 9y z
│ │ │ │                                 2
│ │ │ │                    0 => 2x + 10y z
│ │ │ │                           2           3
│ │ │ │                    1 => 8x y + 10x*y*z
│ │ │ │                             3 2       3
│ │ │ │                    2 => 5x*y z  + 9x*z
│ │ │ │                             3         3
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html
│ │ │ @@ -75,46 +75,31 @@
│ │ │                
    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
│ │ │ +o4 = LineageTable{((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
│ │ │                                          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 +172,23 @@
│ │ │            
    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{((1, 2), 1) => 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 +211,23 @@
│ │ │                          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{((1, 2), 1) => 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
│ │ │ @@ -323,18 +298,18 @@
│ │ │  
    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)
│ │ │ +Adding the following remainder to GB: Adding the following remainder to GB: -1 from lineage Scheduling task for lineage (1,2)
│ │ │ +((0,2),0)
│ │ │  Scheduling task for lineage ((0,2),1)
│ │ │ +-a^3*b-a*b^2*d+c^2 from lineage (0,1)
│ │ │  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)
│ │ │  Found a unit in the Groebner basis; reducing now.
│ │ │  
│ │ │  o14 = LineageTable{(0, 1) => null}
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,46 +39,31 @@
│ │ │ │  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
│ │ │ │ +o4 = LineageTable{((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
│ │ │ │                                          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 +125,23 @@
│ │ │ │  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{((1, 2), 1) => 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 +162,23 @@
│ │ │ │                                 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{((1, 2), 1) => 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
│ │ │ │ @@ -263,18 +238,18 @@
│ │ │ │  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)
│ │ │ │ +1 from lineage Scheduling task for lineage (1,2)
│ │ │ │ +((0,2),0)
│ │ │ │  Scheduling task for lineage ((0,2),1)
│ │ │ │ +-a^3*b-a*b^2*d+c^2 from lineage (0,1)
│ │ │ │  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)
│ │ │ │  Found a unit in the Groebner basis; reducing now.
│ │ │ │  
│ │ │ │  o14 = LineageTable{(0, 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 1.30964s (cpu); 0.859452s (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 4.86552s (cpu); 3.16619s (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 1.30964s (cpu); 0.859452s (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 4.86552s (cpu); 3.16619s (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 1.30964s (cpu); 0.859452s (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 4.86552s (cpu); 3.16619s (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 │ │ │ + -- .115202s 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 │ │ │ + -- 1.02879s 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
      │ │ │ + -- .115202s 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
      │ │ │ + -- 1.02879s 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 │ │ │ │ + -- .115202s 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 │ │ │ │ + -- 1.02879s 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 1.09148s (cpu); 0.723004s (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 0.759405s (cpu); 0.49079s (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 1.09148s (cpu); 0.723004s (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 0.759405s (cpu); 0.49079s (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 1.09148s (cpu); 0.723004s (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 0.759405s (cpu); 0.49079s (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
│ │ │  IGJldHdlZW4gTWFjYXVsYXkyIHNlcnZlciBhbmQgYnJvd3NlciAiLCAibGluZW51bSIgPT4gMjA3
│ │ ├── ./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__Element_cm__Weyl__Group__Element_rp.out
│ │ │ @@ -20,26 +20,26 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o3 : WeylGroupElement
│ │ │  
│ │ │  i4 : myInterval=intervalBruhat(w1,w2)
│ │ │  
│ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |                                            | -1 |        | -1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |                                            |  2 |        |  2 |                                              |  3 |
│ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {}}}}
│ │ │ +                                                          | -2 |        |  2 |       | -1 |                                              | -1 |        | -1 |                                            | -3 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        | -1 |       |  2 |                                              |  2 |        |  2 |                                            |  1 |        |  1 |                                              |  3 |
│ │ │  
│ │ │  o4 : HasseDiagram
│ │ │  
│ │ │  i5 : myInterval#1
│ │ │  
│ │ │ -o5 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}}},
│ │ │ -                                             | -3 |        |  1 |    
│ │ │ -                                             |  1 |        |  1 |    
│ │ │ +o5 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}},
│ │ │ +                                             | -1 |        | -1 |    
│ │ │ +                                             |  2 |        |  2 |    
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}}}
│ │ │ -                                            | -1 |        | -1 |
│ │ │ -                                            |  2 |        |  2 |
│ │ │ +     {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}}}}
│ │ │ +                                            | -3 |        |  1 |
│ │ │ +                                            |  1 |        |  1 |
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_interval__Bruhat_lp__Weyl__Group__Element_cm__Weyl__Group__Element_rp.html
│ │ │ @@ -96,35 +96,35 @@
│ │ │                                             |  1 |
│ │ │  
│ │ │  o3 : WeylGroupElement
      │ │ │ 
      i4 : myInterval=intervalBruhat(w1,w2)
│ │ │  
│ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |                                            | -1 |        | -1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |                                            |  2 |        |  2 |                                              |  3 |
│ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {}}}}
│ │ │ +                                                          | -2 |        |  2 |       | -1 |                                              | -1 |        | -1 |                                            | -3 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        | -1 |       |  2 |                                              |  2 |        |  2 |                                            |  1 |        |  1 |                                              |  3 |
│ │ │  
│ │ │  o4 : HasseDiagram
      │ │ │
      │ │ │
      │ │ │

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

      │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ 
      i5 : myInterval#1
│ │ │  
│ │ │ -o5 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}}},
│ │ │ -                                             | -3 |        |  1 |    
│ │ │ -                                             |  1 |        |  1 |    
│ │ │ +o5 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}},
│ │ │ +                                             | -1 |        | -1 |    
│ │ │ +                                             |  2 |        |  2 |    
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}}}
│ │ │ -                                            | -1 |        | -1 |
│ │ │ -                                            |  2 |        |  2 |
│ │ │ +     {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}}}}
│ │ │ +                                            | -3 |        |  1 |
│ │ │ +                                            |  1 |        |  1 |
│ │ │  
│ │ │  o5 : List
      │ │ │
      │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -37,35 +37,35 @@ │ │ │ │ o3 = WeylGroupElement{RootSystem{...8...}, | -1 |} │ │ │ │ | -2 | │ │ │ │ | 1 | │ │ │ │ │ │ │ │ o3 : WeylGroupElement │ │ │ │ i4 : myInterval=intervalBruhat(w1,w2) │ │ │ │ │ │ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | 0 │ │ │ │ -|}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | - │ │ │ │ -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 |}}}}, { │ │ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | - │ │ │ │ +1 |}, {1, | 0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | │ │ │ │ +0 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | -1 |}}}}, { │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | -1 |}, {}}}} │ │ │ │ - | -2 | | -1 | │ │ │ │ -| 2 | | -3 | | 1 | │ │ │ │ -| -1 | | -1 | | -1 | │ │ │ │ - | 1 | | 2 | │ │ │ │ -| -1 | | 1 | | 1 | │ │ │ │ -| 2 | | 2 | | 3 | │ │ │ │ + | -2 | | 2 | │ │ │ │ +| -1 | | -1 | | -1 | │ │ │ │ +| -3 | | 1 | | -1 | │ │ │ │ + | 1 | | -1 | │ │ │ │ +| 2 | | 2 | | 2 | │ │ │ │ +| 1 | | 1 | | 3 | │ │ │ │ │ │ │ │ o4 : HasseDiagram │ │ │ │ Each row of the Hasse diagram contains the elements of a certain length │ │ │ │ together with their links to the next row. │ │ │ │ i5 : myInterval#1 │ │ │ │ │ │ │ │ -o5 = {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | -1 |}}}, │ │ │ │ - | -3 | | 1 | │ │ │ │ - | 1 | | 1 | │ │ │ │ +o5 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 |}}}, │ │ │ │ + | -1 | | -1 | │ │ │ │ + | 2 | | 2 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 |}}}} │ │ │ │ - | -1 | | -1 | │ │ │ │ - | 2 | | 2 | │ │ │ │ + {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | -1 |}}}} │ │ │ │ + | -3 | | 1 | │ │ │ │ + | 1 | | 1 | │ │ │ │ │ │ │ │ o5 : 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_E_l_e_m_e_n_t_,_W_e_y_l_G_r_o_u_p_E_l_e_m_e_n_t_) -- elements between two │ │ │ │ given ones for the Bruhat order on a Weyl group │ │ ├── ./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 1.91646s (cpu); 1.11515s (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 6.24697s (cpu); 2.9995s (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 7.86239s (cpu); 3.41731s (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 1.91646s (cpu); 1.11515s (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 6.24697s (cpu); 2.9995s (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 7.86239s (cpu); 3.41731s (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 1.91646s (cpu); 1.11515s (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 6.24697s (cpu); 2.9995s (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 7.86239s (cpu); 3.41731s (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/M2-60345-0/172
│ │ │  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/M2-60345-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-60345-0/174
│ │ │  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/M2-60345-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-60345-0/176
│ │ │  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/M2-60345-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-60345-0/178
│ │ │  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/M2-60345-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-60345-0/180
│ │ │  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/M2-60345-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-60345-0/182
│ │ │  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/M2-60345-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-60345-0/184
│ │ │  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/M2-60345-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-60345-0/186
│ │ │  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/M2-60345-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-60345-0/188
│ │ │  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/M2-60345-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-60345-0/190
│ │ │  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/M2-60345-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-60345-0/192
│ │ │  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/M2-60345-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-60345-0/194
│ │ │  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/M2-60345-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-60345-0/196
│ │ │  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/M2-60345-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-60345-0/198
│ │ │  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/M2-60345-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-60345-0/200
│ │ │  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/M2-60345-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-60345-0/202
│ │ │  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/M2-60345-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-60345-0/204
│ │ │  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/M2-60345-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-60345-0/206
│ │ │  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/M2-60345-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-60345-0/208
│ │ │  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/M2-60345-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-60345-0/210
│ │ │  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/M2-60345-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-60345-0/212
│ │ │  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/M2-60345-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-60345-0/214
│ │ │  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/M2-60345-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-60345-0/216
│ │ │  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/M2-60345-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-60345-0/218
│ │ │  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/M2-60345-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-60345-0/220
│ │ │  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/M2-60345-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-60345-0/222
│ │ │  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/M2-60345-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-60345-0/224
│ │ │  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/M2-60345-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-60345-0/226
│ │ │  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/M2-60345-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-60345-0/228
│ │ │  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/M2-60345-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-60345-0/230
│ │ │  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/M2-60345-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-60345-0/232
│ │ │  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/M2-60345-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-60345-0/234
│ │ │  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/M2-60345-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-60345-0/236
│ │ │  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/M2-60345-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-60345-0/238
│ │ │  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/M2-60345-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-60345-0/240
│ │ │  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/M2-60345-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-60345-0/242
│ │ │  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/M2-60345-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-60345-0/244
│ │ │  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/M2-60345-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-60345-0/246
│ │ │  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/M2-60345-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-60345-0/248
│ │ │  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/M2-60345-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-60345-0/250
│ │ │  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/M2-60345-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-60345-0/252
│ │ │  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/M2-60345-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-60345-0/254
│ │ │  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/M2-60345-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-60345-0/256
│ │ │  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/M2-60345-0/256
│ │ │  
│ │ │  i6 : QQ[x,y];
│ │ │  
│ │ │  i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-37469-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-60345-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-37469-0/258
│ │ │ +using temporary file /tmp/M2-60345-0/258
│ │ │  
│ │ │  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/M2-60345-0/172
│ │ │  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/M2-60345-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-60345-0/174
│ │ │  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/M2-60345-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-60345-0/176
│ │ │  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/M2-60345-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-60345-0/178
│ │ │  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/M2-60345-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-60345-0/180
│ │ │  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/M2-60345-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-60345-0/182
│ │ │  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/M2-60345-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-60345-0/184
│ │ │  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/M2-60345-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-60345-0/186
│ │ │  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/M2-60345-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-60345-0/188
│ │ │  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/M2-60345-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-60345-0/190
│ │ │  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/M2-60345-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-60345-0/192
│ │ │  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/M2-60345-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-60345-0/194
│ │ │  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/M2-60345-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-60345-0/196
│ │ │  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/M2-60345-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-60345-0/198
│ │ │  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/M2-60345-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-60345-0/200
│ │ │  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/M2-60345-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-60345-0/202
│ │ │  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/M2-60345-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-60345-0/204
│ │ │  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/M2-60345-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-60345-0/206
│ │ │  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/M2-60345-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-60345-0/208
│ │ │  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/M2-60345-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-60345-0/210
│ │ │  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/M2-60345-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-60345-0/212
│ │ │  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/M2-60345-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-60345-0/214
│ │ │  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/M2-60345-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-60345-0/216
│ │ │  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/M2-60345-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-60345-0/218
│ │ │  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/M2-60345-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-60345-0/220
│ │ │  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/M2-60345-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-60345-0/222
│ │ │  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/M2-60345-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-60345-0/224
│ │ │  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/M2-60345-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-60345-0/226
│ │ │  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/M2-60345-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-60345-0/228
│ │ │  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/M2-60345-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-60345-0/230
│ │ │  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/M2-60345-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-60345-0/232
│ │ │  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/M2-60345-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-60345-0/234
│ │ │  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/M2-60345-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-60345-0/236
│ │ │  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/M2-60345-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-60345-0/238
│ │ │  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/M2-60345-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-60345-0/240
│ │ │  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/M2-60345-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-60345-0/242
│ │ │  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/M2-60345-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-60345-0/244
│ │ │  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/M2-60345-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-60345-0/246
│ │ │  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/M2-60345-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-60345-0/248
│ │ │  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/M2-60345-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-60345-0/250
│ │ │  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/M2-60345-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-60345-0/252
│ │ │  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/M2-60345-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-60345-0/254
│ │ │  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/M2-60345-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-60345-0/256
│ │ │  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/M2-60345-0/256
│ │ │      		          
                  
    i6 : QQ[x,y];
    
│ │ │      		          
                  
    i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-37469-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-60345-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-37469-0/258
    
│ │ │ +using temporary file /tmp/M2-60345-0/258
│ │ │      		          
    
│ │ │          
    
│ │ │            
    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/M2-60345-0/172
│ │ │ │  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/M2-60345-0/172
│ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-60345-0/174
│ │ │ │  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/M2-60345-0/174
│ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-60345-0/176
│ │ │ │  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/M2-60345-0/176
│ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-60345-0/178
│ │ │ │  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/M2-60345-0/178
│ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-60345-0/180
│ │ │ │  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/M2-60345-0/180
│ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-60345-0/182
│ │ │ │  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/M2-60345-0/182
│ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-60345-0/184
│ │ │ │  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/M2-60345-0/184
│ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-60345-0/186
│ │ │ │  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/M2-60345-0/186
│ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-60345-0/188
│ │ │ │  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/M2-60345-0/188
│ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-60345-0/190
│ │ │ │  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/M2-60345-0/190
│ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-60345-0/192
│ │ │ │  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/M2-60345-0/192
│ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-60345-0/194
│ │ │ │  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/M2-60345-0/194
│ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-60345-0/196
│ │ │ │  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/M2-60345-0/196
│ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-60345-0/198
│ │ │ │  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/M2-60345-0/198
│ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-60345-0/200
│ │ │ │  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/M2-60345-0/200
│ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-60345-0/202
│ │ │ │  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/M2-60345-0/202
│ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-60345-0/204
│ │ │ │  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/M2-60345-0/204
│ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-60345-0/206
│ │ │ │  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/M2-60345-0/206
│ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-60345-0/208
│ │ │ │  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/M2-60345-0/208
│ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-60345-0/210
│ │ │ │  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/M2-60345-0/210
│ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-60345-0/212
│ │ │ │  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/M2-60345-0/212
│ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-60345-0/214
│ │ │ │  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/M2-60345-0/214
│ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-60345-0/216
│ │ │ │  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/M2-60345-0/216
│ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-60345-0/218
│ │ │ │  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/M2-60345-0/218
│ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-60345-0/220
│ │ │ │  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/M2-60345-0/220
│ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-60345-0/222
│ │ │ │  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/M2-60345-0/222
│ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-60345-0/224
│ │ │ │  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/M2-60345-0/224
│ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-60345-0/226
│ │ │ │  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/M2-60345-0/226
│ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-60345-0/228
│ │ │ │  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/M2-60345-0/228
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-60345-0/230
│ │ │ │  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/M2-60345-0/230
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-60345-0/232
│ │ │ │  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/M2-60345-0/232
│ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-60345-0/234
│ │ │ │  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/M2-60345-0/234
│ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-60345-0/236
│ │ │ │  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/M2-60345-0/236
│ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-60345-0/238
│ │ │ │  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/M2-60345-0/238
│ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-60345-0/240
│ │ │ │  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/M2-60345-0/240
│ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-60345-0/242
│ │ │ │  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/M2-60345-0/242
│ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-60345-0/244
│ │ │ │  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/M2-60345-0/244
│ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-60345-0/246
│ │ │ │  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/M2-60345-0/246
│ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-60345-0/248
│ │ │ │  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/M2-60345-0/248
│ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-60345-0/250
│ │ │ │  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/M2-60345-0/250
│ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-60345-0/252
│ │ │ │  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/M2-60345-0/252
│ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-60345-0/254
│ │ │ │  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/M2-60345-0/254
│ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-60345-0/256
│ │ │ │  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/M2-60345-0/256
│ │ │ │  i6 : QQ[x,y];
│ │ │ │  i7 : gfan {x,y};
│ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-37469-0/258
│ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-60345-0/258
│ │ │ │  Q[x1,x2]
│ │ │ │  {{
│ │ │ │  x2,
│ │ │ │  x1}
│ │ │ │  }
│ │ │ │ -using temporary file /tmp/M2-37469-0/258
│ │ │ │ +using temporary file /tmp/M2-60345-0/258
│ │ │ │  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 2031 2e34 3539 3034 7320  |.| -- 1.45904s 
│ │ │ │  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                  
│ │ │ │ @@ -6090,15 +6090,15 @@
│ │ │ │  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 
│ │ │ │ +00017d00: 2e37 3136 3331 7320 656c 6170 7365 6420  .71631s 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 7b31          0 => {{1
│ │ │ │ +00006fd0: 7d2c 207b 327d 7d20 7c0a 7c20 2020 2020  }, {2}} |.|     
│ │ │ │  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: 7b31 2c20 327d 7d20 2020 7c0a 7c20 2020  {1, 2}}   |.|   
│ │ │ │  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  +---------------
│ │ │ │ @@ -1850,18 +1850,18 @@
│ │ │ │  00007390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000073a0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │  000073b0: 203d 2041 6273 7472 6163 7453 696d 706c   = AbstractSimpl
│ │ │ │  000073c0: 6963 6961 6c43 6f6d 706c 6578 7b2d 3120  icialComplex{-1 
│ │ │ │  000073d0: 3d3e 207b 7b7d 7d20 2020 2020 7d7c 0a7c  => {{}}     }|.|
│ │ │ │  000073e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000073f0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00007400: 203d 3e20 7b7b 337d 2c20 7b36 7d7d 207c   => {{3}, {6}} |
│ │ │ │ +00007400: 203d 3e20 7b7b 317d 2c20 7b36 7d7d 207c   => {{1}, {6}} |
│ │ │ │  00007410: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00007420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007430: 2031 203d 3e20 7b7b 332c 2036 7d7d 2020   1 => {{3, 6}}  
│ │ │ │ +00007430: 2031 203d 3e20 7b7b 312c 2036 7d7d 2020   1 => {{1, 6}}  
│ │ │ │  00007440: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00007450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007470: 2020 207c 0a7c 6f35 203a 2041 6273 7472     |.|o5 : Abstr
│ │ │ │  00007480: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │  00007490: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │  000074a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ @@ -1941,22 +1941,22 @@
│ │ │ │  00007940: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00007950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007960: 2020 2020 3020 3d3e 207b 7b31 7d2c 207b      0 => {{1}, {
│ │ │ │  00007970: 327d 2c20 7b33 7d2c 207b 347d 2c20 7b35  2}, {3}, {4}, {5
│ │ │ │  00007980: 7d2c 207b 367d 7d20 2020 2020 2020 2020  }, {6}}         
│ │ │ │  00007990: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000079a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000079b0: 2020 2020 3120 3d3e 207b 7b31 2c20 327d      1 => {{1, 2}
│ │ │ │ -000079c0: 2c20 7b31 2c20 357d 2c20 7b32 2c20 337d  , {1, 5}, {2, 3}
│ │ │ │ +000079b0: 2020 2020 3120 3d3e 207b 7b31 2c20 337d      1 => {{1, 3}
│ │ │ │ +000079c0: 2c20 7b31 2c20 347d 2c20 7b31 2c20 367d  , {1, 4}, {1, 6}
│ │ │ │  000079d0: 2c20 7b32 2c20 347d 2c20 7b32 2c20 357d  , {2, 4}, {2, 5}
│ │ │ │  000079e0: 2c20 7c0a 7c20 2020 2020 2020 2020 2020  , |.|           
│ │ │ │  000079f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007a00: 2020 2020 3220 3d3e 207b 7b31 2c20 322c      2 => {{1, 2,
│ │ │ │ -00007a10: 2035 7d2c 207b 322c 2033 2c20 347d 2c20   5}, {2, 3, 4}, 
│ │ │ │ -00007a20: 7b33 2c20 352c 2036 7d7d 2020 2020 2020  {3, 5, 6}}      
│ │ │ │ +00007a00: 2020 2020 3220 3d3e 207b 7b31 2c20 332c      2 => {{1, 3,
│ │ │ │ +00007a10: 2036 7d2c 207b 312c 2034 2c20 367d 2c20   6}, {1, 4, 6}, 
│ │ │ │ +00007a20: 7b32 2c20 342c 2035 7d7d 2020 2020 2020  {2, 4, 5}}      
│ │ │ │  00007a30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00007a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007a80: 2020 7c0a 7c6f 3820 3a20 4162 7374 7261    |.|o8 : Abstra
│ │ │ │  00007a90: 6374 5369 6d70 6c69 6369 616c 436f 6d70  ctSimplicialComp
│ │ │ │ @@ -1965,26 +1965,26 @@
│ │ │ │  00007ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007ad0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  00007ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007b20: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ -00007b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007b40: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +00007b30: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ +00007b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007b70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00007b80: 2020 2020 2020 2020 2020 2020 2020 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}}            
│ │ │ │ +00007bc0: 2020 7c0a 7c7b 332c 2036 7d2c 207b 342c    |.|{3, 6}, {4,
│ │ │ │ +00007bd0: 2035 7d2c 207b 342c 2036 7d7d 2020 2020   5}, {4, 6}}    
│ │ │ │ +00007be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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  ----------------
│ │ │ │  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  ----------------
│ │ │ │ @@ -1995,30 +1995,30 @@
│ │ │ │  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                  
│ │ │ │  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           
│ │ │ │ +00007d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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            
│ │ │ │ +00007d50: 2020 7c0a 7c6f 3920 3d20 5a5a 2020 2020    |.|o9 = ZZ    
│ │ │ │ +00007d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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             
│ │ │ │ +00007e00: 2020 2020 2020 2020 2020 2020 2020 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                  
│ │ │ │  00007e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -2055,18 +2055,18 @@
│ │ │ │  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}} 
│ │ │ │ +000080d0: 3020 3d3e 207b 7b31 7d2c 207b 327d 7d20  0 => {{1}, {2}} 
│ │ │ │  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}}
│ │ │ │ +00008100: 2020 2031 203d 3e20 7b7b 312c 2032 7d7d     1 => {{1, 2}}
│ │ │ │  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           |.+----
│ │ │ │ @@ -2080,15 +2080,15 @@
│ │ │ │  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
│ │ │ │ +00008260: 2020 2020 2020 2020 3020 3d3e 207b 7b31          0 => {{1
│ │ │ │  00008270: 7d7d 2020 2020 2020 7c0a 7c20 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                |.
│ │ │ │ @@ -4592,841 +4592,818 @@
│ │ │ │  00011ef0: 7369 6d70 6c69 6369 616c 2063 6f6d 706c  simplicial compl
│ │ │ │  00011f00: 6578 2077 6974 6820 7665 7274 6963 6573  ex with vertices
│ │ │ │  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                  
│ │ │ │ +00011f60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00011f70: 7365 7452 616e 646f 6d53 6565 6428 6375  setRandomSeed(cu
│ │ │ │ +00011f80: 7272 656e 7454 696d 6528 2929 3b20 2020  rrentTime());   
│ │ │ │ +00011f90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00011fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011fc0: 2d2d 2d2d 2b0a 7c69 3220 3a20 4b20 3d20  ----+.|i2 : K = 
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│ │ │ │ +00011ff0: 3429 7c0a 7c20 2020 2020 2020 2020 2020  4)|.|           
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│ │ │ │ +00012010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012020: 7c0a 7c6f 3220 3d20 4162 7374 7261 6374  |.|o2 = Abstract
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│ │ │ │ -000120b0: 2020 2020 2020 2020 207d 7c0a 7c20 2020           }|.|   
│ │ │ │ -000120c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000120f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ -000121a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000121c0: 3a20 4162 7374 7261 6374 5369 6d70 6c69  : AbstractSimpli
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│ │ │ │ -000121f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
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│ │ │ │ -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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│ │ │ │ -000122a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000122b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00012070: 3020 3d3e 207b 7b34 7d7d 2020 7c0a 7c20  0 => {{4}}  |.| 
│ │ │ │ +00012080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000120a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +000120b0: 3a20 4162 7374 7261 6374 5369 6d70 6c69  : AbstractSimpli
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│ │ │ │ +000120e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012100: 2d2d 2d2d 2d2d 2b0a 0a43 7265 6174 6520  ------+..Create 
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│ │ │ │ +00012150: 746f 2072 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  to r...+--------
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│ │ │ │ +00012170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000121c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000121d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000121e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
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│ │ │ │ +00012230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00012290: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000122c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000122d0: 2020 2020 2031 203d 3e20 7b7b 342c 2036       1 => {{4, 6
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│ │ │ │ -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 :
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│ │ │ │ -00012380: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ -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
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│ │ │ │ -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},
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│ │ │ │ -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           |.|    
│ │ │ │ +00012300: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00012360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +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 2d2d  ----------------
│ │ │ │ +00012480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012490: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7365  ------+.|i5 : se
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│ │ │ │  000124d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000124f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00012540: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00012550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00012570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012580: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4372 6561  ---------+..Crea
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│ │ │ │ -00012660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00012690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000124e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
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│ │ │ │ +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 2d2d 2d2d  ----------------
│ │ │ │ +00012530: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20  ------+.|i6 : M 
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│ │ │ │ +00012550: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +00012560: 7828 362c 332c 3229 2020 2020 2020 2020  x(6,3,2)        
│ │ │ │ +00012570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012580: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00012590: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000125b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000125c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000125d0: 2020 2020 2020 7c0a 7c6f 3620 3d20 4162        |.|o6 = Ab
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│ │ │ │ +00012600: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
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│ │ │ │ +00012630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012640: 2020 2020 2020 2020 3020 3d3e 207b 7b31          0 => {{1
│ │ │ │ +00012650: 7d2c 207b 327d 2c20 7b33 7d2c 207b 347d  }, {2}, {3}, {4}
│ │ │ │ +00012660: 2c20 7b35 7d2c 207b 367d 7d20 2020 2020  , {5}, {6}}     
│ │ │ │ +00012670: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00012680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000126a0: 2c20 337d 2c20 7b31 2c20 357d 2c20 7b31  , 3}, {1, 5}, {1
│ │ │ │ +000126b0: 2c20 367d 2c20 7b32 2c20 337d 2c20 7b32  , 6}, {2, 3}, {2
│ │ │ │ +000126c0: 2c20 347d 2c20 7c0a 7c20 2020 2020 2020  , 4}, |.|       
│ │ │ │  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      |.+---------
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│ │ │ │ -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      |.|         
│ │ │ │ +000126e0: 2020 2020 2020 2020 3220 3d3e 207b 7b31          2 => {{1
│ │ │ │ +000126f0: 2c20 332c 2035 7d2c 207b 312c 2033 2c20  , 3, 5}, {1, 3, 
│ │ │ │ +00012700: 367d 2c20 7b32 2c20 332c 2034 7d7d 2020  6}, {2, 3, 4}}  
│ │ │ │ +00012710: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00012720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012760: 2020 2020 2020 7c0a 7c6f 3620 3a20 4162        |.|o6 : Ab
│ │ │ │ +00012770: 7374 7261 6374 5369 6d70 6c69 6369 616c  stractSimplicial
│ │ │ │ +00012780: 436f 6d70 6c65 7820 2020 2020 2020 2020  Complex         
│ │ │ │ +00012790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000127b0: 2020 2020 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 => {{}}
│ │ │ │ +000127b0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +000127c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000127d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000127e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000127f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012800: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  00012810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012830: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00012820: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00012830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012850: 2020 2020 2020 3020 3d3e 207b 7b31 7d2c        0 => {{1},
│ │ │ │ -00012860: 207b 327d 2c20 7b33 7d2c 207b 347d 2c20   {2}, {3}, {4}, 
│ │ │ │ -00012870: 7b35 7d2c 207b 367d 7d20 2020 2020 2020  {5}, {6}}       
│ │ │ │ -00012880: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00012850: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00012860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012870: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00012890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000128a0: 2020 2020 2020 3120 3d3e 207b 7b31 2c20        1 => {{1, 
│ │ │ │ -000128b0: 337d 2c20 7b31 2c20 357d 2c20 7b32 2c20  3}, {1, 5}, {2, 
│ │ │ │ -000128c0: 347d 2c20 7b32 2c20 357d 2c20 7b33 2c20  4}, {2, 5}, {3, 
│ │ │ │ -000128d0: 357d 2c20 7c0a 7c20 2020 2020 2020 2020  5}, |.|         
│ │ │ │ +000128a0: 2020 2020 2020 7c0a 7c7b 332c 2034 7d2c        |.|{3, 4},
│ │ │ │ +000128b0: 207b 332c 2035 7d2c 207b 332c 2036 7d7d   {3, 5}, {3, 6}}
│ │ │ │ +000128c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000128d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000128e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000128f0: 2020 2020 2020 3220 3d3e 207b 7b31 2c20        2 => {{1, 
│ │ │ │ -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                  
│ │ │ │ -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           
│ │ │ │ -000129a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000129b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000129c0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ -000129d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000129e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000129f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012a10: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ -00012a20: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ -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      |.|         
│ │ │ │ -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  .|              
│ │ │ │ -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   |.|            
│ │ │ │ -00013110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013130: 0a7c 6f33 203d 2041 6273 7472 6163 7453  .|o3 = AbstractS
│ │ │ │ -00013140: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ -00013150: 7b2d 3120 3d3e 207b 7b7d 7d7d 207c 0a7c  {-1 => {{}}} |.|
│ │ │ │ -00013160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013180: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00013190: 203a 2041 6273 7472 6163 7453 696d 706c   : AbstractSimpl
│ │ │ │ -000131a0: 6963 6961 6c43 6f6d 706c 6578 2020 2020  icialComplex    
│ │ │ │ -000131b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000131c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000131d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000131e0: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074  -------+..Ways t
│ │ │ │ -000131f0: 6f20 7573 6520 7261 6e64 6f6d 5375 6253  o use randomSubS
│ │ │ │ -00013200: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ -00013210: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -00013220: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00013230: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -00013240: 7261 6e64 6f6d 5375 6253 696d 706c 6963  randomSubSimplic
│ │ │ │ -00013250: 6961 6c43 6f6d 706c 6578 2841 6273 7472  ialComplex(Abstr
│ │ │ │ -00013260: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │ -00013270: 706c 6578 2922 0a0a 466f 7220 7468 6520  plex)"..For the 
│ │ │ │ -00013280: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00013290: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -000132a0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -000132b0: 7261 6e64 6f6d 5375 6253 696d 706c 6963  randomSubSimplic
│ │ │ │ -000132c0: 6961 6c43 6f6d 706c 6578 3a20 7261 6e64  ialComplex: rand
│ │ │ │ -000132d0: 6f6d 5375 6253 696d 706c 6963 6961 6c43  omSubSimplicialC
│ │ │ │ -000132e0: 6f6d 706c 6578 2c20 6973 2061 0a2a 6e6f  omplex, is a.*no
│ │ │ │ -000132f0: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ -00013300: 6f6e 3a20 284d 6163 6175 6c61 7932 446f  on: (Macaulay2Do
│ │ │ │ -00013310: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -00013320: 2c2e 0a1f 0a46 696c 653a 2041 6273 7472  ,....File: Abstr
│ │ │ │ -00013330: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │ -00013340: 706c 6578 6573 2e69 6e66 6f2c 204e 6f64  plexes.info, Nod
│ │ │ │ -00013350: 653a 2072 6564 7563 6564 5369 6d70 6c69  e: reducedSimpli
│ │ │ │ -00013360: 6369 616c 4368 6169 6e43 6f6d 706c 6578  cialChainComplex
│ │ │ │ -00013370: 2c20 4e65 7874 3a20 7369 6d70 6c69 6369  , Next: simplici
│ │ │ │ -00013380: 616c 4368 6169 6e43 6f6d 706c 6578 2c20  alChainComplex, 
│ │ │ │ -00013390: 5072 6576 3a20 7261 6e64 6f6d 5375 6253  Prev: randomSubS
│ │ │ │ -000133a0: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ -000133b0: 2c20 5570 3a20 546f 700a 0a72 6564 7563  , Up: Top..reduc
│ │ │ │ -000133c0: 6564 5369 6d70 6c69 6369 616c 4368 6169  edSimplicialChai
│ │ │ │ -000133d0: 6e43 6f6d 706c 6578 202d 2d20 5468 6520  nComplex -- The 
│ │ │ │ -000133e0: 7265 6475 6365 6420 686f 6d6f 6c6f 6769  reduced homologi
│ │ │ │ -000133f0: 6361 6c20 6368 6169 6e20 636f 6d70 6c65  cal chain comple
│ │ │ │ -00013400: 7820 7468 6174 2069 7320 6465 7465 726d  x that is determ
│ │ │ │ -00013410: 696e 6564 2062 7920 616e 2061 6273 7472  ined by an abstr
│ │ │ │ -00013420: 6163 7420 7369 6d70 6c69 6369 616c 2063  act simplicial c
│ │ │ │ -00013430: 6f6d 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a  omplex.*********
│ │ │ │ -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  ****************
│ │ │ │ -000134b0: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a  **..Description.
│ │ │ │ -000134c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ -000134d0: 7320 6d65 7468 6f64 2072 6574 7572 6e73  s method returns
│ │ │ │ -000134e0: 2074 6865 2072 6564 7563 6564 2068 6f6d   the reduced hom
│ │ │ │ -000134f0: 6f6c 6f67 6963 616c 2063 6861 696e 2063  ological chain c
│ │ │ │ -00013500: 6f6d 706c 6578 2028 692e 652e 2c20 7468  omplex (i.e., th
│ │ │ │ -00013510: 6572 6520 6973 2061 0a6e 6f6e 7a65 726f  ere is a.nonzero
│ │ │ │ -00013520: 2074 6572 6d20 696e 2068 6f6d 6f6c 6f67   term in homolog
│ │ │ │ -00013530: 6963 616c 2064 6567 7265 6520 2d31 2074  ical degree -1 t
│ │ │ │ -00013540: 6861 7420 636f 7272 6573 706f 6e64 7320  hat corresponds 
│ │ │ │ -00013550: 746f 2074 6865 2065 6d70 7479 2066 6163  to the empty fac
│ │ │ │ -00013560: 6529 2074 6861 740a 6973 2061 7373 6f63  e) that.is assoc
│ │ │ │ -00013570: 6961 7465 6420 746f 2061 6e20 6162 7374  iated to an abst
│ │ │ │ -00013580: 7261 6374 2073 696d 706c 6963 6961 6c20  ract simplicial 
│ │ │ │ -00013590: 636f 6d70 6c65 782e 2020 5468 6520 6368  complex.  The ch
│ │ │ │ -000135a0: 6169 6e20 636f 6d70 6c65 7820 6973 2064  ain complex is d
│ │ │ │ -000135b0: 6566 696e 6564 0a6f 7665 7220 7468 6520  efined.over the 
│ │ │ │ -000135c0: 696e 7465 6765 7273 2e0a 0a2b 2d2d 2d2d  integers...+----
│ │ │ │ -000135d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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  ----------------
│ │ │ │ -00013610: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -00013620: 204b 203d 2061 6273 7472 6163 7453 696d   K = abstractSim
│ │ │ │ -00013630: 706c 6963 6961 6c43 6f6d 706c 6578 287b  plicialComplex({
│ │ │ │ -00013640: 7b31 2c32 2c33 7d2c 7b32 2c34 2c39 7d2c  {1,2,3},{2,4,9},
│ │ │ │ -00013650: 7b31 2c32 2c33 2c35 2c37 2c38 7d2c 7b33  {1,2,3,5,7,8},{3
│ │ │ │ -00013660: 2c34 7d7d 2920 2020 207c 0a7c 2020 2020  ,4}})    |.|    
│ │ │ │ -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                  
│ │ │ │ +000128f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00012900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012940: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ +00012950: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +00012960: 2a6e 6f74 6520 7261 6e64 6f6d 3a20 284d  *note random: (M
│ │ │ │ +00012970: 6163 6175 6c61 7932 446f 6329 7261 6e64  acaulay2Doc)rand
│ │ │ │ +00012980: 6f6d 2c20 2d2d 2067 6574 2061 2072 616e  om, -- get a ran
│ │ │ │ +00012990: 646f 6d20 6f62 6a65 6374 0a20 202a 2072  dom object.  * r
│ │ │ │ +000129a0: 616e 646f 6d53 7175 6172 6546 7265 654d  andomSquareFreeM
│ │ │ │ +000129b0: 6f6e 6f6d 6961 6c49 6465 616c 2028 6d69  onomialIdeal (mi
│ │ │ │ +000129c0: 7373 696e 6720 646f 6375 6d65 6e74 6174  ssing documentat
│ │ │ │ +000129d0: 696f 6e29 0a0a 5761 7973 2074 6f20 7573  ion)..Ways to us
│ │ │ │ +000129e0: 6520 7261 6e64 6f6d 4162 7374 7261 6374  e randomAbstract
│ │ │ │ +000129f0: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +00012a00: 783a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  x:.=============
│ │ │ │ +00012a10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00012a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00012a30: 0a20 202a 2022 7261 6e64 6f6d 4162 7374  .  * "randomAbst
│ │ │ │ +00012a40: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ +00012a50: 6d70 6c65 7828 5a5a 2922 0a20 202a 2022  mplex(ZZ)".  * "
│ │ │ │ +00012a60: 7261 6e64 6f6d 4162 7374 7261 6374 5369  randomAbstractSi
│ │ │ │ +00012a70: 6d70 6c69 6369 616c 436f 6d70 6c65 7828  mplicialComplex(
│ │ │ │ +00012a80: 5a5a 2c5a 5a29 220a 2020 2a20 2272 616e  ZZ,ZZ)".  * "ran
│ │ │ │ +00012a90: 646f 6d41 6273 7472 6163 7453 696d 706c  domAbstractSimpl
│ │ │ │ +00012aa0: 6963 6961 6c43 6f6d 706c 6578 285a 5a2c  icialComplex(ZZ,
│ │ │ │ +00012ab0: 5a5a 2c5a 5a29 220a 0a46 6f72 2074 6865  ZZ,ZZ)"..For the
│ │ │ │ +00012ac0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +00012ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00012ae0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +00012af0: 2072 616e 646f 6d41 6273 7472 6163 7453   randomAbstractS
│ │ │ │ +00012b00: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ +00012b10: 3a0a 7261 6e64 6f6d 4162 7374 7261 6374  :.randomAbstract
│ │ │ │ +00012b20: 5369 6d70 6c69 6369 616c 436f 6d70 6c65  SimplicialComple
│ │ │ │ +00012b30: 782c 2069 7320 6120 2a6e 6f74 6520 6d65  x, is a *note me
│ │ │ │ +00012b40: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ +00012b50: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +00012b60: 686f 6446 756e 6374 696f 6e2c 2e0a 1f0a  hodFunction,....
│ │ │ │ +00012b70: 4669 6c65 3a20 4162 7374 7261 6374 5369  File: AbstractSi
│ │ │ │ +00012b80: 6d70 6c69 6369 616c 436f 6d70 6c65 7865  mplicialComplexe
│ │ │ │ +00012b90: 732e 696e 666f 2c20 4e6f 6465 3a20 7261  s.info, Node: ra
│ │ │ │ +00012ba0: 6e64 6f6d 5375 6253 696d 706c 6963 6961  ndomSubSimplicia
│ │ │ │ +00012bb0: 6c43 6f6d 706c 6578 2c20 4e65 7874 3a20  lComplex, Next: 
│ │ │ │ +00012bc0: 7265 6475 6365 6453 696d 706c 6963 6961  reducedSimplicia
│ │ │ │ +00012bd0: 6c43 6861 696e 436f 6d70 6c65 782c 2050  lChainComplex, P
│ │ │ │ +00012be0: 7265 763a 2072 616e 646f 6d41 6273 7472  rev: randomAbstr
│ │ │ │ +00012bf0: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │ +00012c00: 706c 6578 2c20 5570 3a20 546f 700a 0a72  plex, Up: Top..r
│ │ │ │ +00012c10: 616e 646f 6d53 7562 5369 6d70 6c69 6369  andomSubSimplici
│ │ │ │ +00012c20: 616c 436f 6d70 6c65 7820 2d2d 2043 7265  alComplex -- Cre
│ │ │ │ +00012c30: 6174 6520 6120 7261 6e64 6f6d 2073 7562  ate a random sub
│ │ │ │ +00012c40: 2d73 696d 706c 6963 6961 6c20 636f 6d70  -simplicial comp
│ │ │ │ +00012c50: 6c65 780a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  lex.************
│ │ │ │ +00012c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012c90: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269  ********..Descri
│ │ │ │ +00012ca0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +00012cb0: 3d0a 0a43 7265 6174 6573 2061 2072 616e  =..Creates a ran
│ │ │ │ +00012cc0: 646f 6d20 7375 622d 7369 6d70 6c69 6369  dom sub-simplici
│ │ │ │ +00012cd0: 616c 2063 6f6d 706c 6578 206f 6620 6120  al complex of a 
│ │ │ │ +00012ce0: 6769 7665 6e20 7369 6d70 6c69 6369 616c  given simplicial
│ │ │ │ +00012cf0: 2063 6f6d 706c 6578 2e0a 0a2b 2d2d 2d2d   complex...+----
│ │ │ │ +00012d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00012d30: 203a 2073 6574 5261 6e64 6f6d 5365 6564   : setRandomSeed
│ │ │ │ +00012d40: 2863 7572 7265 6e74 5469 6d65 2829 293b  (currentTime());
│ │ │ │ +00012d50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00012d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00012d90: 0a7c 6932 203a 204b 203d 2072 616e 646f  .|i2 : K = rando
│ │ │ │ +00012da0: 6d41 6273 7472 6163 7453 696d 706c 6963  mAbstractSimplic
│ │ │ │ +00012db0: 6961 6c43 6f6d 706c 6578 2834 2920 2020  ialComplex(4)   
│ │ │ │ +00012dc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00012dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012df0: 2020 207c 0a7c 6f32 203d 2041 6273 7472     |.|o2 = Abstr
│ │ │ │ +00012e00: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │ +00012e10: 706c 6578 7b2d 3120 3d3e 207b 7b7d 7d20  plex{-1 => {{}} 
│ │ │ │ +00012e20: 2020 2020 7d7c 0a7c 2020 2020 2020 2020      }|.|        
│ │ │ │ +00012e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012e40: 2020 2020 2020 2030 203d 3e20 7b7b 327d         0 => {{2}
│ │ │ │ +00012e50: 2c20 7b33 7d7d 207c 0a7c 2020 2020 2020  , {3}} |.|      
│ │ │ │ +00012e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012e70: 2020 2020 2020 2020 2031 203d 3e20 7b7b           1 => {{
│ │ │ │ +00012e80: 322c 2033 7d7d 2020 207c 0a7c 2020 2020  2, 3}}   |.|    
│ │ │ │ +00012e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012eb0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00012ec0: 203a 2041 6273 7472 6163 7453 696d 706c   : AbstractSimpl
│ │ │ │ +00012ed0: 6963 6961 6c43 6f6d 706c 6578 2020 2020  icialComplex    
│ │ │ │ +00012ee0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00012ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00012f20: 0a7c 6933 203a 204a 203d 2072 616e 646f  .|i3 : J = rando
│ │ │ │ +00012f30: 6d53 7562 5369 6d70 6c69 6369 616c 436f  mSubSimplicialCo
│ │ │ │ +00012f40: 6d70 6c65 7828 4b29 2020 2020 2020 2020  mplex(K)        
│ │ │ │ +00012f50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00012f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012f80: 2020 207c 0a7c 6f33 203d 2041 6273 7472     |.|o3 = Abstr
│ │ │ │ +00012f90: 6163 7453 696d 706c 6963 6961 6c43 6f6d  actSimplicialCom
│ │ │ │ +00012fa0: 706c 6578 7b2d 3120 3d3e 207b 7b7d 7d7d  plex{-1 => {{}}}
│ │ │ │ +00012fb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00012fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012fd0: 2020 2020 2020 2030 203d 3e20 7b7b 327d         0 => {{2}
│ │ │ │ +00012fe0: 7d20 2020 2020 207c 0a7c 2020 2020 2020  }      |.|      
│ │ │ │ +00012ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013010: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +00013020: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ +00013030: 6961 6c43 6f6d 706c 6578 2020 2020 2020  ialComplex      
│ │ │ │ +00013040: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00013050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00013080: 5761 7973 2074 6f20 7573 6520 7261 6e64  Ways to use rand
│ │ │ │ +00013090: 6f6d 5375 6253 696d 706c 6963 6961 6c43  omSubSimplicialC
│ │ │ │ +000130a0: 6f6d 706c 6578 3a0a 3d3d 3d3d 3d3d 3d3d  omplex:.========
│ │ │ │ +000130b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000130c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +000130d0: 0a20 202a 2022 7261 6e64 6f6d 5375 6253  .  * "randomSubS
│ │ │ │ +000130e0: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ +000130f0: 2841 6273 7472 6163 7453 696d 706c 6963  (AbstractSimplic
│ │ │ │ +00013100: 6961 6c43 6f6d 706c 6578 2922 0a0a 466f  ialComplex)"..Fo
│ │ │ │ +00013110: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00013120: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00013130: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00013140: 2a6e 6f74 6520 7261 6e64 6f6d 5375 6253  *note randomSubS
│ │ │ │ +00013150: 696d 706c 6963 6961 6c43 6f6d 706c 6578  implicialComplex
│ │ │ │ +00013160: 3a20 7261 6e64 6f6d 5375 6253 696d 706c  : randomSubSimpl
│ │ │ │ +00013170: 6963 6961 6c43 6f6d 706c 6578 2c20 6973  icialComplex, is
│ │ │ │ +00013180: 2061 0a2a 6e6f 7465 206d 6574 686f 6420   a.*note method 
│ │ │ │ +00013190: 6675 6e63 7469 6f6e 3a20 284d 6163 6175  function: (Macau
│ │ │ │ +000131a0: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ +000131b0: 6e63 7469 6f6e 2c2e 0a1f 0a46 696c 653a  nction,....File:
│ │ │ │ +000131c0: 2041 6273 7472 6163 7453 696d 706c 6963   AbstractSimplic
│ │ │ │ +000131d0: 6961 6c43 6f6d 706c 6578 6573 2e69 6e66  ialComplexes.inf
│ │ │ │ +000131e0: 6f2c 204e 6f64 653a 2072 6564 7563 6564  o, Node: reduced
│ │ │ │ +000131f0: 5369 6d70 6c69 6369 616c 4368 6169 6e43  SimplicialChainC
│ │ │ │ +00013200: 6f6d 706c 6578 2c20 4e65 7874 3a20 7369  omplex, Next: si
│ │ │ │ +00013210: 6d70 6c69 6369 616c 4368 6169 6e43 6f6d  mplicialChainCom
│ │ │ │ +00013220: 706c 6578 2c20 5072 6576 3a20 7261 6e64  plex, Prev: rand
│ │ │ │ +00013230: 6f6d 5375 6253 696d 706c 6963 6961 6c43  omSubSimplicialC
│ │ │ │ +00013240: 6f6d 706c 6578 2c20 5570 3a20 546f 700a  omplex, Up: Top.
│ │ │ │ +00013250: 0a72 6564 7563 6564 5369 6d70 6c69 6369  .reducedSimplici
│ │ │ │ +00013260: 616c 4368 6169 6e43 6f6d 706c 6578 202d  alChainComplex -
│ │ │ │ +00013270: 2d20 5468 6520 7265 6475 6365 6420 686f  - The reduced ho
│ │ │ │ +00013280: 6d6f 6c6f 6769 6361 6c20 6368 6169 6e20  mological chain 
│ │ │ │ +00013290: 636f 6d70 6c65 7820 7468 6174 2069 7320  complex that is 
│ │ │ │ +000132a0: 6465 7465 726d 696e 6564 2062 7920 616e  determined by an
│ │ │ │ +000132b0: 2061 6273 7472 6163 7420 7369 6d70 6c69   abstract simpli
│ │ │ │ +000132c0: 6369 616c 2063 6f6d 706c 6578 0a2a 2a2a  cial complex.***
│ │ │ │ +000132d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000132e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00013920: 7d2c 207b 322c 2035 7d2c 207b 322c 2037  }, {2, 5}, {2, 7
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│ │ │ │ +00013970: 2c20 7b31 2c20 332c 2037 7d2c 207b 312c  , {1, 3, 7}, {1,
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│ │ │ │ +00013b90: 0a7c 2035 7d2c 207b 322c 2033 2c20 377d  .| 5}, {2, 3, 7}
│ │ │ │ +00013ba0: 2c20 7b32 2c20 332c 2038 7d2c 207b 322c  , {2, 3, 8}, {2,
│ │ │ │ +00013bb0: 2034 2c20 397d 2c20 7b32 2c20 352c 2037   4, 9}, {2, 5, 7
│ │ │ │ +00013bc0: 7d2c 207b 322c 2035 2c20 2020 2020 2020  }, {2, 5,       
│ │ │ │ +00013bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013be0: 0a7c 332c 2037 2c20 387d 2c20 7b31 2c20  .|3, 7, 8}, {1, 
│ │ │ │ +00013bf0: 352c 2037 2c20 387d 2c20 7b32 2c20 332c  5, 7, 8}, {2, 3,
│ │ │ │ +00013c00: 2035 2c20 377d 2c20 7b32 2c20 332c 2035   5, 7}, {2, 3, 5
│ │ │ │ +00013c10: 2c20 387d 2c20 7b32 2c20 2020 2020 2020  , 8}, {2,       
│ │ │ │ +00013c20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013c30: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00013c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00013c80: 0a7c 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                  
│ │ │ │ +00013ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013cb0: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ +00013cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013cd0: 0a7c 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  ----------------
│ │ │ │ -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  ---------|.|    
│ │ │ │ +00013cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013d20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00013d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013d70: 0a7c 387d 2c20 7b32 2c20 372c 2038 7d2c  .|8}, {2, 7, 8},
│ │ │ │ +00013d80: 207b 332c 2035 2c20 377d 2c20 7b33 2c20   {3, 5, 7}, {3, 
│ │ │ │ +00013d90: 352c 2038 7d2c 207b 332c 2037 2c20 387d  5, 8}, {3, 7, 8}
│ │ │ │ +00013da0: 2c20 7b35 2c20 372c 2038 7d7d 2020 2020  , {5, 7, 8}}    
│ │ │ │ +00013db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013dc0: 0a7c 332c 2037 2c20 387d 2c20 7b32 2c20  .|3, 7, 8}, {2, 
│ │ │ │ +00013dd0: 352c 2037 2c20 387d 2c20 7b33 2c20 352c  5, 7, 8}, {3, 5,
│ │ │ │ +00013de0: 2037 2c20 387d 7d20 2020 2020 2020 2020   7, 8}}         
│ │ │ │  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 7d20 2020 2020 2020 2020        }         
│ │ │ │ -00013e30: 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020                  
│ │ │ │ -00013e80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00013e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013e10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00013e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00013e60: 0a7c 6932 203a 2072 6564 7563 6564 5369  .|i2 : reducedSi
│ │ │ │ +00013e70: 6d70 6c69 6369 616c 4368 6169 6e43 6f6d  mplicialChainCom
│ │ │ │ +00013e80: 706c 6578 284b 2920 2020 2020 2020 2020  plex(K)         
│ │ │ │  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                  
│ │ │ │ +00013ea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013eb0: 0a7c 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  ----------------
│ │ │ │ -00013fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013fc0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -00013fd0: 2072 6564 7563 6564 5369 6d70 6c69 6369   reducedSimplici
│ │ │ │ -00013fe0: 616c 4368 6169 6e43 6f6d 706c 6578 284b  alChainComplex(K
│ │ │ │ -00013ff0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00014000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014010: 2020 2020 2020 2020 207c 0a7c 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                  
│ │ │ │ +00013ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013ef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013f00: 0a7c 2020 2020 2020 2031 2020 2020 2020  .|       1      
│ │ │ │ +00013f10: 2038 2020 2020 2020 2031 3920 2020 2020   8       19     
│ │ │ │ +00013f20: 2020 3231 2020 2020 2020 2031 3520 2020    21       15   
│ │ │ │ +00013f30: 2020 2020 3620 2020 2020 2020 3120 2020      6       1   
│ │ │ │ +00013f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013f50: 0a7c 6f32 203d 205a 5a20 203c 2d2d 205a  .|o2 = ZZ  <-- Z
│ │ │ │ +00013f60: 5a20 203c 2d2d 205a 5a20 2020 3c2d 2d20  Z  <-- ZZ   <-- 
│ │ │ │ +00013f70: 5a5a 2020 203c 2d2d 205a 5a20 2020 3c2d  ZZ   <-- ZZ   <-
│ │ │ │ +00013f80: 2d20 5a5a 2020 3c2d 2d20 5a5a 2020 2020  - ZZ  <-- ZZ    
│ │ │ │ +00013f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00013fb0: 2020 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 207c                 |
│ │ │ │ +00013ff0: 0a7c 2020 2020 202d 3120 2020 2020 2030  .|     -1      0
│ │ │ │ +00014000: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +00014010: 3220 2020 2020 2020 2033 2020 2020 2020  2        3      
│ │ │ │ +00014020: 2020 3420 2020 2020 2020 3520 2020 2020    4       5     
│ │ │ │ +00014030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00014050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014060: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014070: 2020 2031 2020 2020 2020 2038 2020 2020     1       8    
│ │ │ │ -00014080: 2020 2031 3920 2020 2020 2020 3231 2020     19       21  
│ │ │ │ -00014090: 2020 2020 2031 3520 2020 2020 2020 3620       15       6 
│ │ │ │ -000140a0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -000140b0: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ -000140c0: 205a 5a20 203c 2d2d 205a 5a20 203c 2d2d   ZZ  <-- ZZ  <--
│ │ │ │ -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          
│ │ │ │ -00014100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014150: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014160: 202d 3120 2020 2020 2030 2020 2020 2020   -1      0      
│ │ │ │ -00014170: 2031 2020 2020 2020 2020 3220 2020 2020   1        2     
│ │ │ │ -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                  
│ │ │ │ -000141f0: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ -00014200: 2043 6f6d 706c 6578 2020 2020 2020 2020   Complex        
│ │ │ │ -00014210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014240: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -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  ----------------
│ │ │ │ -00014290: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973  ---------+..Ways
│ │ │ │ -000142a0: 2074 6f20 7573 6520 7265 6475 6365 6453   to use reducedS
│ │ │ │ -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
│ │ │ │ -00014300: 696d 706c 6963 6961 6c43 6861 696e 436f  implicialChainCo
│ │ │ │ -00014310: 6d70 6c65 7828 4162 7374 7261 6374 5369  mplex(AbstractSi
│ │ │ │ -00014320: 6d70 6c69 6369 616c 436f 6d70 6c65 7829  mplicialComplex)
│ │ │ │ -00014330: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00014340: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -00014350: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
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│ │ │ │ -00014370: 6564 5369 6d70 6c69 6369 616c 4368 6169  edSimplicialChai
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│ │ │ │ -000143a0: 436f 6d70 6c65 782c 0a69 7320 6120 2a6e  Complex,.is a *n
│ │ │ │ -000143b0: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -000143c0: 696f 6e3a 2028 4d61 6361 756c 6179 3244  ion: (Macaulay2D
│ │ │ │ -000143d0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -000143e0: 6e2c 2e0a 1f0a 4669 6c65 3a20 4162 7374  n,....File: Abst
│ │ │ │ -000143f0: 7261 6374 5369 6d70 6c69 6369 616c 436f  ractSimplicialCo
│ │ │ │ -00014400: 6d70 6c65 7865 732e 696e 666f 2c20 4e6f  mplexes.info, No
│ │ │ │ -00014410: 6465 3a20 7369 6d70 6c69 6369 616c 4368  de: simplicialCh
│ │ │ │ -00014420: 6169 6e43 6f6d 706c 6578 2c20 5072 6576  ainComplex, Prev
│ │ │ │ -00014430: 3a20 7265 6475 6365 6453 696d 706c 6963  : reducedSimplic
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│ │ │ │ -00014450: 2055 703a 2054 6f70 0a0a 7369 6d70 6c69   Up: Top..simpli
│ │ │ │ -00014460: 6369 616c 4368 6169 6e43 6f6d 706c 6578  cialChainComplex
│ │ │ │ -00014470: 202d 2d20 5468 6520 6e6f 6e2d 7265 6475   -- The non-redu
│ │ │ │ -00014480: 6365 6420 686f 6d6f 6c6f 6769 6361 6c20  ced homological 
│ │ │ │ -00014490: 6368 6169 6e20 636f 6d70 6c65 7820 7468  chain complex th
│ │ │ │ -000144a0: 6174 2069 7320 6465 7465 726d 696e 6564  at is determined
│ │ │ │ -000144b0: 2062 7920 616e 2061 6273 7472 6163 7420   by an abstract 
│ │ │ │ -000144c0: 7369 6d70 6c69 6369 616c 2063 6f6d 706c  simplicial compl
│ │ │ │ -000144d0: 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ex.*************
│ │ │ │ -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
│ │ │ │ -00014570: 6420 7265 7475 726e 7320 7468 6520 286e  d returns the (n
│ │ │ │ -00014580: 6f6e 2d72 6564 7563 6564 2920 686f 6d6f  on-reduced) homo
│ │ │ │ -00014590: 6c6f 6769 6361 6c20 6368 6169 6e20 636f  logical chain co
│ │ │ │ -000145a0: 6d70 6c65 7820 2869 2e65 2e2c 2074 6865  mplex (i.e., the
│ │ │ │ -000145b0: 7265 2069 730a 6e6f 206e 6f6e 7a65 726f  re is.no nonzero
│ │ │ │ -000145c0: 2074 6572 6d20 696e 2068 6f6d 6f6c 6f67   term in homolog
│ │ │ │ -000145d0: 6963 616c 2064 6567 7265 6520 2d31 2074  ical degree -1 t
│ │ │ │ -000145e0: 6861 7420 636f 7272 6573 706f 6e64 7320  hat corresponds 
│ │ │ │ -000145f0: 746f 2074 6865 2065 6d70 7479 2066 6163  to the empty fac
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│ │ │ │ -00014610: 6961 7465 6420 746f 2061 6e20 6162 7374  iated to an abst
│ │ │ │ -00014620: 7261 6374 2073 696d 706c 6963 6961 6c20  ract simplicial 
│ │ │ │ -00014630: 636f 6d70 6c65 782e 2020 5468 6520 6368  complex.  The ch
│ │ │ │ -00014640: 6169 6e20 636f 6d70 6c65 7820 6973 0a64  ain complex is.d
│ │ │ │ -00014650: 6566 696e 6564 206f 7665 7220 7468 6520  efined over the 
│ │ │ │ -00014660: 696e 7465 6765 7273 2e0a 0a2b 2d2d 2d2d  integers...+----
│ │ │ │ -00014670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000146b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
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│ │ │ │  000147b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000147f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ -00014840: 207b 322c 2033 7d2c 207c 0a7c 2020 2020   {2, 3}, |.|    
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│ │ │ │ -00014890: 2c20 7b32 2c20 342c 207c 0a7c 2020 2020  , {2, 4, |.|    
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│ │ │ │ -000148b0: 2020 2020 2020 2020 2020 2033 203d 3e20             3 => 
│ │ │ │ -000148c0: 7b7b 322c 2034 2c20 352c 2037 7d7d 2020  {{2, 4, 5, 7}}  
│ │ │ │ +00014810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +000148b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  000148d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000148e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00014950: 6961 6c43 6f6d 706c 6578 2020 2020 2020  ialComplex      
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│ │ │ │ -000149b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00014920: 2c20 7b32 2c20 377d 2c20 7b34 2c20 357d  , {2, 7}, {4, 5}
│ │ │ │ +00014930: 2c20 7b34 2c20 377d 2c20 7b35 2c20 377d  , {4, 7}, {5, 7}
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│ │ │ │ -000149e0: 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           |.|    
│ │ │ │ +000149d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000149e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000149f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014a00: 0a7c 6932 203a 2043 203d 2073 696d 706c  .|i2 : C = simpl
│ │ │ │ +00014a10: 6963 6961 6c43 6861 696e 436f 6d70 6c65  icialChainComple
│ │ │ │ +00014a20: 7828 4b29 2020 2020 2020 2020 2020 2020  x(K)            
│ │ │ │  00014a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00014a50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00014a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00014a80: 347d 2c20 7b32 2c20 357d 2c20 7b32 2c20  4}, {2, 5}, {2, 
│ │ │ │ -00014a90: 377d 2c20 7b34 2c20 357d 2c20 7b34 2c20  7}, {4, 5}, {4, 
│ │ │ │ -00014aa0: 377d 2c20 7b35 2c20 377d 7d20 2020 2020  7}, {5, 7}}     
│ │ │ │ -00014ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00014ad0: 7b32 2c20 352c 2037 7d2c 207b 342c 2035  {2, 5, 7}, {4, 5
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│ │ │ │ -00014af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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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│ │ │ │ -00014b80: 6861 696e 436f 6d70 6c65 7828 4b29 2020  hainComplex(K)  
│ │ │ │ -00014b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00015130: 6f64 653a 206e 6577 2041 6273 7472 6163  ode: new Abstrac
│ │ │ │ +00015140: 7453 696d 706c 6963 6961 6c43 6f6d 706c  tSimplicialCompl
│ │ │ │ +00015150: 6578 7f37 3237 3631 0a4e 6f64 653a 2072  ex.72761.Node: r
│ │ │ │ +00015160: 616e 646f 6d41 6273 7472 6163 7453 696d  andomAbstractSim
│ │ │ │ +00015170: 706c 6963 6961 6c43 6f6d 706c 6578 7f37  plicialComplex.7
│ │ │ │ +00015180: 3331 3132 0a4e 6f64 653a 2072 616e 646f  3112.Node: rando
│ │ │ │ +00015190: 6d53 7562 5369 6d70 6c69 6369 616c 436f  mSubSimplicialCo
│ │ │ │ +000151a0: 6d70 6c65 787f 3736 3635 340a 4e6f 6465  mplex.76654.Node
│ │ │ │ +000151b0: 3a20 7265 6475 6365 6453 696d 706c 6963  : reducedSimplic
│ │ │ │ +000151c0: 6961 6c43 6861 696e 436f 6d70 6c65 787f  ialChainComplex.
│ │ │ │ +000151d0: 3738 3236 350a 4e6f 6465 3a20 7369 6d70  78265.Node: simp
│ │ │ │ +000151e0: 6c69 6369 616c 4368 6169 6e43 6f6d 706c  licialChainCompl
│ │ │ │ +000151f0: 6578 7f38 3235 3534 0a1f 0a45 6e64 2054  ex.82554...End T
│ │ │ │ +00015200: 6167 2054 6162 6c65 0a                   ag Table.
│ │ ├── ./usr/share/info/AdjunctionForSurfaces.info.gz
│ │ │ ├── AdjunctionForSurfaces.info
│ │ │ │ @@ -685,16 +685,16 @@
│ │ │ │  00002ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00002ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │  00002af0: 3020 3a20 656c 6170 7365 6454 696d 6520  0 : elapsedTime 
│ │ │ │  00002b00: 6649 3d72 6573 2049 2020 2020 2020 2020  fI=res I        
│ │ │ │  00002b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002b30: 7c0a 7c20 2d2d 202e 3037 3938 3536 7320  |.| -- .079856s 
│ │ │ │ -00002b40: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │ +00002b30: 7c0a 7c20 2d2d 202e 3032 3633 3738 3773  |.| -- .0263787s
│ │ │ │ +00002b40: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
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│ │ │ │  00002b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00002bb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ @@ -1527,15 +1527,15 @@
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│ │ │ │  00005f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00005f90: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 656c  -----+.|i15 : el
│ │ │ │  00005fa0: 6170 7365 6454 696d 6520 6265 7474 6928  apsedTime betti(
│ │ │ │  00005fb0: 4927 3d74 7269 6d20 6b65 7220 7068 6929  I'=trim ker phi)
│ │ │ │  00005fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005fd0: 7c0a 7c20 2d2d 202e 3639 3934 3834 7320  |.| -- .699484s 
│ │ │ │ +00005fd0: 7c0a 7c20 2d2d 202e 3532 3534 3338 7320  |.| -- .525438s 
│ │ │ │  00005fe0: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00005ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006000: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  00006030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 352e 3230  l H; |.| -- 5.20
│ │ │ │ +00006350: 3632 3773 2065 6c61 7073 6564 2020 2020  627s 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 3031 3434 3437 3973 2065 6c61  -- .0144479s 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 2e30 3634       |.| -- .064
│ │ │ │ +00009f50: 3438 3731 7320 656c 6170 7365 6420 2020  4871s 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 
│ │ │ │ @@ -2594,16 +2594,16 @@
│ │ │ │  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
│ │ │ │ -0000a290: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │ +0000a280: 0a7c 202d 2d20 312e 3538 3335 7320 656c  .| -- 1.5835s el
│ │ │ │ +0000a290: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  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
│ │ │ │  0000a300: 6c6c 7920 6170 706c 7928 6261 7365 5074  lly apply(basePt
│ │ ├── ./usr/share/info/BGG.info.gz
│ │ │ ├── BGG.info
│ │ │ │ @@ -4154,1015 +4154,1012 @@
│ │ │ │  00010390: 2074 6865 2070 726f 6a65 6374 6976 6520   the projective 
│ │ │ │  000103a0: 7370 6163 6573 2066 726f 6d20 7768 6f73  spaces from whos
│ │ │ │  000103b0: 650a 7072 6f64 7563 7420 7765 2061 7265  e.product we are
│ │ │ │  000103c0: 2070 726f 6a65 6374 696e 672e 290a 0a2b   projecting.)..+
│ │ │ │  000103d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000103e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000103f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010400: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 4120  -----+.|i12 : A 
│ │ │ │ -00010410: 3d20 6b6b 5b61 2c62 5d20 2020 2020 2020  = kk[a,b]       
│ │ │ │ +00010400: 2d2d 2d2b 0a7c 6931 3220 3a20 4120 3d20  ---+.|i12 : A = 
│ │ │ │ +00010410: 6b6b 5b61 2c62 5d20 2020 2020 2020 2020  kk[a,b]         
│ │ │ │  00010420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00010430: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00010440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010470: 2020 2020 207c 0a7c 6f31 3220 3d20 4120       |.|o12 = A 
│ │ │ │ +00010460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00010470: 0a7c 6f31 3220 3d20 4120 2020 2020 2020  .|o12 = A       
│ │ │ │  00010480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000104a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000104a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000104b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000104c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000104d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000104e0: 2020 2020 207c 0a7c 6f31 3220 3a20 506f       |.|o12 : Po
│ │ │ │ -000104f0: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +000104d0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000104e0: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ +000104f0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  00010500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010510: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00010510: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00010520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00010550: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 4d20  -----+.|i13 : M 
│ │ │ │ -00010560: 3d20 7261 6e64 6f6d 2841 5e34 2c20 415e  = random(A^4, A^
│ │ │ │ -00010570: 7b34 3a2d 317d 2920 2020 2020 2020 2020  {4:-1})         
│ │ │ │ -00010580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00010540: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ +00010550: 4d20 3d20 7261 6e64 6f6d 2841 5e34 2c20  M = random(A^4, 
│ │ │ │ +00010560: 415e 7b34 3a2d 317d 2920 2020 2020 2020  A^{4:-1})       
│ │ │ │ +00010570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00010580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000105a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000105b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000105c0: 2020 2020 207c 0a7c 6f31 3320 3d20 7c20       |.|o13 = | 
│ │ │ │ -000105d0: 3234 612d 3336 6220 202d 3861 2d32 3262  24a-36b  -8a-22b
│ │ │ │ -000105e0: 2020 3334 612b 3139 6220 202d 3238 612d    34a+19b  -28a-
│ │ │ │ -000105f0: 3437 6220 7c20 2020 2020 2020 207c 0a7c  47b |        |.|
│ │ │ │ -00010600: 2020 2020 2020 7c20 2d33 3061 2d32 3962        | -30a-29b
│ │ │ │ -00010610: 202d 3239 612d 3234 6220 2d34 3761 2d33   -29a-24b -47a-3
│ │ │ │ -00010620: 3962 2033 3861 2b32 6220 2020 7c20 2020  9b 38a+2b   |   
│ │ │ │ -00010630: 2020 2020 207c 0a7c 2020 2020 2020 7c20       |.|      | 
│ │ │ │ -00010640: 3139 612b 3139 6220 202d 3338 612d 3136  19a+19b  -38a-16
│ │ │ │ -00010650: 6220 2d31 3861 2d31 3362 2031 3661 2b32  b -18a-13b 16a+2
│ │ │ │ -00010660: 3262 2020 7c20 2020 2020 2020 207c 0a7c  2b  |        |.|
│ │ │ │ -00010670: 2020 2020 2020 7c20 2d31 3061 2d32 3962        | -10a-29b
│ │ │ │ -00010680: 2033 3961 2b32 3162 2020 2d34 3361 2d31   39a+21b  -43a-1
│ │ │ │ -00010690: 3562 2034 3561 2d33 3462 2020 7c20 2020  5b 45a-34b  |   
│ │ │ │ -000106a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000105b0: 2020 207c 0a7c 6f31 3320 3d20 7c20 3234     |.|o13 = | 24
│ │ │ │ +000105c0: 612d 3336 6220 202d 3861 2d32 3262 2020  a-36b  -8a-22b  
│ │ │ │ +000105d0: 3334 612b 3139 6220 202d 3238 612d 3437  34a+19b  -28a-47
│ │ │ │ +000105e0: 6220 7c20 2020 2020 207c 0a7c 2020 2020  b |      |.|    
│ │ │ │ +000105f0: 2020 7c20 2d33 3061 2d32 3962 202d 3239    | -30a-29b -29
│ │ │ │ +00010600: 612d 3234 6220 2d34 3761 2d33 3962 2033  a-24b -47a-39b 3
│ │ │ │ +00010610: 3861 2b32 6220 2020 7c20 2020 2020 207c  8a+2b   |      |
│ │ │ │ +00010620: 0a7c 2020 2020 2020 7c20 3139 612b 3139  .|      | 19a+19
│ │ │ │ +00010630: 6220 202d 3338 612d 3136 6220 2d31 3861  b  -38a-16b -18a
│ │ │ │ +00010640: 2d31 3362 2031 3661 2b32 3262 2020 7c20  -13b 16a+22b  | 
│ │ │ │ +00010650: 2020 2020 207c 0a7c 2020 2020 2020 7c20       |.|      | 
│ │ │ │ +00010660: 2d31 3061 2d32 3962 2033 3961 2b32 3162  -10a-29b 39a+21b
│ │ │ │ +00010670: 2020 2d34 3361 2d31 3562 2034 3561 2d33    -43a-15b 45a-3
│ │ │ │ +00010680: 3462 2020 7c20 2020 2020 207c 0a7c 2020  4b  |      |.|  
│ │ │ │ +00010690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000106a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000106b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000106c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000106d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000106e0: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ -000106f0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ -00010700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010710: 2020 2020 207c 0a7c 6f31 3320 3a20 4d61       |.|o13 : Ma
│ │ │ │ -00010720: 7472 6978 2041 2020 3c2d 2d20 4120 2020  trix A  <-- A   
│ │ │ │ -00010730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010740: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000106c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000106d0: 2020 3420 2020 2020 2034 2020 2020 2020    4      4      
│ │ │ │ +000106e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000106f0: 2020 2020 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ +00010700: 4d61 7472 6978 2041 2020 3c2d 2d20 4120  Matrix A  <-- A 
│ │ │ │ +00010710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010720: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00010730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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
│ │ │ │ -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              
│ │ │ │ +00010760: 2d2d 2d2b 0a7c 6931 3420 3a20 7469 6d65  ---+.|i14 : time
│ │ │ │ +00010770: 2062 6574 7469 2028 4620 3d20 7075 7265   betti (F = pure
│ │ │ │ +00010780: 5265 736f 6c75 7469 6f6e 284d 2c7b 302c  Resolution(M,{0,
│ │ │ │ +00010790: 322c 347d 2929 2020 207c 0a7c 202d 2d20  2,4}))   |.| -- 
│ │ │ │ +000107a0: 7573 6564 2030 2e35 3637 3337 3273 2028  used 0.567372s (
│ │ │ │ +000107b0: 6370 7529 3b20 302e 3336 3936 7320 2874  cpu); 0.3696s (t
│ │ │ │ +000107c0: 6872 6561 6429 3b20 3073 2028 6763 297c  hread); 0s (gc)|
│ │ │ │ +000107d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000107e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000107f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010800: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00010810: 2020 2020 2030 2031 2032 2020 2020 2020       0 1 2      
│ │ │ │ +00010820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010830: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00010840: 3420 3d20 746f 7461 6c3a 2033 2036 2033  4 = total: 3 6 3
│ │ │ │  00010850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010860: 2020 2020 207c 0a7c 6f31 3420 3d20 746f       |.|o14 = to
│ │ │ │ -00010870: 7461 6c3a 2033 2036 2033 2020 2020 2020  tal: 3 6 3      
│ │ │ │ -00010880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000108a0: 2020 2020 2020 2020 2020 303a 2033 202e            0: 3 .
│ │ │ │ -000108b0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +00010860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010870: 207c 0a7c 2020 2020 2020 2020 2020 303a   |.|          0:
│ │ │ │ +00010880: 2033 202e 202e 2020 2020 2020 2020 2020   3 . .          
│ │ │ │ +00010890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000108a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000108b0: 2020 2020 313a 202e 2036 202e 2020 2020      1: . 6 .    
│ │ │ │  000108c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000108d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000108e0: 2020 313a 202e 2036 202e 2020 2020 2020    1: . 6 .      
│ │ │ │ -000108f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010900: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010910: 2020 2020 2020 2020 2020 323a 202e 202e            2: . .
│ │ │ │ -00010920: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +000108d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000108e0: 2020 2020 2020 2020 2020 323a 202e 202e            2: . .
│ │ │ │ +000108f0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00010900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010910: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00010920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010940: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00010950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010940: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ +00010950: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │  00010960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010970: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00010980: 6f31 3420 3a20 4265 7474 6954 616c 6c79  o14 : BettiTally
│ │ │ │ -00010990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000109a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000109b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000109c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000109d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000109e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -000109f0: 5769 7468 2074 6865 2066 6f72 6d20 7075  With the form pu
│ │ │ │ -00010a00: 7265 5265 736f 6c75 7469 6f6e 2870 2c71  reResolution(p,q
│ │ │ │ -00010a10: 2c44 2920 7765 2063 616e 2064 6972 6563  ,D) we can direc
│ │ │ │ -00010a20: 746c 7920 6372 6561 7465 2074 6865 2073  tly create the s
│ │ │ │ -00010a30: 6974 7561 7469 6f6e 206f 660a 7075 7265  ituation of.pure
│ │ │ │ -00010a40: 5265 736f 6c75 7469 6f6e 284d 2c44 2920  Resolution(M,D) 
│ │ │ │ -00010a50: 7768 6572 6520 4d20 6973 2067 656e 6572  where M is gener
│ │ │ │ -00010a60: 6963 2070 726f 6475 6374 286d 5f69 2b31  ic product(m_i+1
│ │ │ │ -00010a70: 2920 7820 2344 2d31 2b73 756d 286d 5f69  ) x #D-1+sum(m_i
│ │ │ │ -00010a80: 2920 6d61 7472 6978 206f 660a 6c69 6e65  ) matrix of.line
│ │ │ │ -00010a90: 6172 2066 6f72 6d73 2064 6566 696e 6564  ar forms defined
│ │ │ │ -00010aa0: 206f 7665 7220 6120 7269 6e67 2077 6974   over a ring wit
│ │ │ │ -00010ab0: 6820 7072 6f64 7563 7428 6d5f 692b 3129  h product(m_i+1)
│ │ │ │ -00010ac0: 202a 2023 442d 312b 7375 6d28 6d5f 6929   * #D-1+sum(m_i)
│ │ │ │ -00010ad0: 2076 6172 6961 626c 6573 0a6f 6620 6368   variables.of ch
│ │ │ │ -00010ae0: 6172 6163 7465 7269 7374 6963 2070 2c20  aracteristic p, 
│ │ │ │ -00010af0: 6372 6561 7465 6420 6279 2074 6865 2073  created by the s
│ │ │ │ -00010b00: 6372 6970 742e 2046 6f72 2061 2067 6976  cript. For a giv
│ │ │ │ -00010b10: 656e 206e 756d 6265 7220 6f66 2076 6172  en number of var
│ │ │ │ -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)|.|        
│ │ │ │ -00010c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00010980: 0a2b 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 2d2d 2d2b 0a0a 5769 7468 2074 6865  -----+..With the
│ │ │ │ +000109c0: 2066 6f72 6d20 7075 7265 5265 736f 6c75   form pureResolu
│ │ │ │ +000109d0: 7469 6f6e 2870 2c71 2c44 2920 7765 2063  tion(p,q,D) we c
│ │ │ │ +000109e0: 616e 2064 6972 6563 746c 7920 6372 6561  an directly crea
│ │ │ │ +000109f0: 7465 2074 6865 2073 6974 7561 7469 6f6e  te the situation
│ │ │ │ +00010a00: 206f 660a 7075 7265 5265 736f 6c75 7469   of.pureResoluti
│ │ │ │ +00010a10: 6f6e 284d 2c44 2920 7768 6572 6520 4d20  on(M,D) where M 
│ │ │ │ +00010a20: 6973 2067 656e 6572 6963 2070 726f 6475  is generic produ
│ │ │ │ +00010a30: 6374 286d 5f69 2b31 2920 7820 2344 2d31  ct(m_i+1) x #D-1
│ │ │ │ +00010a40: 2b73 756d 286d 5f69 2920 6d61 7472 6978  +sum(m_i) matrix
│ │ │ │ +00010a50: 206f 660a 6c69 6e65 6172 2066 6f72 6d73   of.linear forms
│ │ │ │ +00010a60: 2064 6566 696e 6564 206f 7665 7220 6120   defined over a 
│ │ │ │ +00010a70: 7269 6e67 2077 6974 6820 7072 6f64 7563  ring with produc
│ │ │ │ +00010a80: 7428 6d5f 692b 3129 202a 2023 442d 312b  t(m_i+1) * #D-1+
│ │ │ │ +00010a90: 7375 6d28 6d5f 6929 2076 6172 6961 626c  sum(m_i) variabl
│ │ │ │ +00010aa0: 6573 0a6f 6620 6368 6172 6163 7465 7269  es.of characteri
│ │ │ │ +00010ab0: 7374 6963 2070 2c20 6372 6561 7465 6420  stic p, created 
│ │ │ │ +00010ac0: 6279 2074 6865 2073 6372 6970 742e 2046  by the script. F
│ │ │ │ +00010ad0: 6f72 2061 2067 6976 656e 206e 756d 6265  or a given numbe
│ │ │ │ +00010ae0: 7220 6f66 2076 6172 6961 626c 6573 2069  r of variables i
│ │ │ │ +00010af0: 6e0a 4120 7468 6973 2072 756e 7320 6d75  n.A this runs mu
│ │ │ │ +00010b00: 6368 2066 6173 7465 7220 7468 616e 2074  ch faster than t
│ │ │ │ +00010b10: 616b 696e 6720 6120 7261 6e64 6f6d 206d  aking a random m
│ │ │ │ +00010b20: 6174 7269 7820 4d2e 0a0a 2b2d 2d2d 2d2d  atrix M...+-----
│ │ │ │ +00010b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00010b60: 0a7c 6931 3520 3a20 7469 6d65 2062 6574  .|i15 : time bet
│ │ │ │ +00010b70: 7469 2028 4620 3d20 7075 7265 5265 736f  ti (F = pureReso
│ │ │ │ +00010b80: 6c75 7469 6f6e 2831 312c 342c 7b30 2c32  lution(11,4,{0,2
│ │ │ │ +00010b90: 2c34 7d29 2920 7c0a 7c20 2d2d 2075 7365  ,4})) |.| -- use
│ │ │ │ +00010ba0: 6420 302e 3630 3833 3373 2028 6370 7529  d 0.60833s (cpu)
│ │ │ │ +00010bb0: 3b20 302e 3430 3031 3036 7320 2874 6872  ; 0.400106s (thr
│ │ │ │ +00010bc0: 6561 6429 3b20 3073 2028 6763 297c 0a7c  ead); 0s (gc)|.|
│ │ │ │ +00010bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010c00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00010c10: 2020 2020 3020 3120 3220 2020 2020 2020      0 1 2       
│ │ │ │  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 207c 0a7c 6f31             |.|o1
│ │ │ │ +00010c40: 3520 3d20 746f 7461 6c3a 2033 2036 2033  5 = total: 3 6 3
│ │ │ │ +00010c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020 2030    |.|          0
│ │ │ │ +00010c80: 3a20 3320 2e20 2e20 2020 2020 2020 2020  : 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 . . 
│ │ │ │ +00010ca0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00010cb0: 2020 2020 2020 313a 202e 2036 202e 2020        1: . 6 .  
│ │ │ │  00010cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ +00010cf0: 2e20 2e20 3320 2020 2020 2020 2020 2020  . . 3           
│ │ │ │  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   
│ │ │ │ +00010d10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00010d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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  .|              
│ │ │ │ -00010d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010d40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00010d50: 7c6f 3135 203a 2042 6574 7469 5461 6c6c  |o15 : BettiTall
│ │ │ │ +00010d60: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  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  ----------------
│ │ │ │ -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             
│ │ │ │ +00010d80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00010d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00010dc0: 3136 203a 2072 696e 6720 4620 2020 2020  16 : ring F     
│ │ │ │ +00010dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010df0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00010e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020            |.|   
│ │ │ │ +00010e30: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  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: 207c 0a7c 6f31 3620 3d20 2d2d 5b61 202e   |.|o16 = --[a .
│ │ │ │ +00010e70: 2e61 2020 5d20 2020 2020 2020 2020 2020  .a  ]           
│ │ │ │  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  ]   
│ │ │ │ +00010e90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00010ea0: 2031 3120 2030 2020 2031 3520 2020 2020   11  0   15     
│ │ │ │  00010eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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             
│ │ │ │ +00010ec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00010ed0: 0a7c 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 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00010f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00010f00: 2020 2020 2020 7c0a 7c6f 3136 203a 2050        |.|o16 : P
│ │ │ │ +00010f10: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │  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  ----------------
│ │ │ │ -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: 
│ │ │ │ -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
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│ │ │ │ -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 
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│ │ │ │ -00011170: 7469 6f6e 3a20 7075 7265 5265 736f 6c75  tion: pureResolu
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│ │ │ │ -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,.
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│ │ │ │ -00011240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00011250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -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  ================
│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -00011420: 6174 6552 6573 6f6c 7574 696f 6e2c 2050  ateResolution, P
│ │ │ │ -00011430: 7265 763a 2052 6567 756c 6172 6974 792c  rev: Regularity,
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│ │ │ │ -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  ****************
│ │ │ │ -000114a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000114b0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
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│ │ │ │ -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 *
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│ │ │ │ -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
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│ │ │ │ -000116c0: 6b20 6f66 0a61 7320 6120 7072 6573 656e  k of.as a presen
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│ │ │ │ -000116f0: 7261 6465 6420 532d 6d6f 6475 6c65 204d  raded S-module M
│ │ │ │ -00011700: 206d 6174 7269 7820 7265 7072 6573 656e   matrix represen
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│ │ │ │ -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  ----------------
│ │ │ │ -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  ----------------
│ │ │ │ -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  ----------------
│ │ │ │ -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  ----------------
│ │ │ │ -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                  
│ │ │ │ +00010f30: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00010f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00010f70: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ +00010f80: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00010f90: 6f74 6520 6469 7265 6374 496d 6167 6543  ote directImageC
│ │ │ │ +00010fa0: 6f6d 706c 6578 3a20 6469 7265 6374 496d  omplex: directIm
│ │ │ │ +00010fb0: 6167 6543 6f6d 706c 6578 2c20 2d2d 2064  ageComplex, -- d
│ │ │ │ +00010fc0: 6972 6563 7420 696d 6167 6520 636f 6d70  irect image comp
│ │ │ │ +00010fd0: 6c65 780a 2020 2a20 2a6e 6f74 6520 756e  lex.  * *note un
│ │ │ │ +00010fe0: 6976 6572 7361 6c45 7874 656e 7369 6f6e  iversalExtension
│ │ │ │ +00010ff0: 3a20 756e 6976 6572 7361 6c45 7874 656e  : universalExten
│ │ │ │ +00011000: 7369 6f6e 2c20 2d2d 2055 6e69 7665 7273  sion, -- Univers
│ │ │ │ +00011010: 616c 2065 7874 656e 7369 6f6e 206f 660a  al extension of.
│ │ │ │ +00011020: 2020 2020 7665 6374 6f72 2062 756e 646c      vector bundl
│ │ │ │ +00011030: 6573 206f 6e20 505e 310a 0a57 6179 7320  es on P^1..Ways 
│ │ │ │ +00011040: 746f 2075 7365 2070 7572 6552 6573 6f6c  to use pureResol
│ │ │ │ +00011050: 7574 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d  ution:.=========
│ │ │ │ +00011060: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00011070: 3d3d 0a0a 2020 2a20 2270 7572 6552 6573  ==..  * "pureRes
│ │ │ │ +00011080: 6f6c 7574 696f 6e28 4d61 7472 6978 2c4c  olution(Matrix,L
│ │ │ │ +00011090: 6973 7429 220a 2020 2a20 2270 7572 6552  ist)".  * "pureR
│ │ │ │ +000110a0: 6573 6f6c 7574 696f 6e28 5269 6e67 2c4c  esolution(Ring,L
│ │ │ │ +000110b0: 6973 7429 220a 2020 2a20 2270 7572 6552  ist)".  * "pureR
│ │ │ │ +000110c0: 6573 6f6c 7574 696f 6e28 5a5a 2c4c 6973  esolution(ZZ,Lis
│ │ │ │ +000110d0: 7429 220a 2020 2a20 2270 7572 6552 6573  t)".  * "pureRes
│ │ │ │ +000110e0: 6f6c 7574 696f 6e28 5a5a 2c5a 5a2c 4c69  olution(ZZ,ZZ,Li
│ │ │ │ +000110f0: 7374 2922 0a0a 466f 7220 7468 6520 7072  st)"..For the pr
│ │ │ │ +00011100: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +00011110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +00011120: 206f 626a 6563 7420 2a6e 6f74 6520 7075   object *note pu
│ │ │ │ +00011130: 7265 5265 736f 6c75 7469 6f6e 3a20 7075  reResolution: pu
│ │ │ │ +00011140: 7265 5265 736f 6c75 7469 6f6e 2c20 6973  reResolution, is
│ │ │ │ +00011150: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ +00011160: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ +00011170: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
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│ │ │ │ +00011190: 2042 4747 2e69 6e66 6f2c 204e 6f64 653a   BGG.info, Node:
│ │ │ │ +000111a0: 2052 6567 756c 6172 6974 792c 204e 6578   Regularity, Nex
│ │ │ │ +000111b0: 743a 2073 796d 4578 742c 2050 7265 763a  t: symExt, Prev:
│ │ │ │ +000111c0: 2070 7572 6552 6573 6f6c 7574 696f 6e2c   pureResolution,
│ │ │ │ +000111d0: 2055 703a 2054 6f70 0a0a 5265 6775 6c61   Up: Top..Regula
│ │ │ │ +000111e0: 7269 7479 202d 2d20 4f70 7469 6f6e 2066  rity -- Option f
│ │ │ │ +000111f0: 6f72 2064 6972 6563 7449 6d61 6765 436f  or directImageCo
│ │ │ │ +00011200: 6d70 6c65 780a 2a2a 2a2a 2a2a 2a2a 2a2a  mplex.**********
│ │ │ │ +00011210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011230: 2a0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d  *..Caveat.======
│ │ │ │ +00011240: 0a0a 4375 7272 656e 746c 7920 6e6f 7420  ..Currently not 
│ │ │ │ +00011250: 7375 7070 6f72 7465 640a 0a46 756e 6374  supported..Funct
│ │ │ │ +00011260: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +00011270: 616c 2061 7267 756d 656e 7420 6e61 6d65  al argument name
│ │ │ │ +00011280: 6420 5265 6775 6c61 7269 7479 3a0a 3d3d  d Regularity:.==
│ │ │ │ +00011290: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000112a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000112b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000112c0: 0a0a 2020 2a20 2264 6972 6563 7449 6d61  ..  * "directIma
│ │ │ │ +000112d0: 6765 436f 6d70 6c65 7828 2e2e 2e2c 5265  geComplex(...,Re
│ │ │ │ +000112e0: 6775 6c61 7269 7479 3d3e 2e2e 2e29 2220  gularity=>...)" 
│ │ │ │ +000112f0: 2d2d 2073 6565 202a 6e6f 7465 2064 6972  -- see *note dir
│ │ │ │ +00011300: 6563 7449 6d61 6765 436f 6d70 6c65 783a  ectImageComplex:
│ │ │ │ +00011310: 0a20 2020 2064 6972 6563 7449 6d61 6765  .    directImage
│ │ │ │ +00011320: 436f 6d70 6c65 782c 202d 2d20 6469 7265  Complex, -- dire
│ │ │ │ +00011330: 6374 2069 6d61 6765 2063 6f6d 706c 6578  ct image complex
│ │ │ │ +00011340: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +00011350: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +00011360: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +00011370: 6563 7420 2a6e 6f74 6520 5265 6775 6c61  ect *note Regula
│ │ │ │ +00011380: 7269 7479 3a20 2852 6567 756c 6172 6974  rity: (Regularit
│ │ │ │ +00011390: 7929 546f 702c 2069 7320 6120 2a6e 6f74  y)Top, is a *not
│ │ │ │ +000113a0: 6520 7379 6d62 6f6c 3a0a 284d 6163 6175  e symbol:.(Macau
│ │ │ │ +000113b0: 6c61 7932 446f 6329 5379 6d62 6f6c 2c2e  lay2Doc)Symbol,.
│ │ │ │ +000113c0: 0a1f 0a46 696c 653a 2042 4747 2e69 6e66  ...File: BGG.inf
│ │ │ │ +000113d0: 6f2c 204e 6f64 653a 2073 796d 4578 742c  o, Node: symExt,
│ │ │ │ +000113e0: 204e 6578 743a 2074 6174 6552 6573 6f6c   Next: tateResol
│ │ │ │ +000113f0: 7574 696f 6e2c 2050 7265 763a 2052 6567  ution, Prev: Reg
│ │ │ │ +00011400: 756c 6172 6974 792c 2055 703a 2054 6f70  ularity, Up: Top
│ │ │ │ +00011410: 0a0a 7379 6d45 7874 202d 2d20 7468 6520  ..symExt -- the 
│ │ │ │ +00011420: 6669 7273 7420 6469 6666 6572 656e 7469  first differenti
│ │ │ │ +00011430: 616c 206f 6620 7468 6520 636f 6d70 6c65  al of the comple
│ │ │ │ +00011440: 7820 5228 4d29 0a2a 2a2a 2a2a 2a2a 2a2a  x R(M).*********
│ │ │ │ +00011450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ +00011480: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ +00011490: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +000114a0: 2020 2020 7379 6d45 7874 286d 2c45 290a      symExt(m,E).
│ │ │ │ +000114b0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +000114c0: 2020 2a20 6d2c 2061 202a 6e6f 7465 206d    * m, a *note m
│ │ │ │ +000114d0: 6174 7269 783a 2028 4d61 6361 756c 6179  atrix: (Macaulay
│ │ │ │ +000114e0: 3244 6f63 294d 6174 7269 782c 2c20 6120  2Doc)Matrix,, a 
│ │ │ │ +000114f0: 7072 6573 656e 7461 7469 6f6e 206d 6174  presentation mat
│ │ │ │ +00011500: 7269 7820 666f 7220 610a 2020 2020 2020  rix for a.      
│ │ │ │ +00011510: 2020 706f 7369 7469 7665 6c79 2067 7261    positively gra
│ │ │ │ +00011520: 6465 6420 6d6f 6475 6c65 204d 206f 7665  ded module M ove
│ │ │ │ +00011530: 7220 6120 706f 6c79 6e6f 6d69 616c 2072  r a polynomial r
│ │ │ │ +00011540: 696e 670a 2020 2020 2020 2a20 452c 2061  ing.      * E, a
│ │ │ │ +00011550: 202a 6e6f 7465 2070 6f6c 796e 6f6d 6961   *note polynomia
│ │ │ │ +00011560: 6c20 7269 6e67 3a20 284d 6163 6175 6c61  l ring: (Macaula
│ │ │ │ +00011570: 7932 446f 6329 506f 6c79 6e6f 6d69 616c  y2Doc)Polynomial
│ │ │ │ +00011580: 5269 6e67 2c2c 2065 7874 6572 696f 720a  Ring,, exterior.
│ │ │ │ +00011590: 2020 2020 2020 2020 616c 6765 6272 610a          algebra.
│ │ │ │ +000115a0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +000115b0: 2020 202a 2061 202a 6e6f 7465 206d 6174     * a *note mat
│ │ │ │ +000115c0: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ +000115d0: 6f63 294d 6174 7269 782c 2c20 6120 6d61  oc)Matrix,, a ma
│ │ │ │ +000115e0: 7472 6978 2072 6570 7265 7365 6e74 696e  trix representin
│ │ │ │ +000115f0: 6720 7468 6520 6d61 700a 2020 2020 2020  g the map.      
│ │ │ │ +00011600: 2020 4d5f 3120 2a2a 206f 6d65 6761 5f45    M_1 ** omega_E
│ │ │ │ +00011610: 203c 2d2d 204d 5f30 202a 2a20 6f6d 6567   <-- M_0 ** omeg
│ │ │ │ +00011620: 615f 450a 0a44 6573 6372 6970 7469 6f6e  a_E..Description
│ │ │ │ +00011630: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ +00011640: 6973 2066 756e 6374 696f 6e20 7461 6b65  is function take
│ │ │ │ +00011650: 7320 6173 2069 6e70 7574 2061 206d 6174  s as input a mat
│ │ │ │ +00011660: 7269 7820 6d20 7769 7468 206c 696e 6561  rix m with linea
│ │ │ │ +00011670: 7220 656e 7472 6965 732c 2077 6869 6368  r entries, which
│ │ │ │ +00011680: 2077 6520 7468 696e 6b20 6f66 0a61 7320   we think of.as 
│ │ │ │ +00011690: 6120 7072 6573 656e 7461 7469 6f6e 206d  a presentation m
│ │ │ │ +000116a0: 6174 7269 7820 666f 7220 6120 706f 7369  atrix for a posi
│ │ │ │ +000116b0: 7469 7665 6c79 2067 7261 6465 6420 532d  tively graded S-
│ │ │ │ +000116c0: 6d6f 6475 6c65 204d 206d 6174 7269 7820  module M matrix 
│ │ │ │ +000116d0: 7265 7072 6573 656e 7469 6e67 0a74 6865  representing.the
│ │ │ │ +000116e0: 206d 6170 204d 5f31 202a 2a20 6f6d 6567   map M_1 ** omeg
│ │ │ │ +000116f0: 615f 4520 3c2d 2d20 4d5f 3020 2a2a 206f  a_E <-- M_0 ** o
│ │ │ │ +00011700: 6d65 6761 5f45 2077 6869 6368 2069 7320  mega_E which is 
│ │ │ │ +00011710: 7468 6520 6669 7273 7420 6469 6666 6572  the first differ
│ │ │ │ +00011720: 656e 7469 616c 206f 660a 7468 6520 636f  ential of.the co
│ │ │ │ +00011730: 6d70 6c65 7820 5228 4d29 2e0a 2b2d 2d2d  mplex R(M)..+---
│ │ │ │ +00011740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011770: 2b0a 7c69 3120 3a20 5320 3d20 5a5a 2f33  +.|i1 : S = ZZ/3
│ │ │ │ +00011780: 3230 3033 5b78 5f30 2e2e 785f 325d 3b20  2003[x_0..x_2]; 
│ │ │ │ +00011790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000117a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000117b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000117c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000117d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000117e0: 3220 3a20 4520 3d20 5a5a 2f33 3230 3033  2 : E = ZZ/32003
│ │ │ │ +000117f0: 5b65 5f30 2e2e 655f 322c 2053 6b65 7743  [e_0..e_2, SkewC
│ │ │ │ +00011800: 6f6d 6d75 7461 7469 7665 3d3e 7472 7565  ommutative=>true
│ │ │ │ +00011810: 5d3b 7c0a 2b2d 2d2d 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 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +00011850: 4d20 3d20 636f 6b65 7220 6d61 7472 6978  M = coker matrix
│ │ │ │ +00011860: 207b 7b78 5f30 5e32 2c20 785f 315e 327d   {{x_0^2, x_1^2}
│ │ │ │ +00011870: 7d3b 2020 2020 2020 2020 2020 2020 7c0a  };            |.
│ │ │ │ +00011880: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00011890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000118a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000118b0: 2d2d 2d2d 2b0a 7c69 3420 3a20 6d20 3d20  ----+.|i4 : m = 
│ │ │ │ +000118c0: 7072 6573 656e 7461 7469 6f6e 2074 7275  presentation tru
│ │ │ │ +000118d0: 6e63 6174 6528 7265 6775 6c61 7269 7479  ncate(regularity
│ │ │ │ +000118e0: 204d 2c4d 293b 2020 2020 7c0a 7c20 2020   M,M);    |.|   
│ │ │ │ +000118f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011920: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00011930: 3420 2020 2020 2038 2020 2020 2020 2020  4      8        
│ │ │ │  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
│ │ │ │ +00011950: 2020 2020 2020 7c0a 7c6f 3420 3a20 4d61        |.|o4 : Ma
│ │ │ │ +00011960: 7472 6978 2053 2020 3c2d 2d20 5320 2020  trix S  <-- S   
│ │ │ │  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  ----------------
│ │ │ │ -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)   
│ │ │ │ +00011980: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00011990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000119a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000119b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000119c0: 2d2d 2b0a 7c69 3520 3a20 7379 6d45 7874  --+.|i5 : symExt
│ │ │ │ +000119d0: 286d 2c45 2920 2020 2020 2020 2020 2020  (m,E)           
│ │ │ │ +000119e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000119f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00011a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00011a20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00011a30: 7c6f 3520 3d20 7b2d 317d 207c 2065 5f32  |o5 = {-1} | e_2
│ │ │ │ +00011a40: 2030 2020 2030 2020 2030 2020 207c 2020   0   0   0   |  
│ │ │ │  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   |    
│ │ │ │ +00011a60: 2020 2020 7c0a 7c20 2020 2020 7b2d 317d      |.|     {-1}
│ │ │ │ +00011a70: 207c 2065 5f31 2065 5f32 2030 2020 2030   | e_1 e_2 0   0
│ │ │ │ +00011a80: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ +00011a90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00011aa0: 2020 7b2d 317d 207c 2065 5f30 2030 2020    {-1} | e_0 0  
│ │ │ │ +00011ab0: 2065 5f32 2030 2020 207c 2020 2020 2020   e_2 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                  
│ │ │ │ +00011ad0: 7c0a 7c20 2020 2020 7b2d 317d 207c 2030  |.|     {-1} | 0
│ │ │ │ +00011ae0: 2020 2065 5f30 2065 5f31 2065 5f32 207c     e_0 e_1 e_2 |
│ │ │ │ +00011af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011b00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00011b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00011b40: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +00011b50: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │  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    
│ │ │ │ +00011b70: 2020 7c0a 7c6f 3520 3a20 4d61 7472 6978    |.|o5 : Matrix
│ │ │ │ +00011b80: 2045 2020 3c2d 2d20 4520 2020 2020 2020   E  <-- E       
│ │ │ │  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  ----------------
│ │ │ │ -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  ****************
│ │ │ │ -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  ----------------
│ │ │ │ -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  ----------------
│ │ │ │ -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  ----------------
│ │ │ │ -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}};    
│ │ │ │ +00011ba0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00011bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00011be0: 0a43 6176 6561 740a 3d3d 3d3d 3d3d 0a0a  .Caveat.======..
│ │ │ │ +00011bf0: 5468 6973 2066 756e 6374 696f 6e20 6973  This function is
│ │ │ │ +00011c00: 2061 2071 7569 636b 2d61 6e64 2d64 6972   a quick-and-dir
│ │ │ │ +00011c10: 7479 2074 6f6f 6c20 7768 6963 6820 7265  ty tool which re
│ │ │ │ +00011c20: 7175 6972 6573 206c 6974 746c 6520 636f  quires little co
│ │ │ │ +00011c30: 6d70 7574 6174 696f 6e2e 0a48 6f77 6576  mputation..Howev
│ │ │ │ +00011c40: 6572 2069 6620 6974 2069 7320 6361 6c6c  er if it is call
│ │ │ │ +00011c50: 6564 206f 6e20 7477 6f20 7375 6363 6573  ed on two succes
│ │ │ │ +00011c60: 7369 7665 2074 7275 6e63 6174 696f 6e73  sive truncations
│ │ │ │ +00011c70: 206f 6620 6120 6d6f 6475 6c65 2c20 7468   of a module, th
│ │ │ │ +00011c80: 656e 2074 6865 0a6d 6170 7320 6974 2070  en the.maps it p
│ │ │ │ +00011c90: 726f 6475 6365 7320 6d61 7920 4e4f 5420  roduces may NOT 
│ │ │ │ +00011ca0: 636f 6d70 6f73 6520 746f 207a 6572 6f20  compose to zero 
│ │ │ │ +00011cb0: 6265 6361 7573 6520 7468 6520 6368 6f69  because the choi
│ │ │ │ +00011cc0: 6365 206f 6620 6261 7365 7320 6973 206e  ce of bases is n
│ │ │ │ +00011cd0: 6f74 0a63 6f6e 7369 7374 656e 742e 0a0a  ot.consistent...
│ │ │ │ +00011ce0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +00011cf0: 3d0a 0a20 202a 202a 6e6f 7465 2062 6767  =..  * *note bgg
│ │ │ │ +00011d00: 3a20 6267 672c 202d 2d20 7468 6520 6974  : bgg, -- the it
│ │ │ │ +00011d10: 6820 6469 6666 6572 656e 7469 616c 206f  h differential o
│ │ │ │ +00011d20: 6620 7468 6520 636f 6d70 6c65 7820 5228  f the complex R(
│ │ │ │ +00011d30: 4d29 0a0a 5761 7973 2074 6f20 7573 6520  M)..Ways to use 
│ │ │ │ +00011d40: 7379 6d45 7874 3a0a 3d3d 3d3d 3d3d 3d3d  symExt:.========
│ │ │ │ +00011d50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00011d60: 2022 7379 6d45 7874 284d 6174 7269 782c   "symExt(Matrix,
│ │ │ │ +00011d70: 506f 6c79 6e6f 6d69 616c 5269 6e67 2922  PolynomialRing)"
│ │ │ │ +00011d80: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +00011d90: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +00011da0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +00011db0: 6563 7420 2a6e 6f74 6520 7379 6d45 7874  ect *note symExt
│ │ │ │ +00011dc0: 3a20 7379 6d45 7874 2c20 6973 2061 202a  : symExt, is a *
│ │ │ │ +00011dd0: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +00011de0: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ +00011df0: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +00011e00: 6f6e 2c2e 0a1f 0a46 696c 653a 2042 4747  on,....File: BGG
│ │ │ │ +00011e10: 2e69 6e66 6f2c 204e 6f64 653a 2074 6174  .info, Node: tat
│ │ │ │ +00011e20: 6552 6573 6f6c 7574 696f 6e2c 204e 6578  eResolution, Nex
│ │ │ │ +00011e30: 743a 2075 6e69 7665 7273 616c 4578 7465  t: universalExte
│ │ │ │ +00011e40: 6e73 696f 6e2c 2050 7265 763a 2073 796d  nsion, Prev: sym
│ │ │ │ +00011e50: 4578 742c 2055 703a 2054 6f70 0a0a 7461  Ext, Up: Top..ta
│ │ │ │ +00011e60: 7465 5265 736f 6c75 7469 6f6e 202d 2d20  teResolution -- 
│ │ │ │ +00011e70: 6669 6e69 7465 2070 6965 6365 206f 6620  finite piece of 
│ │ │ │ +00011e80: 7468 6520 5461 7465 2072 6573 6f6c 7574  the Tate resolut
│ │ │ │ +00011e90: 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ion.************
│ │ │ │ +00011ea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011eb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00011ec0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ +00011ed0: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ +00011ee0: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00011ef0: 2020 7461 7465 5265 736f 6c75 7469 6f6e    tateResolution
│ │ │ │ +00011f00: 286d 2c45 2c6c 2c68 290a 2020 2a20 496e  (m,E,l,h).  * In
│ │ │ │ +00011f10: 7075 7473 3a0a 2020 2020 2020 2a20 6d2c  puts:.      * m,
│ │ │ │ +00011f20: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ +00011f30: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +00011f40: 6174 7269 782c 2c20 6120 7072 6573 656e  atrix,, a presen
│ │ │ │ +00011f50: 7461 7469 6f6e 206d 6174 7269 7820 666f  tation matrix fo
│ │ │ │ +00011f60: 7220 610a 2020 2020 2020 2020 6d6f 6475  r a.        modu
│ │ │ │ +00011f70: 6c65 0a20 2020 2020 202a 2045 2c20 6120  le.      * E, a 
│ │ │ │ +00011f80: 2a6e 6f74 6520 706f 6c79 6e6f 6d69 616c  *note polynomial
│ │ │ │ +00011f90: 2072 696e 673a 2028 4d61 6361 756c 6179   ring: (Macaulay
│ │ │ │ +00011fa0: 3244 6f63 2950 6f6c 796e 6f6d 6961 6c52  2Doc)PolynomialR
│ │ │ │ +00011fb0: 696e 672c 2c20 6578 7465 7269 6f72 0a20  ing,, exterior. 
│ │ │ │ +00011fc0: 2020 2020 2020 2061 6c67 6562 7261 0a20         algebra. 
│ │ │ │ +00011fd0: 2020 2020 202a 206c 2c20 616e 202a 6e6f       * l, an *no
│ │ │ │ +00011fe0: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +00011ff0: 6175 6c61 7932 446f 6329 5a5a 2c2c 206c  aulay2Doc)ZZ,, l
│ │ │ │ +00012000: 6f77 6572 2063 6f68 6f6d 6f6c 6f67 6963  ower cohomologic
│ │ │ │ +00012010: 616c 2064 6567 7265 650a 2020 2020 2020  al degree.      
│ │ │ │ +00012020: 2a20 682c 2061 6e20 2a6e 6f74 6520 696e  * h, an *note in
│ │ │ │ +00012030: 7465 6765 723a 2028 4d61 6361 756c 6179  teger: (Macaulay
│ │ │ │ +00012040: 3244 6f63 295a 5a2c 2c20 7570 7065 7220  2Doc)ZZ,, upper 
│ │ │ │ +00012050: 626f 756e 6420 6f6e 2074 6865 0a20 2020  bound on the.   
│ │ │ │ +00012060: 2020 2020 2063 6f68 6f6d 6f6c 6f67 6963       cohomologic
│ │ │ │ +00012070: 616c 2064 6567 7265 650a 2020 2a20 4f75  al degree.  * Ou
│ │ │ │ +00012080: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ +00012090: 202a 6e6f 7465 2063 6861 696e 2063 6f6d   *note chain com
│ │ │ │ +000120a0: 706c 6578 3a20 284d 6163 6175 6c61 7932  plex: (Macaulay2
│ │ │ │ +000120b0: 446f 6329 4368 6169 6e43 6f6d 706c 6578  Doc)ChainComplex
│ │ │ │ +000120c0: 2c2c 2061 2066 696e 6974 6520 7069 6563  ,, a finite piec
│ │ │ │ +000120d0: 6520 6f66 0a20 2020 2020 2020 2074 6865  e of.        the
│ │ │ │ +000120e0: 2054 6174 6520 7265 736f 6c75 7469 6f6e   Tate resolution
│ │ │ │ +000120f0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00012100: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320  =========..This 
│ │ │ │ +00012110: 6675 6e63 7469 6f6e 2074 616b 6573 2061  function takes a
│ │ │ │ +00012120: 7320 696e 7075 7420 6120 7072 6573 656e  s input a presen
│ │ │ │ +00012130: 7461 7469 6f6e 206d 6174 7269 7820 6d20  tation matrix m 
│ │ │ │ +00012140: 6f66 2061 2066 696e 6974 656c 7920 6765  of a finitely ge
│ │ │ │ +00012150: 6e65 7261 7465 640a 6772 6164 6564 2053  nerated.graded S
│ │ │ │ +00012160: 2d6d 6f64 756c 6520 4d20 616e 2065 7874  -module M an ext
│ │ │ │ +00012170: 6572 696f 7220 616c 6765 6272 6120 4520  erior algebra E 
│ │ │ │ +00012180: 616e 6420 7477 6f20 696e 7465 6765 7273  and two integers
│ │ │ │ +00012190: 206c 2061 6e64 2068 2e20 4966 2072 2069   l and h. If r i
│ │ │ │ +000121a0: 7320 7468 650a 7265 6775 6c61 7269 7479  s the.regularity
│ │ │ │ +000121b0: 206f 6620 4d2c 2074 6865 6e20 7468 6973   of M, then this
│ │ │ │ +000121c0: 2066 756e 6374 696f 6e20 636f 6d70 7574   function comput
│ │ │ │ +000121d0: 6573 2074 6865 2070 6965 6365 206f 6620  es the piece of 
│ │ │ │ +000121e0: 7468 6520 5461 7465 2072 6573 6f6c 7574  the Tate resolut
│ │ │ │ +000121f0: 696f 6e0a 6672 6f6d 2063 6f68 6f6d 6f6c  ion.from cohomol
│ │ │ │ +00012200: 6f67 6963 616c 2064 6567 7265 6520 6c20  ogical degree l 
│ │ │ │ +00012210: 746f 2063 6f68 6f6d 6f6c 6f67 6963 616c  to cohomological
│ │ │ │ +00012220: 2064 6567 7265 6520 6d61 7828 722b 322c   degree max(r+2,
│ │ │ │ +00012230: 6829 2e20 466f 7220 696e 7374 616e 6365  h). For instance
│ │ │ │ +00012240: 2c0a 666f 7220 7468 6520 686f 6d6f 6765  ,.for the homoge
│ │ │ │ +00012250: 6e65 6f75 7320 636f 6f72 6469 6e61 7465  neous coordinate
│ │ │ │ +00012260: 2072 696e 6720 6f66 2061 2070 6f69 6e74   ring of a point
│ │ │ │ +00012270: 2069 6e20 7468 6520 7072 6f6a 6563 7469   in the projecti
│ │ │ │ +00012280: 7665 2070 6c61 6e65 3a0a 2b2d 2d2d 2d2d  ve plane:.+-----
│ │ │ │ +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 2b0a  --------------+.
│ │ │ │ +000122c0: 7c69 3120 3a20 5320 3d20 5a5a 2f33 3230  |i1 : S = ZZ/320
│ │ │ │ +000122d0: 3033 5b78 5f30 2e2e 785f 325d 3b20 2020  03[x_0..x_2];   
│ │ │ │ +000122e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000122f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00012300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00012330: 3a20 4520 3d20 5a5a 2f33 3230 3033 5b65  : E = ZZ/32003[e
│ │ │ │ +00012340: 5f30 2e2e 655f 322c 2053 6b65 7743 6f6d  _0..e_2, SkewCom
│ │ │ │ +00012350: 6d75 7461 7469 7665 3d3e 7472 7565 5d3b  mutative=>true];
│ │ │ │ +00012360: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00012370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012390: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 6d20  ------+.|i3 : m 
│ │ │ │ +000123a0: 3d20 6d61 7472 6978 7b7b 785f 302c 785f  = matrix{{x_0,x_
│ │ │ │ +000123b0: 317d 7d3b 2020 2020 2020 2020 2020 2020  1}};            
│ │ │ │ +000123c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +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 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00012410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012400: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00012410: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │  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  ----------------
│ │ │ │ -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          
│ │ │ │ +00012430: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +00012440: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +00012450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012460: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00012470: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00012480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000124a0: 2d2d 2d2d 2b0a 7c69 3420 3a20 7265 6775  ----+.|i4 : regu
│ │ │ │ +000124b0: 6c61 7269 7479 2063 6f6b 6572 206d 2020  larity coker m  
│ │ │ │ +000124c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000124d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000124e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000124f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012510: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00012510: 7c0a 7c6f 3420 3d20 3020 2020 2020 2020  |.|o4 = 0       
│ │ │ │  00012520: 2020 2020 2020 2020 2020 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  ----------------
│ │ │ │ -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                  
│ │ │ │ -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                  
│ │ │ │ -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                  
│ │ │ │ +00012540: 2020 2020 2020 7c0a 2b2d 2d2d 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 2b0a 7c69  ------------+.|i
│ │ │ │ +00012580: 3520 3a20 5420 3d20 7461 7465 5265 736f  5 : T = tateReso
│ │ │ │ +00012590: 6c75 7469 6f6e 286d 2c45 2c2d 322c 3429  lution(m,E,-2,4)
│ │ │ │ +000125a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000125b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000125c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000125d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000125e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000125f0: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ +00012600: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00012610: 2020 2020 3120 2020 2020 2031 2020 7c0a      1      1  |.
│ │ │ │ +00012620: 7c6f 3520 3d20 4520 203c 2d2d 2045 2020  |o5 = E  <-- E  
│ │ │ │ +00012630: 3c2d 2d20 4520 203c 2d2d 2045 2020 3c2d  <-- E  <-- E  <-
│ │ │ │ +00012640: 2d20 4520 203c 2d2d 2045 2020 3c2d 2d20  - E  <-- E  <-- 
│ │ │ │ +00012650: 4520 2020 7c0a 7c20 2020 2020 2020 2020  E   |.|         
│ │ │ │ +00012660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012680: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00012690: 2020 3020 2020 2020 2031 2020 2020 2020    0      1      
│ │ │ │ +000126a0: 3220 2020 2020 2033 2020 2020 2020 3420  2      3      4 
│ │ │ │ +000126b0: 2020 2020 2035 2020 2020 2020 3620 2020       5      6   
│ │ │ │ +000126c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000126d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000126e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000126f0: 2020 2020 2020 7c0a 7c6f 3520 3a20 4368        |.|o5 : Ch
│ │ │ │ +00012700: 6169 6e43 6f6d 706c 6578 2020 2020 2020  ainComplex      
│ │ │ │  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  ----------------
│ │ │ │ -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       
│ │ │ │ +00012720: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00012730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012760: 2d2d 2b0a 7c69 3620 3a20 6265 7474 6920  --+.|i6 : betti 
│ │ │ │ +00012770: 5420 2020 2020 2020 2020 2020 2020 2020  T               
│ │ │ │ +00012780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012790: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000127a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +000127c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000127d0: 7c20 2020 2020 2020 2020 2020 2030 2031  |            0 1
│ │ │ │ +000127e0: 2032 2033 2034 2035 2036 2020 2020 2020   2 3 4 5 6      
│ │ │ │  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        
│ │ │ │ +00012800: 2020 2020 7c0a 7c6f 3620 3d20 746f 7461      |.|o6 = tota
│ │ │ │ +00012810: 6c3a 2031 2031 2031 2031 2031 2031 2031  l: 1 1 1 1 1 1 1
│ │ │ │ +00012820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012830: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00012840: 2020 2020 202d 343a 2031 2031 2031 2031       -4: 1 1 1 1
│ │ │ │ +00012850: 2031 2031 2031 2020 2020 2020 2020 2020   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  
│ │ │ │ +00012870: 7c0a 7c20 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 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000128b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000128a0: 2020 2020 2020 7c0a 7c6f 3620 3a20 4265        |.|o6 : Be
│ │ │ │ +000128b0: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │  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  ----------------
│ │ │ │ -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        
│ │ │ │ +000128d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000128e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000128f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012910: 2d2d 2b0a 7c69 3720 3a20 542e 6464 5f31  --+.|i7 : T.dd_1
│ │ │ │ +00012920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012940: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00012950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +00012970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00012980: 7c6f 3720 3d20 7b2d 347d 207c 2065 5f32  |o7 = {-4} | e_2
│ │ │ │ +00012990: 207c 2020 2020 2020 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 |      
│ │ │ │ +000129b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000129c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000129d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000129e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000129f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000129e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000129f0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +00012a00: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  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      
│ │ │ │ +00012a20: 7c0a 7c6f 3720 3a20 4d61 7472 6978 2045  |.|o7 : Matrix E
│ │ │ │ +00012a30: 2020 3c2d 2d20 4520 2020 2020 2020 2020    <-- E         
│ │ │ │  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  ----------------
│ │ │ │ -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  ****************
│ │ │ │ -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...+
│ │ │ │ +00012a50: 2020 2020 2020 7c0a 2b2d 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 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +00012a90: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +00012aa0: 0a0a 2020 2a20 2a6e 6f74 6520 7379 6d45  ..  * *note symE
│ │ │ │ +00012ab0: 7874 3a20 7379 6d45 7874 2c20 2d2d 2074  xt: symExt, -- t
│ │ │ │ +00012ac0: 6865 2066 6972 7374 2064 6966 6665 7265  he first differe
│ │ │ │ +00012ad0: 6e74 6961 6c20 6f66 2074 6865 2063 6f6d  ntial of the com
│ │ │ │ +00012ae0: 706c 6578 2052 284d 290a 0a57 6179 7320  plex R(M)..Ways 
│ │ │ │ +00012af0: 746f 2075 7365 2074 6174 6552 6573 6f6c  to use tateResol
│ │ │ │ +00012b00: 7574 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d  ution:.=========
│ │ │ │ +00012b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00012b20: 3d3d 0a0a 2020 2a20 2274 6174 6552 6573  ==..  * "tateRes
│ │ │ │ +00012b30: 6f6c 7574 696f 6e28 4d61 7472 6978 2c50  olution(Matrix,P
│ │ │ │ +00012b40: 6f6c 796e 6f6d 6961 6c52 696e 672c 5a5a  olynomialRing,ZZ
│ │ │ │ +00012b50: 2c5a 5a29 220a 0a46 6f72 2074 6865 2070  ,ZZ)"..For the p
│ │ │ │ +00012b60: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +00012b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +00012b80: 6520 6f62 6a65 6374 202a 6e6f 7465 2074  e object *note t
│ │ │ │ +00012b90: 6174 6552 6573 6f6c 7574 696f 6e3a 2074  ateResolution: t
│ │ │ │ +00012ba0: 6174 6552 6573 6f6c 7574 696f 6e2c 2069  ateResolution, i
│ │ │ │ +00012bb0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +00012bc0: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ +00012bd0: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ +00012be0: 756e 6374 696f 6e2c 2e0a 1f0a 4669 6c65  unction,....File
│ │ │ │ +00012bf0: 3a20 4247 472e 696e 666f 2c20 4e6f 6465  : BGG.info, Node
│ │ │ │ +00012c00: 3a20 756e 6976 6572 7361 6c45 7874 656e  : universalExten
│ │ │ │ +00012c10: 7369 6f6e 2c20 5072 6576 3a20 7461 7465  sion, Prev: tate
│ │ │ │ +00012c20: 5265 736f 6c75 7469 6f6e 2c20 5570 3a20  Resolution, Up: 
│ │ │ │ +00012c30: 546f 700a 0a75 6e69 7665 7273 616c 4578  Top..universalEx
│ │ │ │ +00012c40: 7465 6e73 696f 6e20 2d2d 2055 6e69 7665  tension -- Unive
│ │ │ │ +00012c50: 7273 616c 2065 7874 656e 7369 6f6e 206f  rsal extension o
│ │ │ │ +00012c60: 6620 7665 6374 6f72 2062 756e 646c 6573  f vector bundles
│ │ │ │ +00012c70: 206f 6e20 505e 310a 2a2a 2a2a 2a2a 2a2a   on P^1.********
│ │ │ │ +00012c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00012cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ +00012cc0: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ +00012cd0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +00012ce0: 2020 2045 203d 2075 6e69 7665 7273 616c     E = universal
│ │ │ │ +00012cf0: 4578 7465 6e73 696f 6e28 4c61 2c20 4c62  Extension(La, Lb
│ │ │ │ +00012d00: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ +00012d10: 2020 2020 2a20 4c61 2c20 6120 2a6e 6f74      * La, a *not
│ │ │ │ +00012d20: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ +00012d30: 7932 446f 6329 4c69 7374 2c2c 206f 6620  y2Doc)List,, of 
│ │ │ │ +00012d40: 696e 7465 6765 7273 0a20 2020 2020 202a  integers.      *
│ │ │ │ +00012d50: 204c 622c 2061 202a 6e6f 7465 206c 6973   Lb, a *note lis
│ │ │ │ +00012d60: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ +00012d70: 294c 6973 742c 2c20 6f66 2069 6e74 6567  )List,, of integ
│ │ │ │ +00012d80: 6572 730a 2020 2a20 4f75 7470 7574 733a  ers.  * Outputs:
│ │ │ │ +00012d90: 0a20 2020 2020 202a 2045 2c20 6120 2a6e  .      * E, a *n
│ │ │ │ +00012da0: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +00012db0: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +00012dc0: 2c2c 2072 6570 7265 7365 6e74 696e 6720  ,, representing 
│ │ │ │ +00012dd0: 7468 6520 6578 7465 6e73 696f 6e0a 0a44  the extension..D
│ │ │ │ +00012de0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00012df0: 3d3d 3d3d 3d3d 0a0a 4576 6572 7920 7665  ======..Every ve
│ │ │ │ +00012e00: 6374 6f72 2062 756e 646c 6520 4520 6f6e  ctor bundle E on
│ │ │ │ +00012e10: 2024 5c50 505e 3124 2073 706c 6974 7320   $\PP^1$ splits 
│ │ │ │ +00012e20: 6173 2061 2073 756d 206f 6620 6c69 6e65  as a sum of line
│ │ │ │ +00012e30: 2062 756e 646c 6573 204f 4f28 615f 6929   bundles OO(a_i)
│ │ │ │ +00012e40: 2e20 4966 204c 610a 6973 2061 206c 6973  . If La.is a lis
│ │ │ │ +00012e50: 7420 6f66 2069 6e74 6567 6572 732c 2077  t of integers, w
│ │ │ │ +00012e60: 6520 7772 6974 6520 4528 4c61 2920 666f  e write E(La) fo
│ │ │ │ +00012e70: 7220 7468 6520 6469 7265 6374 2073 756d  r the direct sum
│ │ │ │ +00012e80: 206f 6620 7468 6520 6c69 6e65 2062 756e   of the line bun
│ │ │ │ +00012e90: 646c 650a 4f4f 284c 615f 6929 2e20 2047  dle.OO(La_i).  G
│ │ │ │ +00012ea0: 6976 656e 2074 776f 2073 7563 6820 6275  iven two such bu
│ │ │ │ +00012eb0: 6e64 6c65 7320 7370 6563 6966 6965 6420  ndles specified 
│ │ │ │ +00012ec0: 6279 2074 6865 206c 6973 7473 204c 6120  by the lists La 
│ │ │ │ +00012ed0: 616e 6420 4c62 2074 6869 7320 7363 7269  and Lb this scri
│ │ │ │ +00012ee0: 7074 0a63 6f6e 7374 7275 6374 7320 6120  pt.constructs a 
│ │ │ │ +00012ef0: 6d6f 6475 6c65 2072 6570 7265 7365 6e74  module represent
│ │ │ │ +00012f00: 696e 6720 7468 6520 756e 6976 6572 7361  ing the universa
│ │ │ │ +00012f10: 6c20 6578 7465 6e73 696f 6e20 6f66 2045  l extension of E
│ │ │ │ +00012f20: 284c 6229 2062 7920 4528 4c61 292e 2049  (Lb) by E(La). I
│ │ │ │ +00012f30: 740a 6973 2064 6566 696e 6564 206f 6e20  t.is defined on 
│ │ │ │ +00012f40: 7468 6520 7072 6f64 7563 7420 7661 7269  the product vari
│ │ │ │ +00012f50: 6574 7920 4578 745e 3128 4528 4c61 292c  ety Ext^1(E(La),
│ │ │ │ +00012f60: 2045 284c 6229 2920 7820 245c 5050 5e31   E(Lb)) x $\PP^1
│ │ │ │ +00012f70: 242c 2061 6e64 0a72 6570 7265 7365 6e74  $, and.represent
│ │ │ │ +00012f80: 6564 2068 6572 6520 6279 2061 2067 7261  ed here by a gra
│ │ │ │ +00012f90: 6465 6420 6d6f 6475 6c65 206f 7665 7220  ded module over 
│ │ │ │ +00012fa0: 7468 6520 636f 6f72 6469 6e61 7465 2072  the coordinate r
│ │ │ │ +00012fb0: 696e 6720 5320 3d20 415b 795f 302c 795f  ing S = A[y_0,y_
│ │ │ │ +00012fc0: 315d 206f 660a 7468 6973 2076 6172 6965  1] of.this varie
│ │ │ │ +00012fd0: 7479 3b20 6865 7265 2041 2069 7320 7468  ty; here A is th
│ │ │ │ +00012fe0: 6520 636f 6f72 6469 6e61 7465 2072 696e  e coordinate rin
│ │ │ │ +00012ff0: 6720 6f66 2045 7874 5e31 2845 284c 6129  g of Ext^1(E(La)
│ │ │ │ +00013000: 2c20 4528 4c62 2929 2c20 7768 6963 6820  , E(Lb)), which 
│ │ │ │ +00013010: 6973 2061 0a70 6f6c 796e 6f6d 6961 6c20  is a.polynomial 
│ │ │ │ +00013020: 7269 6e67 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ring...+--------
│ │ │ │ +00013030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013070: 2d2d 2d2d 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})          
│ │ │ │ +00013070: 2d2d 2d2d 2d2b 0a7c 6931 203a 204d 203d  -----+.|i1 : M =
│ │ │ │ +00013080: 2075 6e69 7665 7273 616c 4578 7465 6e73   universalExtens
│ │ │ │ +00013090: 696f 6e28 7b2d 327d 2c20 7b32 7d29 2020  ion({-2}, {2})  
│ │ │ │ +000130a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000130b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000130c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000130d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000130e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000130f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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                  
│ │ │ │ -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 |            
│ │ │ │ +00013110: 2020 2020 207c 0a7c 6f31 203d 2063 6f6b       |.|o1 = cok
│ │ │ │ +00013120: 6572 6e65 6c20 7b32 2c20 307d 207c 2078  ernel {2, 0} | x
│ │ │ │ +00013130: 5f30 2078 5f31 2078 5f32 207c 2020 2020  _0 x_1 x_2 |    
│ │ │ │ +00013140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013170: 2020 2020 2020 7b31 2c20 317d 207c 2079        {1, 1} | y
│ │ │ │ +00013180: 5f30 2030 2020 2030 2020 207c 2020 2020  _0 0   0   |    
│ │ │ │ +00013190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000131a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000131b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000131c0: 2020 2020 2020 7b31 2c20 317d 207c 2079        {1, 1} | y
│ │ │ │ +000131d0: 5f31 2079 5f30 2030 2020 207c 2020 2020  _1 y_0 0   |    
│ │ │ │ +000131e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000131f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013210: 2020 2020 2020 7b31 2c20 317d 207c 2030        {1, 1} | 0
│ │ │ │ +00013220: 2020 2079 5f31 2079 5f30 207c 2020 2020     y_1 y_0 |    
│ │ │ │ +00013230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013260: 2020 2020 2020 7b31 2c20 317d 207c 2030        {1, 1} | 0
│ │ │ │ +00013270: 2020 2030 2020 2079 5f31 207c 2020 2020     0   y_1 |    
│ │ │ │ +00013280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000132a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000132b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000132c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000132d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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: 2020 2020 207c 0a7c 2020 2020 2020 5a5a       |.|      ZZ
│ │ │ │  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               |.+
│ │ │ │ +00013320: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +00013330: 2020 2020 2020 2020 2020 3520 2020 2020            5     
│ │ │ │ +00013340: 2020 2020 207c 0a7c 6f31 203a 202d 2d2d       |.|o1 : ---
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│ │ │ │ +00013360: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +00013370: 7420 6f66 2028 2d2d 2d5b 7820 2e2e 7820  t of (---[x ..x 
│ │ │ │ +00013380: 5d5b 7920 2e2e 7920 5d29 2020 2020 2020  ][y ..y ])      
│ │ │ │ +00013390: 2020 2020 207c 0a7c 2020 2020 2031 3031       |.|     101
│ │ │ │ +000133a0: 2020 3020 2020 3220 2020 3020 2020 3120    0   2   0   1 
│ │ │ │ +000133b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000133c0: 2020 2020 2020 3130 3120 2030 2020 2032        101  0   2
│ │ │ │ +000133d0: 2020 2030 2020 2031 2020 2020 2020 2020     0   1        
│ │ │ │ +000133e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000133f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -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})     
│ │ │ │ +00013430: 2d2d 2d2d 2d2b 0a7c 6932 203a 204d 203d  -----+.|i2 : M =
│ │ │ │ +00013440: 2075 6e69 7665 7273 616c 4578 7465 6e73   universalExtens
│ │ │ │ +00013450: 696f 6e28 7b2d 322c 2d33 7d2c 207b 322c  ion({-2,-3}, {2,
│ │ │ │ +00013460: 337d 2920 2020 2020 2020 2020 2020 2020  3})             
│ │ │ │ +00013470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013480: 2020 2020 207c 0a7c 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 207c 0a7c               |.|
│ │ │ │ +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                  
│ │ │ │ -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           |.|
│ │ │ │ +000134d0: 2020 2020 207c 0a7c 6f32 203d 2063 6f6b       |.|o2 = cok
│ │ │ │ +000134e0: 6572 6e65 6c20 7b32 2c20 307d 207c 2078  ernel {2, 0} | x
│ │ │ │ +000134f0: 5f30 795f 3120 785f 3179 5f31 2078 5f32  _0y_1 x_1y_1 x_2
│ │ │ │ +00013500: 795f 3120 785f 3379 5f31 2078 5f34 795f  y_1 x_3y_1 x_4y_
│ │ │ │ +00013510: 3120 785f 3579 5f31 2078 5f36 795f 3120  1 x_5y_1 x_6y_1 
│ │ │ │ +00013520: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013530: 2020 2020 2020 7b33 2c20 307d 207c 2078        {3, 0} | x
│ │ │ │ +00013540: 5f39 2020 2020 785f 3130 2020 2078 5f31  _9    x_10   x_1
│ │ │ │ +00013550: 3120 2020 785f 3132 2020 2078 5f31 3320  1   x_12   x_13 
│ │ │ │ +00013560: 2020 785f 3134 2020 2078 5f31 3520 2020    x_14   x_15   
│ │ │ │ +00013570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013580: 2020 2020 2020 7b32 2c20 317d 207c 2079        {2, 1} | y
│ │ │ │ +00013590: 5f30 2020 2020 3020 2020 2020 2030 2020  _0    0      0  
│ │ │ │ +000135a0: 2020 2020 3020 2020 2020 2030 2020 2020      0      0    
│ │ │ │ +000135b0: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +000135c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000135d0: 2020 2020 2020 7b32 2c20 317d 207c 2079        {2, 1} | y
│ │ │ │ +000135e0: 5f31 2020 2020 795f 3020 2020 2030 2020  _1    y_0    0  
│ │ │ │ +000135f0: 2020 2020 3020 2020 2020 2030 2020 2020      0      0    
│ │ │ │ +00013600: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013620: 2020 2020 2020 7b32 2c20 317d 207c 2030        {2, 1} | 0
│ │ │ │ +00013630: 2020 2020 2020 795f 3120 2020 2079 5f30        y_1    y_0
│ │ │ │ +00013640: 2020 2020 3020 2020 2020 2030 2020 2020      0      0    
│ │ │ │ +00013650: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013660: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013670: 2020 2020 2020 7b32 2c20 317d 207c 2030        {2, 1} | 0
│ │ │ │ +00013680: 2020 2020 2020 3020 2020 2020 2079 5f31        0      y_1
│ │ │ │ +00013690: 2020 2020 795f 3020 2020 2030 2020 2020      y_0    0    
│ │ │ │ +000136a0: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +000136b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000136c0: 2020 2020 2020 7b32 2c20 317d 207c 2030        {2, 1} | 0
│ │ │ │ +000136d0: 2020 2020 2020 3020 2020 2020 2030 2020        0      0  
│ │ │ │ +000136e0: 2020 2020 795f 3120 2020 2030 2020 2020      y_1    0    
│ │ │ │ +000136f0: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013710: 2020 2020 2020 7b32 2c20 317d 207c 2030        {2, 1} | 0
│ │ │ │ +00013720: 2020 2020 2020 3020 2020 2020 2030 2020        0      0  
│ │ │ │ +00013730: 2020 2020 3020 2020 2020 2079 5f30 2020      0      y_0  
│ │ │ │ +00013740: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013750: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013760: 2020 2020 2020 7b32 2c20 317d 207c 2030        {2, 1} | 0
│ │ │ │ +00013770: 2020 2020 2020 3020 2020 2020 2030 2020        0      0  
│ │ │ │ +00013780: 2020 2020 3020 2020 2020 2079 5f31 2020      0      y_1  
│ │ │ │ +00013790: 2020 795f 3020 2020 2030 2020 2020 2020    y_0    0      
│ │ │ │ +000137a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000137b0: 2020 2020 2020 7b32 2c20 317d 207c 2030        {2, 1} | 0
│ │ │ │ +000137c0: 2020 2020 2020 3020 2020 2020 2030 2020        0      0  
│ │ │ │ +000137d0: 2020 2020 3020 2020 2020 2030 2020 2020      0      0    
│ │ │ │ +000137e0: 2020 795f 3120 2020 2079 5f30 2020 2020    y_1    y_0    
│ │ │ │ +000137f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013800: 2020 2020 2020 7b32 2c20 317d 207c 2030        {2, 1} | 0
│ │ │ │ +00013810: 2020 2020 2020 3020 2020 2020 2030 2020        0      0  
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│ │ │ │ +00013830: 2020 3020 2020 2020 2079 5f31 2020 2020    0      y_1    
│ │ │ │ +00013840: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013850: 2020 2020 2020 7b32 2c20 317d 207c 2030        {2, 1} | 0
│ │ │ │ +00013860: 2020 2020 2020 3020 2020 2020 2030 2020        0      0  
│ │ │ │ +00013870: 2020 2020 3020 2020 2020 2030 2020 2020      0      0    
│ │ │ │ +00013880: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +00013890: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000138a0: 2020 2020 2020 7b32 2c20 317d 207c 2030        {2, 1} | 0
│ │ │ │ +000138b0: 2020 2020 2020 3020 2020 2020 2030 2020        0      0  
│ │ │ │ +000138c0: 2020 2020 3020 2020 2020 2030 2020 2020      0      0    
│ │ │ │ +000138d0: 2020 3020 2020 2020 2030 2020 2020 2020    0      0      
│ │ │ │ +000138e0: 2020 2020 207c 0a7c 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 2020 2020                  
│ │ │ │  00013920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013930: 2020 2020 207c 0a7c 2020 2020 2020 5a5a       |.|      ZZ
│ │ │ │  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                  
│ │ │ │ -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           |.|
│ │ │ │ +00013960: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +00013970: 2020 2020 2020 2020 2020 2020 3133 2020              13  
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│ │ │ │ +000139b0: 6e74 206f 6620 282d 2d2d 5b78 202e 2e78  nt of (---[x ..x
│ │ │ │ +000139c0: 2020 5d5b 7920 2e2e 7920 5d29 2020 2020    ][y ..y ])    
│ │ │ │ +000139d0: 2020 2020 207c 0a7c 2020 2020 2031 3031       |.|     101
│ │ │ │ +000139e0: 2020 3020 2020 3137 2020 2030 2020 2031    0   17   0   1
│ │ │ │ +000139f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013a00: 2020 2020 2020 2031 3031 2020 3020 2020         101  0   
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│ │ │ │ +00013a20: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00013a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00013a60: 2d2d 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 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                  
│ │ │ │ +00013a70: 2d2d 2d2d 2d7c 0a7c 785f 3779 5f31 2078  -----|.|x_7y_1 x
│ │ │ │ +00013a80: 5f38 795f 3120 7c20 2020 2020 2020 2020  _8y_1 |         
│ │ │ │ +00013a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013ac0: 2020 2020 207c 0a7c 785f 3136 2020 2078       |.|x_16   x
│ │ │ │ +00013ad0: 5f31 3720 2020 7c20 2020 2020 2020 2020  _17   |         
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│ │ │ │  00013b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00013c10: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │  00013c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00013d50: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
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│ │ │ │ -00013dc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00013e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00013e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ ├── ./usr/share/info/Benchmark.info.gz
│ │ │ ├── Benchmark.info
│ │ │ │ @@ -146,59 +146,64 @@
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│ │ │ │ -00000cb0: 2054 6162 6c65 0a                         Table.
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│ │ │ │ +00000d00: 2054 6162 6c65 0a                         Table.
│ │ ├── ./usr/share/info/Bertini.info.gz
│ │ │ ├── Bertini.info
│ │ │ │ @@ -2113,7845 +2113,7845 @@
│ │ │ │  00008400: 6e75 6d62 6572 0a20 2020 2020 2020 206f  number.        o
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│ │ │ │ -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
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│ │ │ │ -000084f0: 6f73 653d 3e5f 7064 5f70 645f 7064 5f72  ose=>_pd_pd_pd_r
│ │ │ │ -00008500: 700a 2020 2020 2020 2020 2c20 3d3e 202e  p.        , => .
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -00008710: 7061 7261 6d65 7465 7220 686f 6d6f 746f  parameter homoto
│ │ │ │ -00008720: 7079 0a6d 6574 686f 6420 6172 6520 6361  py.method are ca
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│ │ │ │ -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 
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│ │ │ │ -000087c0: 7220 7468 6973 2069 6e69 7469 616c 2070  r this initial p
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│ │ │ │ -00008810: 6574 6572 7320 7573 696e 6720 6120 6e75  eters using a nu
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│ │ │ │ -00008860: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │ +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.        , => 
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│ │ │ │ +00008520: 7565 2066 616c 7365 2c20 4f70 7469 6f6e  ue false, Option
│ │ │ │ +00008530: 2074 6f20 7369 6c65 6e63 6520 6164 6469   to silence addi
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│ │ │ │ +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
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│ │ │ │ +00008620: 7379 7374 656d 7320 6672 6f6d 2061 2070  systems from a p
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│ │ │ │ +000086d0: 2066 6f72 0a77 6869 6368 2073 6f6c 7574   for.which solut
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│ │ │ │ +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  .+--------------
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│ │ │ │ +00009350: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00009360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -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                 |
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│ │ │ │ +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
│ │ │ │ +0000a7d0: 7075 7453 7479 6c65 3d3e 224f 7574 536f  putStyle=>"OutSo
│ │ │ │ +0000a7e0: 6c75 7469 6f6e 7322 2920 2020 2020 2020  lutions")       
│ │ │ │ +0000a7f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0000a800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000a840: 0a7c 6931 3820 3a20 636c 6173 7320 6250  .|i18 : class bP
│ │ │ │ -0000a850: 4831 5f30 5f30 2020 2020 2020 2020 2020  H1_0_0          
│ │ │ │ +0000a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000a840: 2b0a 7c69 3138 203a 2063 6c61 7373 2062  +.|i18 : class b
│ │ │ │ +0000a850: 5048 315f 305f 3020 2020 2020 2020 2020  PH1_0_0         
│ │ │ │  0000a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a880: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a890: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a890: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a8d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a8e0: 0a7c 6f31 3820 3d20 506f 696e 7420 2020  .|o18 = Point   
│ │ │ │ +0000a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a8e0: 7c0a 7c6f 3138 203d 2050 6f69 6e74 2020  |.|o18 = Point  
│ │ │ │  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 207c                 |
│ │ │ │ -0000a930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a930: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a980: 0a7c 6f31 3820 3a20 5479 7065 2020 2020  .|o18 : Type    
│ │ │ │ +0000a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a980: 7c0a 7c6f 3138 203a 2054 7970 6520 2020  |.|o18 : Type   
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a9d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000a9d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0000a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000aa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000aa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000aa20: 0a7c 6931 3920 3a20 636c 6173 7320 6250  .|i19 : class bP
│ │ │ │ -0000aa30: 4832 5f30 5f30 2020 2020 2020 2020 2020  H2_0_0          
│ │ │ │ +0000aa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000aa20: 2b0a 7c69 3139 203a 2063 6c61 7373 2062  +.|i19 : class b
│ │ │ │ +0000aa30: 5048 325f 305f 3020 2020 2020 2020 2020  PH2_0_0         
│ │ │ │  0000aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aa60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000aa70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aa70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000aab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000aac0: 0a7c 6f31 3920 3d20 4c69 7374 2020 2020  .|o19 = List    
│ │ │ │ +0000aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000aac0: 7c0a 7c6f 3139 203d 204c 6973 7420 2020  |.|o19 = List   
│ │ │ │  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 207c                 |
│ │ │ │ -0000ab10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0000ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ab10: 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000ab60: 0a7c 6f31 3920 3a20 5479 7065 2020 2020  .|o19 : Type    
│ │ │ │ +0000ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ab60: 7c0a 7c6f 3139 203a 2054 7970 6520 2020  |.|o19 : Type   
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000abb0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000abb0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0000abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000abe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000ac00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000ac00: 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +.+-------------
│ │ │ │  0000ac10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ac20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ac30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ac40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000ac50: 0a7c 6932 3020 3a20 6469 7231 203a 3d20  .|i20 : dir1 := 
│ │ │ │ -0000ac60: 7465 6d70 6f72 6172 7946 696c 654e 616d  temporaryFileNam
│ │ │ │ -0000ac70: 6528 293b 202d 2d20 6275 696c 6420 6120  e(); -- build a 
│ │ │ │ -0000ac80: 6469 7265 6374 6f72 7920 746f 2073 746f  directory to sto
│ │ │ │ -0000ac90: 7265 2074 656d 706f 7261 7279 2020 207c  re temporary   |
│ │ │ │ -0000aca0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +0000ac40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000ac50: 2b0a 7c69 3230 203a 2064 6972 3120 3a3d  +.|i20 : dir1 :=
│ │ │ │ +0000ac60: 2074 656d 706f 7261 7279 4669 6c65 4e61   temporaryFileNa
│ │ │ │ +0000ac70: 6d65 2829 3b20 2d2d 2062 7569 6c64 2061  me(); -- build a
│ │ │ │ +0000ac80: 2064 6972 6563 746f 7279 2074 6f20 7374   directory to st
│ │ │ │ +0000ac90: 6f72 6520 7465 6d70 6f72 6172 7920 2020  ore temporary   
│ │ │ │ +0000aca0: 7c0a 7c2d 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0000acf0: 0a7c 6461 7461 2020 2020 2020 2020 2020  .|data          
│ │ │ │ +0000ace0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000acf0: 7c0a 7c64 6174 6120 2020 2020 2020 2020  |.|data         
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000ad40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ad40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0000ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000ad90: 0a7c 6932 3120 3a20 6d61 6b65 4469 7265  .|i21 : makeDire
│ │ │ │ -0000ada0: 6374 6f72 7920 6469 7231 3b20 2020 2020  ctory dir1;     
│ │ │ │ +0000ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0000ad90: 2b0a 7c69 3231 203a 206d 616b 6544 6972  +.|i21 : makeDir
│ │ │ │ +0000ada0: 6563 746f 7279 2064 6972 313b 2020 2020  ectory dir1;    
│ │ │ │  0000adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000add0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000ade0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0000add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0000ade0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  0000adf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ae00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ae10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000ae20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0000ae30: 0a7c 6932 3220 3a20 6250 4835 3d62 6572  .|i22 : bPH5=ber
│ │ │ │ -0000ae40: 7469 6e69 5061 7261 6d65 7465 7248 6f6d  tiniParameterHom
│ │ │ │ -0000ae50: 6f74 6f70 7928 207b 2278 5e32 2d75 3122  otopy( {"x^2-u1"
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│ │ │ │ -0000b710: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
│ │ │ │ -0000b720: 7074 696f 6e61 6c20 696e 7075 7473 2c3a  ptional inputs,:
│ │ │ │ -0000b730: 0a20 2020 2020 202a 2042 6572 7469 6e69  .      * Bertini
│ │ │ │ -0000b740: 496e 7075 7443 6f6e 6669 6775 7261 7469  InputConfigurati
│ │ │ │ -0000b750: 6f6e 2028 6d69 7373 696e 6720 646f 6375  on (missing docu
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│ │ │ │ -0000b770: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -0000b780: 0a20 2020 2020 2020 207b 7d2c 0a20 2020  .        {},.   
│ │ │ │ -0000b790: 2020 202a 202a 6e6f 7465 2049 7350 726f     * *note IsPro
│ │ │ │ -0000b7a0: 6a65 6374 6976 653a 2049 7350 726f 6a65  jective: IsProje
│ │ │ │ -0000b7b0: 6374 6976 652c 203d 3e20 2e2e 2e2c 2064  ctive, => ..., d
│ │ │ │ -0000b7c0: 6566 6175 6c74 2076 616c 7565 202d 312c  efault value -1,
│ │ │ │ -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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│ │ │ │ -0000b800: 2075 7365 2068 6f6d 6f67 656e 656f 7573   use homogeneous
│ │ │ │ -0000b810: 2063 6f6f 7264 696e 6174 6573 0a20 2020   coordinates.   
│ │ │ │ -0000b820: 2020 202a 202a 6e6f 7465 2056 6572 626f     * *note Verbo
│ │ │ │ -0000b830: 7365 3a20 6265 7274 696e 6954 7261 636b  se: bertiniTrack
│ │ │ │ -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, 
│ │ │ │ -0000b8d0: 6120 2a6e 6f74 6520 6e75 6d65 7269 6361  a *note numerica
│ │ │ │ -0000b8e0: 6c20 7661 7269 6574 793a 2028 4e41 4774  l variety: (NAGt
│ │ │ │ -0000b8f0: 7970 6573 294e 756d 6572 6963 616c 5661  ypes)NumericalVa
│ │ │ │ -0000b900: 7269 6574 792c 2c20 6120 6e75 6d65 7269  riety,, a numeri
│ │ │ │ -0000b910: 6361 6c0a 2020 2020 2020 2020 6972 7265  cal.        irre
│ │ │ │ -0000b920: 6475 6369 626c 6520 6465 636f 6d70 6f73  ducible decompos
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│ │ │ │ -0000b950: 460a 0a44 6573 6372 6970 7469 6f6e 0a3d  F..Description.=
│ │ │ │ -0000b960: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -0000b970: 6d65 7468 6f64 2062 6572 7469 6e69 506f  method bertiniPo
│ │ │ │ -0000b980: 7344 696d 536f 6c76 6520 6361 6c6c 7320  sDimSolve calls 
│ │ │ │ -0000b990: 2042 6572 7469 6e69 2074 6f20 6669 6e64   Bertini to find
│ │ │ │ -0000b9a0: 2061 206e 756d 6572 6963 616c 2069 7272   a numerical irr
│ │ │ │ -0000b9b0: 6564 7563 6962 6c65 0a64 6563 6f6d 706f  educible.decompo
│ │ │ │ -0000b9c0: 7369 7469 6f6e 206f 6620 7468 6520 7a65  sition of the ze
│ │ │ │ -0000b9d0: 726f 2d73 6574 206f 6620 462e 2020 5468  ro-set of F.  Th
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│ │ │ │ -0000b9f0: 6973 2072 6574 7572 6e65 6420 6173 2074  is returned as t
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│ │ │ │ -0000ba20: 7970 6573 294e 756d 6572 6963 616c 5661  ypes)NumericalVa
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│ │ │ │ -0000ba80: 2046 3d30 2e20 4265 7274 696e 6920 2831   F=0. Bertini (1
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│ │ │ │ -0000bab0: 2066 696c 6573 2c20 2832 2920 696e 766f   files, (2) invo
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│ │ │ │ -0000bb10: 2066 696e 6420 7769 746e 6573 7320 7375   find witness su
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│ │ │ │ -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
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│ │ │ │ +0000ba80: 6d20 463d 302e 2042 6572 7469 6e69 2028  m F=0. Bertini (
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│ │ │ │ +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
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│ │ │ │ +0000bb70: 6c6f 6361 6c20 6469 6d65 6e73 696f 6e20  local dimension 
│ │ │ │ +0000bb80: 7465 7374 2c20 2835 290a 6465 666c 6174  test, (5).deflat
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│ │ │ │ +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
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│ │ │ │ +0000bbf0: 6e65 6172 2074 7261 6365 2074 6573 740a  near trace test.
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│ │ │ │  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                |.
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│ │ │ │  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.
│ │ │ │ -0000c510: 6c69 7374 2070 6f73 6974 696f 6e2e 0a0a  list position...
│ │ │ │ -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  +---------------
│ │ │ │ +0000c6f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0000c700: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0000c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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...}   
│ │ │ │ +0000c7e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +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...}  
│ │ │ │  0000c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000c830: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -0000d1e0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
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│ │ │ │ -0000d240: 652c 2050 7265 763a 2062 6572 7469 6e69  e, Prev: bertini
│ │ │ │ -0000d250: 506f 7344 696d 536f 6c76 652c 2055 703a  PosDimSolve, Up:
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│ │ │ │ +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
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│ │ │ │ +0000d200: 1f0a 4669 6c65 3a20 4265 7274 696e 692e  ..File: Bertini.
│ │ │ │ +0000d210: 696e 666f 2c20 4e6f 6465 3a20 6265 7274  info, Node: bert
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│ │ │ │ +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
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│ │ │ │  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.
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│ │ │ │ -0000d310: 6167 653a 200a 2020 2020 2020 2020 5320  age: .        S 
│ │ │ │ -0000d320: 3d20 6265 7274 696e 6952 6566 696e 6553  = bertiniRefineS
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│ │ │ │ -0000d350: 6265 7274 696e 6952 6566 696e 6553 6f6c  bertiniRefineSol
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│ │ │ │ -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
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│ │ │ │ -0000d700: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
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│ │ │ │ -0000d720: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
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│ │ │ │ -0000d760: 6465 6661 756c 7420 7661 6c75 6520 6661  default value fa
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│ │ │ │ -0000d830: 736f 6c75 7469 6f6e 7320 6672 6f6d 2061  solutions from a
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│ │ │ │ +0000d3a0: 2020 2020 2a20 4946 442c 2061 202a 6e6f      * IFD, a *no
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│ │ │ │ +0000d620: 2020 2020 202a 202a 6e6f 7465 2049 7350       * *note IsP
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│ │ │ │ +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 
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│ │ │ │ +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         , => ...,
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│ │ │ │ +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
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│ │ │ │ +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
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│ │ │ │ +0000d840: 6120 6265 7274 696e 695a 6572 6f44 696d  a bertiniZeroDim
│ │ │ │ +0000d850: 536f 6c76 6520 616e 640a 7368 6172 7065  Solve and.sharpe
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│ │ │ │ +0000d870: 6974 7320 7573 696e 6720 7468 6520 7368  its using the sh
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│ │ │ │ +0000d890: 6f66 2042 6572 7469 6e69 2e20 5768 656e  of Bertini. When
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│ │ │ │ +0000d8c0: 6973 2070 756c 6c65 6420 6672 6f6d 2074  is pulled from t
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│ │ │ │ +0000d8e0: 6669 7273 7420 706f 696e 7420 696e 2057  first point in W
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│ │ │ │  0000d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0000d980: 3d20 4343 5b78 2c79 5d3b 2020 2020 2020  = CC[x,y];      
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│ │ │ │ +0000d980: 203d 2043 435b 782c 795d 3b20 2020 2020   = CC[x,y];     
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│ │ │ │  0000d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  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                  
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│ │ │ │ +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  ----------------
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│ │ │ │ +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                  
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│ │ │ │  0000dbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000dbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0000dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0000dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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                  
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│ │ │ │  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  ----------------
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│ │ │ │ +0000dd30: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2053  -------+.|i4 : S
│ │ │ │ +0000dd40: 203d 2062 6572 7469 6e69 5265 6669 6e65   = bertiniRefine
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│ │ │ │  0000dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0000dd80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0000dd80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │ -0000df20: 7374 2020 2020 2020 2020 2020 2020 2020  st              
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│ │ │ │  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                  
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│ │ │ │ -0000e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +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                  
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│ │ │ │ +0000e0a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0000e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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:.=============
│ │ │ │  0000e480: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0000e490: 0a0a 2020 2a20 2262 6572 7469 6e69 5265  ..  * "bertiniRe
│ │ │ │ -0000e4a0: 6669 6e65 536f 6c73 2853 7472 696e 672c  fineSols(String,
│ │ │ │ -0000e4b0: 5a5a 2c4c 6973 742c 5374 7269 6e67 2922  ZZ,List,String)"
│ │ │ │ -0000e4c0: 0a20 202a 2022 6265 7274 696e 6952 6566  .  * "bertiniRef
│ │ │ │ -0000e4d0: 696e 6553 6f6c 7328 5a5a 2c4c 6973 7429  ineSols(ZZ,List)
│ │ │ │ -0000e4e0: 220a 2020 2a20 2262 6572 7469 6e69 5265  ".  * "bertiniRe
│ │ │ │ -0000e4f0: 6669 6e65 536f 6c73 285a 5a2c 4c69 7374  fineSols(ZZ,List
│ │ │ │ -0000e500: 2c53 7472 696e 6729 220a 0a46 6f72 2074  ,String)"..For t
│ │ │ │ -0000e510: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +0000e490: 3d0a 0a20 202a 2022 6265 7274 696e 6952  =..  * "bertiniR
│ │ │ │ +0000e4a0: 6566 696e 6553 6f6c 7328 5374 7269 6e67  efineSols(String
│ │ │ │ +0000e4b0: 2c5a 5a2c 4c69 7374 2c53 7472 696e 6729  ,ZZ,List,String)
│ │ │ │ +0000e4c0: 220a 2020 2a20 2262 6572 7469 6e69 5265  ".  * "bertiniRe
│ │ │ │ +0000e4d0: 6669 6e65 536f 6c73 285a 5a2c 4c69 7374  fineSols(ZZ,List
│ │ │ │ +0000e4e0: 2922 0a20 202a 2022 6265 7274 696e 6952  )".  * "bertiniR
│ │ │ │ +0000e4f0: 6566 696e 6553 6f6c 7328 5a5a 2c4c 6973  efineSols(ZZ,Lis
│ │ │ │ +0000e500: 742c 5374 7269 6e67 2922 0a0a 466f 7220  t,String)"..For 
│ │ │ │ +0000e510: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │  0000e520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0000e530: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -0000e540: 7465 2062 6572 7469 6e69 5265 6669 6e65  te bertiniRefine
│ │ │ │ -0000e550: 536f 6c73 3a20 6265 7274 696e 6952 6566  Sols: bertiniRef
│ │ │ │ -0000e560: 696e 6553 6f6c 732c 2069 7320 6120 2a6e  ineSols, is a *n
│ │ │ │ -0000e570: 6f74 6520 6d65 7468 6f64 0a66 756e 6374  ote method.funct
│ │ │ │ -0000e580: 696f 6e20 7769 7468 206f 7074 696f 6e73  ion with options
│ │ │ │ -0000e590: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -0000e5a0: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769  MethodFunctionWi
│ │ │ │ -0000e5b0: 7468 4f70 7469 6f6e 732c 2e0a 1f0a 4669  thOptions,....Fi
│ │ │ │ -0000e5c0: 6c65 3a20 4265 7274 696e 692e 696e 666f  le: Bertini.info
│ │ │ │ -0000e5d0: 2c20 4e6f 6465 3a20 6265 7274 696e 6953  , Node: bertiniS
│ │ │ │ -0000e5e0: 616d 706c 652c 204e 6578 743a 2062 6572  ample, Next: ber
│ │ │ │ -0000e5f0: 7469 6e69 5472 6163 6b48 6f6d 6f74 6f70  tiniTrackHomotop
│ │ │ │ -0000e600: 792c 2050 7265 763a 2062 6572 7469 6e69  y, Prev: bertini
│ │ │ │ -0000e610: 5265 6669 6e65 536f 6c73 2c20 5570 3a20  RefineSols, Up: 
│ │ │ │ -0000e620: 546f 700a 0a62 6572 7469 6e69 5361 6d70  Top..bertiniSamp
│ │ │ │ -0000e630: 6c65 202d 2d20 6120 6d61 696e 206d 6574  le -- a main met
│ │ │ │ -0000e640: 686f 6420 746f 2073 616d 706c 6520 706f  hod to sample po
│ │ │ │ -0000e650: 696e 7473 2066 726f 6d20 616e 2069 7272  ints from an irr
│ │ │ │ -0000e660: 6564 7563 6962 6c65 2063 6f6d 706f 6e65  educible compone
│ │ │ │ -0000e670: 6e74 206f 6620 6120 7661 7269 6574 790a  nt of a variety.
│ │ │ │ -0000e680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0000e530: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +0000e540: 6f74 6520 6265 7274 696e 6952 6566 696e  ote bertiniRefin
│ │ │ │ +0000e550: 6553 6f6c 733a 2062 6572 7469 6e69 5265  eSols: bertiniRe
│ │ │ │ +0000e560: 6669 6e65 536f 6c73 2c20 6973 2061 202a  fineSols, is a *
│ │ │ │ +0000e570: 6e6f 7465 206d 6574 686f 640a 6675 6e63  note method.func
│ │ │ │ +0000e580: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e  tion with option
│ │ │ │ +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
│ │ │ │ +0000e5f0: 7274 696e 6954 7261 636b 486f 6d6f 746f  rtiniTrackHomoto
│ │ │ │ +0000e600: 7079 2c20 5072 6576 3a20 6265 7274 696e  py, Prev: bertin
│ │ │ │ +0000e610: 6952 6566 696e 6553 6f6c 732c 2055 703a  iRefineSols, Up:
│ │ │ │ +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
│ │ │ │ +0000e660: 7265 6475 6369 626c 6520 636f 6d70 6f6e  reducible compon
│ │ │ │ +0000e670: 656e 7420 6f66 2061 2076 6172 6965 7479  ent of a variety
│ │ │ │ +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
│ │ │ │ -0000e6e0: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ -0000e6f0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -0000e700: 2020 2056 203d 2062 6572 7469 6e69 5361     V = bertiniSa
│ │ │ │ -0000e710: 6d70 6c65 2028 6e2c 2057 290a 2020 2a20  mple (n, W).  * 
│ │ │ │ -0000e720: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -0000e730: 6e2c 2061 6e20 2a6e 6f74 6520 696e 7465  n, an *note inte
│ │ │ │ -0000e740: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ -0000e750: 6f63 295a 5a2c 2c20 616e 2069 6e74 6567  oc)ZZ,, an integ
│ │ │ │ -0000e760: 6572 2073 7065 6369 6679 696e 6720 7468  er specifying th
│ │ │ │ -0000e770: 650a 2020 2020 2020 2020 6e75 6d62 6572  e.        number
│ │ │ │ -0000e780: 206f 6620 6465 7369 7265 6420 7361 6d70   of desired samp
│ │ │ │ -0000e790: 6c65 2070 6f69 6e74 730a 2020 2020 2020  le points.      
│ │ │ │ -0000e7a0: 2a20 572c 2061 202a 6e6f 7465 2077 6974  * W, a *note wit
│ │ │ │ -0000e7b0: 6e65 7373 2073 6574 3a20 284e 4147 7479  ness set: (NAGty
│ │ │ │ -0000e7c0: 7065 7329 5769 746e 6573 7353 6574 2c2c  pes)WitnessSet,,
│ │ │ │ -0000e7d0: 2061 2077 6974 6e65 7373 2073 6574 2066   a witness set f
│ │ │ │ -0000e7e0: 6f72 2061 6e0a 2020 2020 2020 2020 6972  or an.        ir
│ │ │ │ -0000e7f0: 7265 6475 6369 626c 6520 636f 6d70 6f6e  reducible compon
│ │ │ │ -0000e800: 656e 740a 2020 2a20 2a6e 6f74 6520 4f70  ent.  * *note Op
│ │ │ │ -0000e810: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ -0000e820: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ -0000e830: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ -0000e840: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ -0000e850: 732c 3a0a 2020 2020 2020 2a20 4265 7274  s,:.      * Bert
│ │ │ │ -0000e860: 696e 6949 6e70 7574 436f 6e66 6967 7572  iniInputConfigur
│ │ │ │ -0000e870: 6174 696f 6e20 286d 6973 7369 6e67 2064  ation (missing d
│ │ │ │ -0000e880: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e  ocumentation) =>
│ │ │ │ -0000e890: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -0000e8a0: 6c75 650a 2020 2020 2020 2020 7b7d 2c0a  lue.        {},.
│ │ │ │ -0000e8b0: 2020 2020 2020 2a20 2a6e 6f74 6520 4973        * *note Is
│ │ │ │ -0000e8c0: 5072 6f6a 6563 7469 7665 3a20 4973 5072  Projective: IsPr
│ │ │ │ -0000e8d0: 6f6a 6563 7469 7665 2c20 3d3e 202e 2e2e  ojective, => ...
│ │ │ │ -0000e8e0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -0000e8f0: 2d31 2c20 6f70 7469 6f6e 616c 0a20 2020  -1, optional.   
│ │ │ │ -0000e900: 2020 2020 2061 7267 756d 656e 7420 746f       argument to
│ │ │ │ -0000e910: 2073 7065 6369 6679 2077 6865 7468 6572   specify whether
│ │ │ │ -0000e920: 2074 6f20 7573 6520 686f 6d6f 6765 6e65   to use homogene
│ │ │ │ -0000e930: 6f75 7320 636f 6f72 6469 6e61 7465 730a  ous coordinates.
│ │ │ │ -0000e940: 2020 2020 2020 2a20 2a6e 6f74 6520 5665        * *note Ve
│ │ │ │ -0000e950: 7262 6f73 653a 2062 6572 7469 6e69 5472  rbose: bertiniTr
│ │ │ │ -0000e960: 6163 6b48 6f6d 6f74 6f70 795f 6c70 5f70  ackHomotopy_lp_p
│ │ │ │ -0000e970: 645f 7064 5f70 645f 636d 5665 7262 6f73  d_pd_pd_cmVerbos
│ │ │ │ -0000e980: 653d 3e5f 7064 5f70 645f 7064 5f72 700a  e=>_pd_pd_pd_rp.
│ │ │ │ -0000e990: 2020 2020 2020 2020 2c20 3d3e 202e 2e2e          , => ...
│ │ │ │ -0000e9a0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -0000e9b0: 6661 6c73 652c 204f 7074 696f 6e20 746f  false, Option to
│ │ │ │ -0000e9c0: 2073 696c 656e 6365 2061 6464 6974 696f   silence additio
│ │ │ │ -0000e9d0: 6e61 6c20 6f75 7470 7574 0a20 202a 204f  nal output.  * O
│ │ │ │ -0000e9e0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -0000e9f0: 4c2c 2061 202a 6e6f 7465 206c 6973 743a  L, a *note list:
│ │ │ │ -0000ea00: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -0000ea10: 6973 742c 2c20 6120 6c69 7374 206f 6620  ist,, a list of 
│ │ │ │ -0000ea20: 7361 6d70 6c65 2070 6f69 6e74 730a 0a44  sample points..D
│ │ │ │ -0000ea30: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -0000ea40: 3d3d 3d3d 3d3d 0a0a 5361 6d70 6c65 7320  ======..Samples 
│ │ │ │ -0000ea50: 706f 696e 7473 2066 726f 6d20 616e 2069  points from an i
│ │ │ │ -0000ea60: 7272 6564 7563 6962 6c65 2063 6f6d 706f  rreducible compo
│ │ │ │ -0000ea70: 6e65 6e74 206f 6620 6120 7661 7269 6574  nent of a variet
│ │ │ │ -0000ea80: 7920 7573 696e 6720 4265 7274 696e 692e  y using Bertini.
│ │ │ │ -0000ea90: 2020 5468 650a 6972 7265 6475 6369 626c    The.irreducibl
│ │ │ │ -0000eaa0: 6520 636f 6d70 6f6e 656e 7420 6e65 6564  e component need
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│ │ │ │ +0000e720: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
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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, => ..
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│ │ │ │ +0000e8f0: 202d 312c 206f 7074 696f 6e61 6c0a 2020   -1, optional.  
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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_
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│ │ │ │ +0000eaf0: 5769 746e 6573 7353 6574 2c2e 2020 5468  WitnessSet,.  Th
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│ │ │ │ +0000eb70: 4265 7274 696e 6920 2831 2920 7772 6974  Bertini (1) writ
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│ │ │ │ +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
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│ │ │ │ +0000ecb0: 2835 2920 7061 7273 6573 2061 6e64 206f  (5) parses and o
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│ │ │ │  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                  
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│ │ │ │ -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                  
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│ │ │ │  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 = 
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│ │ │ │ +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
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│ │ │ │ +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                  
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│ │ │ │ -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                  
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│ │ │ │ +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       |.|        
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│ │ │ │  0000f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0000f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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      |.|     ----
│ │ │ │ +0000f590: 2020 2020 207c 0a7c 6f35 203d 207b 7b2d       |.|o5 = {{-
│ │ │ │ +0000f5a0: 332e 3938 3839 3565 2d32 312d 322e 3636  3.98895e-21-2.66
│ │ │ │ +0000f5b0: 3930 3365 2d32 312a 6969 2c20 2d31 2e30  903e-21*ii, -1.0
│ │ │ │ +0000f5c0: 3930 3439 652d 3230 2b32 2e33 3930 3537  9049e-20+2.39057
│ │ │ │ +0000f5d0: 652d 3231 2a69 692c 2020 2020 2020 2020  e-21*ii,        
│ │ │ │ +0000f5e0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │  0000f5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0000f630: 2d2d 2d2d 7c0a 7c20 2020 2020 2e33 3335  ----|.|     .335
│ │ │ │ -0000f640: 3439 372b 2e30 3837 3736 3734 2a69 697d  497+.0877674*ii}
│ │ │ │ -0000f650: 2c20 7b2d 322e 3338 3835 3165 2d32 312b  , {-2.38851e-21+
│ │ │ │ -0000f660: 372e 3433 3036 3665 2d32 312a 6969 2c20  7.43066e-21*ii, 
│ │ │ │ +0000f630: 2d2d 2d2d 2d7c 0a7c 2020 2020 202e 3333  -----|.|     .33
│ │ │ │ +0000f640: 3534 3937 2b2e 3038 3737 3637 342a 6969  5497+.0877674*ii
│ │ │ │ +0000f650: 7d2c 207b 2d32 2e33 3838 3531 652d 3231  }, {-2.38851e-21
│ │ │ │ +0000f660: 2b37 2e34 3330 3636 652d 3231 2a69 692c  +7.43066e-21*ii,
│ │ │ │  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+
│ │ │ │ -0000f700: 2e30 3637 3933 3136 2a69 697d 2c20 2020  .0679316*ii},   
│ │ │ │ +0000f6d0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2037 2e30  -----|.|     7.0
│ │ │ │ +0000f6e0: 3438 3835 652d 3231 2d33 2e33 3134 3335  4885e-21-3.31435
│ │ │ │ +0000f6f0: 652d 3231 2a69 692c 202e 3233 3531 3634  e-21*ii, .235164
│ │ │ │ +0000f700: 2b2e 3036 3739 3331 362a 6969 7d2c 2020  +.0679316*ii},  
│ │ │ │  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                  
│ │ │ │  0000f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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
│ │ │ │ -0000fb20: 6163 6b48 6f6d 6f74 6f70 792c 204e 6578  ackHomotopy, Nex
│ │ │ │ -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.********
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│ │ │ │ +0000faa0: 6953 616d 706c 652c 2069 7320 6120 2a6e  iSample, is a *n
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│ │ │ │ +0000fb00: 6c65 3a20 4265 7274 696e 692e 696e 666f  le: Bertini.info
│ │ │ │ +0000fb10: 2c20 4e6f 6465 3a20 6265 7274 696e 6954  , Node: bertiniT
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│ │ │ │ +0000fb40: 486f 6d6f 746f 7079 5f6c 705f 7064 5f70  Homotopy_lp_pd_p
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│ │ │ │ +0000fba0: 202d 2d20 6120 6d61 696e 206d 6574 686f   -- a main metho
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│ │ │ │ +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
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│ │ │ │ -0000fc80: 202a 2074 2c20 6120 2a6e 6f74 6520 7269   * t, a *note ri
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│ │ │ │ -0000fda0: 2020 2020 7379 7374 656d 0a20 202a 202a      system.  * *
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│ │ │ │ -0000fea0: 6e61 6c0a 2020 2020 2020 2020 6172 6775  nal.        argu
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│ │ │ │ -00010090: 6f66 2073 7461 7274 2073 6f6c 7574 696f  of start solutio
│ │ │ │ -000100a0: 6e73 2053 312e 0a42 6572 7469 6e69 2028  ns S1..Bertini (
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│ │ │ │ -000100d0: 2073 6f6c 7574 696f 6e73 2074 6f20 7465   solutions to te
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│ │ │ │ +0000fc20: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +0000fc30: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
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│ │ │ │ +0000fc50: 3d62 6572 7469 6e69 5472 6163 6b48 6f6d  =bertiniTrackHom
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│ │ │ │ +0000fc70: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0000fc80: 2020 2a20 742c 2061 202a 6e6f 7465 2072    * t, a *note r
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│ │ │ │ +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
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│ │ │ │ +0000fcf0: 4c69 7374 2c2c 2061 206c 6973 7420 706f  List,, a list po
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│ │ │ │ +0000fd10: 6566 696e 650a 2020 2020 2020 2020 7468  efine.        th
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│ │ │ │ +0000fd50: 2020 202a 2053 312c 2061 202a 6e6f 7465     * S1, a *note
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│ │ │ │ +0000fd70: 3244 6f63 294c 6973 742c 2c20 6120 6c69  2Doc)List,, a li
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│ │ │ │ +0000fdb0: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
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│ │ │ │ +0000fe00: 2020 2a20 4265 7274 696e 6949 6e70 7574    * BertiniInput
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│ │ │ │ +0000fe90: 7420 7661 6c75 6520 2d31 2c20 6f70 7469  t value -1, opti
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│ │ │ │ +0000fef0: 2a6e 6f74 6520 5665 7262 6f73 653a 2062  *note Verbose: b
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│ │ │ │ +0000ff50: 7420 7661 6c75 6520 6661 6c73 652c 204f  t value false, O
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│ │ │ │ +0000ff70: 2061 6464 6974 696f 6e61 6c20 6f75 7470   additional outp
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│ │ │ │ +00010060: 2074 6865 2068 6f6d 6f74 6f70 7920 482c   the homotopy H,
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│ │ │ │ +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
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│ │ │ │ +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
│ │ │ │ +00011000: 312c 2073 6f6c 327d 2020 2020 2020 2020  1, sol2}        
│ │ │ │  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                  
│ │ │ │ -000110e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +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
│ │ │ │ -00011140: 7274 696e 6954 7261 636b 486f 6d6f 746f  rtiniTrackHomoto
│ │ │ │ -00011150: 7079 2874 2c20 482c 2053 312c 2049 7350  py(t, H, S1, IsP
│ │ │ │ -00011160: 726f 6a65 6374 6976 653d 3e31 2920 2d2d  rojective=>1) --
│ │ │ │ -00011170: 736f 6c75 7469 6f6e 7320 746f 2048 2077  solutions to H w
│ │ │ │ -00011180: 6865 7c0a 7c20 2020 2020 2020 2020 2020  he|.|           
│ │ │ │ +00011130: 2d2d 2d2b 0a7c 6931 3520 3a20 5330 3d62  ---+.|i15 : S0=b
│ │ │ │ +00011140: 6572 7469 6e69 5472 6163 6b48 6f6d 6f74  ertiniTrackHomot
│ │ │ │ +00011150: 6f70 7928 742c 2048 2c20 5331 2c20 4973  opy(t, H, S1, Is
│ │ │ │ +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
│ │ │ │ -000111e0: 3832 3131 342d 2e36 3436 3030 392a 6969  82114-.646009*ii
│ │ │ │ -000111f0: 2c20 2e32 3135 3034 382d 2e34 3632 3233  , .215048-.46223
│ │ │ │ -00011200: 332a 6969 7d2c 207b 342e 3133 3938 2d34  3*ii}, {4.1398-4
│ │ │ │ -00011210: 2e37 3235 332a 6969 2c20 2020 2020 2020  .7253*ii,       
│ │ │ │ -00011220: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d    |.|      -----
│ │ │ │ +000111d0: 2020 207c 0a7c 6f31 3520 3d20 7b7b 2d2e     |.|o15 = {{-.
│ │ │ │ +000111e0: 3438 3231 3134 2d2e 3634 3630 3039 2a69  482114-.646009*i
│ │ │ │ +000111f0: 692c 202e 3231 3530 3438 2d2e 3436 3232  i, .215048-.4622
│ │ │ │ +00011200: 3333 2a69 697d 2c20 7b34 2e31 3339 382d  33*ii}, {4.1398-
│ │ │ │ +00011210: 342e 3732 3533 2a69 692c 2020 2020 2020  4.7253*ii,      
│ │ │ │ +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
│ │ │ │ -00011280: 392d 332e 3732 3235 332a 6969 7d7d 2020  9-3.72253*ii}}  
│ │ │ │ +00011270: 2d2d 2d7c 0a7c 2020 2020 2020 2d31 2e33  ---|.|      -1.3
│ │ │ │ +00011280: 3839 2d33 2e37 3232 3533 2a69 697d 7d20  89-3.72253*ii}} 
│ │ │ │  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      
│ │ │ │ +000113b0: 2d2d 2d7c 0a7c 6e20 743d 3020 2020 2020  ---|.|n t=0     
│ │ │ │  000113c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011400: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +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  ----------------
│ │ │ │ -00011450: 2d2d 2b0a 0a56 6172 6961 626c 6573 206d  --+..Variables m
│ │ │ │ -00011460: 7573 7420 6265 6769 6e20 7769 7468 2061  ust begin with a
│ │ │ │ -00011470: 206c 6574 7465 7220 286c 6f77 6572 6361   letter (lowerca
│ │ │ │ -00011480: 7365 206f 7220 6361 7069 7461 6c29 2061  se or capital) a
│ │ │ │ -00011490: 6e64 2063 616e 206f 6e6c 7920 636f 6e74  nd can only cont
│ │ │ │ -000114a0: 6169 6e0a 6c65 7474 6572 732c 206e 756d  ain.letters, num
│ │ │ │ -000114b0: 6265 7273 2c20 756e 6465 7273 636f 7265  bers, underscore
│ │ │ │ -000114c0: 732c 2061 6e64 2073 7175 6172 6520 6272  s, and square br
│ │ │ │ -000114d0: 6163 6b65 7473 2e0a 0a57 6179 7320 746f  ackets...Ways to
│ │ │ │ -000114e0: 2075 7365 2062 6572 7469 6e69 5472 6163   use bertiniTrac
│ │ │ │ -000114f0: 6b48 6f6d 6f74 6f70 793a 0a3d 3d3d 3d3d  kHomotopy:.=====
│ │ │ │ +00011450: 2d2d 2d2b 0a0a 5661 7269 6162 6c65 7320  ---+..Variables 
│ │ │ │ +00011460: 6d75 7374 2062 6567 696e 2077 6974 6820  must begin with 
│ │ │ │ +00011470: 6120 6c65 7474 6572 2028 6c6f 7765 7263  a letter (lowerc
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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
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│ │ │ │  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
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│ │ │ │ -00011800: 2020 2020 2020 6265 7274 696e 6952 6566        bertiniRef
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│ │ │ │ -00011830: 2020 2020 2020 6265 7274 696e 6953 616d        bertiniSam
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│ │ │ │ -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(...
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│ │ │ │ -000118f0: 0a20 2020 2020 2020 206d 616b 654d 656d  .        makeMem
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│ │ │ │ -00011950: 2020 2020 2020 2020 6227 5048 4d6f 6e6f          b'PHMono
│ │ │ │ -00011960: 6472 6f6d 7943 6f6c 6c65 6374 282e 2e2e  dromyCollect(...
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│ │ │ │ -00011990: 7449 6e63 6964 656e 6365 4d61 7472 6978  tIncidenceMatrix
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│ │ │ │ -000119b0: 6f6c 6561 6e29 0a20 2020 2020 2020 2069  olean).        i
│ │ │ │ -000119c0: 6d70 6f72 744d 6169 6e44 6174 6146 696c  mportMainDataFil
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│ │ │ │ -000119e0: 6f6f 6c65 616e 290a 2020 2020 2020 2020  oolean).        
│ │ │ │ -000119f0: 696d 706f 7274 536c 6963 6546 696c 6528  importSliceFile(
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│ │ │ │ -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
│ │ │ │ +00012250: 7473 0a20 2020 2064 6f63 756d 656e 7461  ts.    documenta
│ │ │ │ +00012260: 7469 6f6e 0a20 202a 2022 6973 4973 6f6d  tion.  * "isIsom
│ │ │ │ +00012270: 6f72 7068 6963 282e 2e2e 2c56 6572 626f  orphic(...,Verbo
│ │ │ │ +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
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│ │ │ │ +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
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│ │ │ │ -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
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│ │ │ │ -000128b0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
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│ │ │ │ -000128d0: 2746 756e 6374 696f 6e73 2028 6d69 7373  'Functions (miss
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -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: 3334 3939 2d30 2f30 222c 204f 7074 696f  3499-0/0", 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                  
│ │ │ │  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 7c0a 7c6f              |.|o
│ │ │ │ -00013bf0: 3134 203d 207b 736f 6c31 2c20 736f 6c32  14 = {sol1, sol2
│ │ │ │ -00013c00: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00013be0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013bf0: 7c6f 3134 203d 207b 736f 6c31 2c20 736f  |o14 = {sol1, so
│ │ │ │ +00013c00: 6c32 7d20 2020 2020 2020 2020 2020 2020  l2}             
│ │ │ │  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 7c0a 7c20              |.| 
│ │ │ │ -00013c40: 2020 2020 2020 2020 2020 2020 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 7c0a 7c6f              |.|o
│ │ │ │ -00013c90: 3134 203a 204c 6973 7420 2020 2020 2020  14 : List       
│ │ │ │ +00013c80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013c90: 7c6f 3134 203a 204c 6973 7420 2020 2020  |o14 : List     
│ │ │ │  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 7c0a 2b2d              |.+-
│ │ │ │ -00013ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2b0a 7c69  ------------+.|i
│ │ │ │ -00013d30: 3135 203a 2053 303d 6265 7274 696e 6955  15 : S0=bertiniU
│ │ │ │ -00013d40: 7365 7248 6f6d 6f74 6f70 7928 742c 7b61  serHomotopy(t,{a
│ │ │ │ -00013d50: 3d3e 747d 2c20 482c 2053 312c 2048 6f6d  =>t}, H, S1, Hom
│ │ │ │ -00013d60: 5661 7269 6162 6c65 4772 6f75 703d 3e7b  VariableGroup=>{
│ │ │ │ -00013d70: 782c 797d 2920 2020 2020 2020 7c0a 7c20  x,y})       |.| 
│ │ │ │ -00013d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013d30: 7c69 3135 203a 2053 303d 6265 7274 696e  |i15 : S0=bertin
│ │ │ │ +00013d40: 6955 7365 7248 6f6d 6f74 6f70 7928 742c  iUserHomotopy(t,
│ │ │ │ +00013d50: 7b61 3d3e 747d 2c20 482c 2053 312c 2048  {a=>t}, H, S1, H
│ │ │ │ +00013d60: 6f6d 5661 7269 6162 6c65 4772 6f75 703d  omVariableGroup=
│ │ │ │ +00013d70: 3e7b 782c 797d 2920 2020 2020 2020 7c0a  >{x,y})       |.
│ │ │ │ +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 7c0a 7c6f              |.|o
│ │ │ │ -00013dd0: 3135 203d 207b 7b31 2c20 2e33 2b2e 3535  15 = {{1, .3+.55
│ │ │ │ -00013de0: 3637 3736 2a69 697d 2c20 7b31 2c20 2e33  6776*ii}, {1, .3
│ │ │ │ -00013df0: 2d2e 3535 3637 3736 2a69 697d 7d20 2020  -.556776*ii}}   
│ │ │ │ +00013dc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013dd0: 7c6f 3135 203d 207b 7b31 2c20 2e33 2b2e  |o15 = {{1, .3+.
│ │ │ │ +00013de0: 3535 3637 3736 2a69 697d 2c20 7b31 2c20  556776*ii}, {1, 
│ │ │ │ +00013df0: 2e33 2d2e 3535 3637 3736 2a69 697d 7d20  .3-.556776*ii}} 
│ │ │ │  00013e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00013e20: 2020 2020 2020 2020 2020 2020 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 7c0a 7c6f              |.|o
│ │ │ │ -00013e70: 3135 203a 204c 6973 7420 2020 2020 2020  15 : List       
│ │ │ │ +00013e60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013e70: 7c6f 3135 203a 204c 6973 7420 2020 2020  |o15 : List     
│ │ │ │  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 7c0a 7c2d              |.|-
│ │ │ │ -00013ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013eb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013ec0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00013ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d  ------------|.|-
│ │ │ │ -00013f10: 2d73 6f6c 7574 696f 6e73 2074 6f20 4820  -solutions to H 
│ │ │ │ -00013f20: 7768 656e 2074 3d30 2020 2020 2020 2020  when t=0        
│ │ │ │ +00013f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00013f10: 7c2d 2d73 6f6c 7574 696f 6e73 2074 6f20  |--solutions to 
│ │ │ │ +00013f20: 4820 7768 656e 2074 3d30 2020 2020 2020  H when t=0      
│ │ │ │  00013f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013f50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00013f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013f50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00013f60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00013f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57  ------------+..W
│ │ │ │ -00013fb0: 6179 7320 746f 2075 7365 2062 6572 7469  ays to use berti
│ │ │ │ -00013fc0: 6e69 5573 6572 486f 6d6f 746f 7079 3a0a  niUserHomotopy:.
│ │ │ │ -00013fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00013fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00013fb0: 0a57 6179 7320 746f 2075 7365 2062 6572  .Ways to use ber
│ │ │ │ +00013fc0: 7469 6e69 5573 6572 486f 6d6f 746f 7079  tiniUserHomotopy
│ │ │ │ +00013fd0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │  00013fe0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00013ff0: 0a0a 2020 2a20 2262 6572 7469 6e69 5573  ..  * "bertiniUs
│ │ │ │ -00014000: 6572 486f 6d6f 746f 7079 2854 6869 6e67  erHomotopy(Thing
│ │ │ │ -00014010: 2c4c 6973 742c 4c69 7374 2c4c 6973 7429  ,List,List,List)
│ │ │ │ -00014020: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00014030: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -00014040: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -00014050: 6a65 6374 202a 6e6f 7465 2062 6572 7469  ject *note berti
│ │ │ │ -00014060: 6e69 5573 6572 486f 6d6f 746f 7079 3a20  niUserHomotopy: 
│ │ │ │ -00014070: 6265 7274 696e 6955 7365 7248 6f6d 6f74  bertiniUserHomot
│ │ │ │ -00014080: 6f70 792c 2069 7320 6120 2a6e 6f74 6520  opy, is a *note 
│ │ │ │ -00014090: 6d65 7468 6f64 0a66 756e 6374 696f 6e20  method.function 
│ │ │ │ -000140a0: 7769 7468 206f 7074 696f 6e73 3a20 284d  with options: (M
│ │ │ │ -000140b0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -000140c0: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
│ │ │ │ -000140d0: 7469 6f6e 732c 2e0a 1f0a 4669 6c65 3a20  tions,....File: 
│ │ │ │ -000140e0: 4265 7274 696e 692e 696e 666f 2c20 4e6f  Bertini.info, No
│ │ │ │ -000140f0: 6465 3a20 6265 7274 696e 695a 6572 6f44  de: bertiniZeroD
│ │ │ │ -00014100: 696d 536f 6c76 652c 204e 6578 743a 2043  imSolve, Next: C
│ │ │ │ -00014110: 6f70 7942 2746 696c 652c 2050 7265 763a  opyB'File, Prev:
│ │ │ │ -00014120: 2062 6572 7469 6e69 5573 6572 486f 6d6f   bertiniUserHomo
│ │ │ │ -00014130: 746f 7079 2c20 5570 3a20 546f 700a 0a62  topy, Up: Top..b
│ │ │ │ -00014140: 6572 7469 6e69 5a65 726f 4469 6d53 6f6c  ertiniZeroDimSol
│ │ │ │ -00014150: 7665 202d 2d20 6120 6d61 696e 206d 6574  ve -- a main met
│ │ │ │ -00014160: 686f 6420 746f 2073 6f6c 7665 2061 207a  hod to solve a z
│ │ │ │ -00014170: 6572 6f2d 6469 6d65 6e73 696f 6e61 6c20  ero-dimensional 
│ │ │ │ -00014180: 7379 7374 656d 206f 6620 6571 7561 7469  system of equati
│ │ │ │ -00014190: 6f6e 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ons.************
│ │ │ │ +00013ff0: 3d3d 0a0a 2020 2a20 2262 6572 7469 6e69  ==..  * "bertini
│ │ │ │ +00014000: 5573 6572 486f 6d6f 746f 7079 2854 6869  UserHomotopy(Thi
│ │ │ │ +00014010: 6e67 2c4c 6973 742c 4c69 7374 2c4c 6973  ng,List,List,Lis
│ │ │ │ +00014020: 7429 220a 0a46 6f72 2074 6865 2070 726f  t)"..For the pro
│ │ │ │ +00014030: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +00014040: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +00014050: 6f62 6a65 6374 202a 6e6f 7465 2062 6572  object *note ber
│ │ │ │ +00014060: 7469 6e69 5573 6572 486f 6d6f 746f 7079  tiniUserHomotopy
│ │ │ │ +00014070: 3a20 6265 7274 696e 6955 7365 7248 6f6d  : bertiniUserHom
│ │ │ │ +00014080: 6f74 6f70 792c 2069 7320 6120 2a6e 6f74  otopy, is a *not
│ │ │ │ +00014090: 6520 6d65 7468 6f64 0a66 756e 6374 696f  e method.functio
│ │ │ │ +000140a0: 6e20 7769 7468 206f 7074 696f 6e73 3a20  n with options: 
│ │ │ │ +000140b0: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
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│ │ │ │  000141b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000141c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000141d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00014210: 2053 203d 2062 6572 7469 6e69 5a65 726f   S = bertiniZero
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│ │ │ │ -00014240: 726f 4469 6d53 6f6c 7665 2049 0a20 2020  roDimSolve I.   
│ │ │ │ -00014250: 2020 2020 2053 203d 2062 6572 7469 6e69       S = bertini
│ │ │ │ -00014260: 5a65 726f 4469 6d53 6f6c 7665 2849 2c20  ZeroDimSolve(I, 
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│ │ │ │ -000145e0: 6f6d 6f74 6f70 6965 730a 2020 2020 2020  omotopies.      
│ │ │ │ -000145f0: 2a20 4973 5072 6f6a 6563 7469 7665 2028  * IsProjective (
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│ │ │ │ -00014620: 6566 6175 6c74 2076 616c 7565 202d 312c  efault value -1,
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│ │ │ │ -00014680: 204e 616d 654d 6169 6e44 6174 6146 696c   NameMainDataFil
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│ │ │ │ -000147a0: 783a 2042 6572 7469 6e69 2069 6e70 7574  x: Bertini input
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│ │ │ │ -00014880: 720a 2020 2020 2020 2a20 2a6e 6f74 6520  r.      * *note 
│ │ │ │ -00014890: 5261 6e64 6f6d 5265 616c 3a20 4265 7274  RandomReal: Bert
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│ │ │ │ -00014910: 2020 2020 7379 6d62 6f6c 732f 7374 7269      symbols/stri
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -00014a10: 5265 6765 6e65 7261 7469 6f6e 2028 6d69  Regeneration (mi
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│ │ │ │ -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.
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│ │ │ │ -00014ac0: 6661 6c73 652c 204f 7074 696f 6e20 746f  false, Option to
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -00014b80: 5468 6973 206d 6574 686f 6420 6669 6e64  This method find
│ │ │ │ -00014b90: 7320 6973 6f6c 6174 6564 2073 6f6c 7574  s isolated solut
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│ │ │ │ -00014bd0: 6d6f 746f 7079 2063 6f6e 7469 6e75 6174  motopy continuat
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│ │ │ │ -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
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│ │ │ │ -00014c60: 2061 206d 6163 6869 6e65 0a72 6561 6461   a machine.reada
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│ │ │ │ -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
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│ │ │ │ +00014290: 3a0a 2020 2020 2020 2a20 462c 2061 202a  :.      * F, a *
│ │ │ │ +000142a0: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
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│ │ │ │ +000142c0: 6120 6c69 7374 206f 6620 7269 6e67 2065  a list of ring e
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│ │ │ │ +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 
│ │ │ │ +00014850: 7261 6e64 6f6d 2072 6561 6c20 6e75 6d62  random real numb
│ │ │ │ +00014860: 6572 0a20 2020 2020 2020 206f 7220 7261  er.        or ra
│ │ │ │ +00014870: 6e64 6f6d 2063 6f6d 706c 6578 206e 756d  ndom complex num
│ │ │ │ +00014880: 6265 720a 2020 2020 2020 2a20 2a6e 6f74  ber.      * *not
│ │ │ │ +00014890: 6520 5261 6e64 6f6d 5265 616c 3a20 4265  e RandomReal: Be
│ │ │ │ +000148a0: 7274 696e 6920 696e 7075 7420 6669 6c65  rtini input file
│ │ │ │ +000148b0: 2064 6563 6c61 7261 7469 6f6e 735f 636f   declarations_co
│ │ │ │ +000148c0: 2072 616e 646f 6d20 6e75 6d62 6572 732c   random numbers,
│ │ │ │ +000148d0: 203d 3e0a 2020 2020 2020 2020 2e2e 2e2c   =>.        ...,
│ │ │ │ +000148e0: 2064 6566 6175 6c74 2076 616c 7565 207b   default value {
│ │ │ │ +000148f0: 7d2c 2061 6e20 6f70 7469 6f6e 2077 6869  }, an option whi
│ │ │ │ +00014900: 6368 2064 6573 6967 6e61 7465 730a 2020  ch designates.  
│ │ │ │ +00014910: 2020 2020 2020 7379 6d62 6f6c 732f 7374        symbols/st
│ │ │ │ +00014920: 7269 6e67 732f 7661 7269 6162 6c65 7320  rings/variables 
│ │ │ │ +00014930: 7468 6174 2077 696c 6c20 6265 2073 6574  that will be set
│ │ │ │ +00014940: 2074 6f20 6265 2061 2072 616e 646f 6d20   to be a random 
│ │ │ │ +00014950: 7265 616c 206e 756d 6265 720a 2020 2020  real number.    
│ │ │ │ +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
│ │ │ │ +00014990: 7265 6374 6f72 793a 2054 6f70 4469 7265  rectory: TopDire
│ │ │ │ +000149a0: 6374 6f72 792c 203d 3e20 2e2e 2e2c 2064  ctory, => ..., d
│ │ │ │ +000149b0: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ +000149c0: 2020 2020 2022 2f74 6d70 2f4d 322d 3132       "/tmp/M2-12
│ │ │ │ +000149d0: 3334 3939 2d30 2f30 222c 204f 7074 696f  3499-0/0", Optio
│ │ │ │ +000149e0: 6e20 746f 2063 6861 6e67 6520 6469 7265  n to change dire
│ │ │ │ +000149f0: 6374 6f72 7920 666f 7220 6669 6c65 2073  ctory for file s
│ │ │ │ +00014a00: 746f 7261 6765 2e0a 2020 2020 2020 2a20  torage..      * 
│ │ │ │ +00014a10: 5573 6552 6567 656e 6572 6174 696f 6e20  UseRegeneration 
│ │ │ │ +00014a20: 286d 6973 7369 6e67 2064 6f63 756d 656e  (missing documen
│ │ │ │ +00014a30: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20  tation) => ..., 
│ │ │ │ +00014a40: 6465 6661 756c 7420 7661 6c75 6520 2d31  default value -1
│ │ │ │ +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:.     
│ │ │ │ +00014b00: 202a 2053 2c20 6120 2a6e 6f74 6520 6c69   * S, a *note li
│ │ │ │ +00014b10: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +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
│ │ │ │ +00014b40: 7265 0a20 2020 2020 2020 2063 6f6e 7461  re.        conta
│ │ │ │ +00014b50: 696e 6564 2069 6e20 7468 6520 7661 7269  ined in the vari
│ │ │ │ +00014b60: 6574 7920 6f66 2046 0a0a 4465 7363 7269  ety of F..Descri
│ │ │ │ +00014b70: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +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
│ │ │ │ +00014bc0: 7269 6361 6c20 706f 6c79 6e6f 6d69 616c  rical polynomial
│ │ │ │ +00014bd0: 0a68 6f6d 6f74 6f70 7920 636f 6e74 696e  .homotopy contin
│ │ │ │ +00014be0: 7561 7469 6f6e 2062 7920 2831 2920 6275  uation by (1) bu
│ │ │ │ +00014bf0: 696c 6469 6e67 2061 2042 6572 7469 6e69  ilding a Bertini
│ │ │ │ +00014c00: 2069 6e70 7574 2066 696c 6520 6672 6f6d   input file from
│ │ │ │ +00014c10: 2074 6865 2073 7973 7465 6d20 462c 0a28   the system F,.(
│ │ │ │ +00014c20: 3229 2063 616c 6c69 6e67 2042 6572 7469  2) calling Berti
│ │ │ │ +00014c30: 6e69 206f 6e20 7468 6973 2069 6e70 7574  ni on this input
│ │ │ │ +00014c40: 2066 696c 652c 2028 3329 2072 6574 7572   file, (3) retur
│ │ │ │ +00014c50: 6e69 6e67 2073 6f6c 7574 696f 6e73 2066  ning solutions f
│ │ │ │ +00014c60: 726f 6d20 6120 6d61 6368 696e 650a 7265  rom a machine.re
│ │ │ │ +00014c70: 6164 6162 6c65 2066 696c 6520 7468 6174  adable file that
│ │ │ │ +00014c80: 2069 7320 616e 206f 7574 7075 7420 6672   is an output fr
│ │ │ │ +00014c90: 6f6d 2042 6572 7469 6e69 2e0a 0a2b 2d2d  om Bertini...+--
│ │ │ │  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                  
│ │ │ │ +00014d60: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2046  -------+.|i2 : F
│ │ │ │ +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,
│ │ │ │ -00014e80: 2031 2e34 3134 3231 7d2c 207b 312c 202d   1.41421}, {1, -
│ │ │ │ -00014e90: 312e 3431 3432 317d 2c20 7b2d 312c 2031  1.41421}, {-1, 1
│ │ │ │ -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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│ │ │ │ +00015050: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +00015060: 3a20 7065 656b 2053 5f30 2020 2020 2020  : peek S_0      
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│ │ │ │ -00015080: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00015080: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │  00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00015140: 6573 2061 206d 756c 7469 686f 6d6f 6765  es a multihomoge
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│ │ │ │ -000151b0: 202e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d   ...+-----------
│ │ │ │ +00015130: 2d2d 2d2d 2d2d 2b0a 0a42 6572 7469 6e69  ------+..Bertini
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│ │ │ │ +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  ----------------
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│ │ │ │ +00015220: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │  00015240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +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        |.+-------
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│ │ │ │  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 
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│ │ │ │ -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
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│ │ │ │ -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*
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│ │ │ │  00015360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │  000153b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000153c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000153c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  000153d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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
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│ │ │ │ +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    
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│ │ │ │  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       |.+--------
│ │ │ │  00015510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00015530: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a56 6172  ----------+..Var
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│ │ │ │ +00015590: 6572 732c 206e 756d 6265 7273 2c20 756e  ers, numbers, un
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│ │ │ │ +000155b0: 7175 6172 6520 6272 6163 6b65 7473 2e20  quare brackets. 
│ │ │ │ +000155c0: 5265 6765 6e65 7261 7469 6f6e 2069 6e0a  Regeneration in.
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│ │ │ │ +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
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│ │ │ │ -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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│ │ │ │ -00015730: 6e63 7469 6f6e 5769 7468 4f70 7469 6f6e  nctionWithOption
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│ │ │ │ -00015760: 436f 7079 4227 4669 6c65 2c20 4e65 7874  CopyB'File, Next
│ │ │ │ -00015770: 3a20 696d 706f 7274 496e 6369 6465 6e63  : importIncidenc
│ │ │ │ -00015780: 654d 6174 7269 782c 2050 7265 763a 2062  eMatrix, Prev: b
│ │ │ │ -00015790: 6572 7469 6e69 5a65 726f 4469 6d53 6f6c  ertiniZeroDimSol
│ │ │ │ -000157a0: 7665 2c20 5570 3a20 546f 700a 0a43 6f70  ve, Up: Top..Cop
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│ │ │ │ -000157c0: 7469 6f6e 616c 2061 7267 756d 656e 7420  tional argument 
│ │ │ │ -000157d0: 746f 2073 7065 6369 6679 2077 6865 7468  to specify wheth
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│ │ │ │ +00015640: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00015650: 2020 2a20 2262 6572 7469 6e69 5a65 726f    * "bertiniZero
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│ │ │ │  00015ac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00015ad0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00015ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00015af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00015cc0: 2a20 2a6e 6f74 6520 5665 7262 6f73 653a  * *note Verbose:
│ │ │ │ -00015cd0: 2062 6572 7469 6e69 5472 6163 6b48 6f6d   bertiniTrackHom
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│ │ │ │ -00015d90: 4265 7274 696e 6920 7072 6f64 7563 6573  Bertini produces
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│ │ │ │ -00015fb0: 7769 7468 0a61 2064 6966 6665 7265 6e74  with.a different
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│ │ │ │ +00015b00: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
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│ │ │ │ +00015b60: 2061 202a 6e6f 7465 2073 7472 696e 673a   a *note string:
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│ │ │ │ +00015cc0: 2020 202a 202a 6e6f 7465 2056 6572 626f     * *note Verbo
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│ │ │ │ -00016890: 2042 6572 7469 6e69 2072 756e 2e0a 2a2a   Bertini run..**
│ │ │ │ -000168a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000166e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00016710: 7269 6e67 2922 0a0a 466f 7220 7468 6520  ring)"..For the 
│ │ │ │ +00016720: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00016730: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
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│ │ │ │ +00016750: 696d 706f 7274 496e 6369 6465 6e63 654d  importIncidenceM
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│ │ │ │ +00016790: 6675 6e63 7469 6f6e 2077 6974 6820 6f70  function with op
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│ │ │ │ +000168a0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  000168b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000168c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000168d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000168e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000168f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -00016900: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -00016910: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -00016920: 2020 2020 2069 6d70 6f72 744d 6169 6e44       importMainD
│ │ │ │ -00016930: 6174 6146 696c 6528 7468 6544 6972 290a  ataFile(theDir).
│ │ │ │ -00016940: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00016950: 2020 2a20 7468 6544 6972 2c20 6120 2a6e    * theDir, a *n
│ │ │ │ -00016960: 6f74 6520 7374 7269 6e67 3a20 284d 6163  ote string: (Mac
│ │ │ │ -00016970: 6175 6c61 7932 446f 6329 5374 7269 6e67  aulay2Doc)String
│ │ │ │ -00016980: 2c2c 2054 6865 2064 6972 6563 746f 7279  ,, The directory
│ │ │ │ -00016990: 2077 6865 7265 2074 6865 0a20 2020 2020   where the.     
│ │ │ │ -000169a0: 2020 206d 6169 6e5f 6461 7461 2066 696c     main_data fil
│ │ │ │ -000169b0: 6520 6973 206c 6f63 6174 6564 2e0a 2020  e is located..  
│ │ │ │ -000169c0: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c  * *note Optional
│ │ │ │ -000169d0: 2069 6e70 7574 733a 2028 4d61 6361 756c   inputs: (Macaul
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│ │ │ │ -00016a10: 2020 2020 2a20 4d32 5072 6563 6973 696f      * M2Precisio
│ │ │ │ -00016a20: 6e20 286d 6973 7369 6e67 2064 6f63 756d  n (missing docum
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│ │ │ │ -00016a50: 3533 2c20 0a20 2020 2020 202a 204e 616d  53, .      * Nam
│ │ │ │ -00016a60: 654d 6169 6e44 6174 6146 696c 6520 286d  eMainDataFile (m
│ │ │ │ -00016a70: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │ -00016a80: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │ -00016a90: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ -00016aa0: 2020 2020 226d 6169 6e5f 6461 7461 222c      "main_data",
│ │ │ │ -00016ab0: 0a20 2020 2020 202a 2050 6174 684c 6973  .      * PathLis
│ │ │ │ -00016ac0: 7420 286d 6973 7369 6e67 2064 6f63 756d  t (missing docum
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│ │ │ │ -00016ae0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -00016af0: 6e75 6c6c 2c20 0a20 2020 2020 202a 2053  null, .      * S
│ │ │ │ -00016b00: 7065 6369 6679 4469 6d20 286d 6973 7369  pecifyDim (missi
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│ │ │ │ -00016b30: 7420 7661 6c75 6520 6661 6c73 652c 200a  t value false, .
│ │ │ │ -00016b40: 2020 2020 2020 2a20 2a6e 6f74 6520 5665        * *note Ve
│ │ │ │ -00016b50: 7262 6f73 653a 2062 6572 7469 6e69 5472  rbose: bertiniTr
│ │ │ │ -00016b60: 6163 6b48 6f6d 6f74 6f70 795f 6c70 5f70  ackHomotopy_lp_p
│ │ │ │ -00016b70: 645f 7064 5f70 645f 636d 5665 7262 6f73  d_pd_pd_cmVerbos
│ │ │ │ -00016b80: 653d 3e5f 7064 5f70 645f 7064 5f72 700a  e=>_pd_pd_pd_rp.
│ │ │ │ -00016b90: 2020 2020 2020 2020 2c20 3d3e 202e 2e2e          , => ...
│ │ │ │ -00016ba0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -00016bb0: 6661 6c73 652c 204f 7074 696f 6e20 746f  false, Option to
│ │ │ │ -00016bc0: 2073 696c 656e 6365 2061 6464 6974 696f   silence additio
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│ │ │ │ -00016be0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
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│ │ │ │ -00016c50: 6461 7461 2066 696c 652e 2054 6865 7365  data file. These
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│ │ │ │ -00016c70: 636f 6f72 6469 6e61 7465 732c 2063 6f6e  coordinates, con
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│ │ │ │ -00016cd0: 6174 696f 6e20 7761 7320 7573 6564 2061  ation was used a
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│ │ │ │ -00016d10: 6520 3120 6973 2075 7365 642c 2055 4e43  e 1 is used, UNC
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│ │ │ │ -00016d50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000168f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +00016900: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ +00016910: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ +00016920: 2020 2020 2020 2020 696d 706f 7274 4d61          importMa
│ │ │ │ +00016930: 696e 4461 7461 4669 6c65 2874 6865 4469  inDataFile(theDi
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│ │ │ │ +00016950: 2020 2020 202a 2074 6865 4469 722c 2061       * theDir, a
│ │ │ │ +00016960: 202a 6e6f 7465 2073 7472 696e 673a 2028   *note string: (
│ │ │ │ +00016970: 4d61 6361 756c 6179 3244 6f63 2953 7472  Macaulay2Doc)Str
│ │ │ │ +00016980: 696e 672c 2c20 5468 6520 6469 7265 6374  ing,, The direct
│ │ │ │ +00016990: 6f72 7920 7768 6572 6520 7468 650a 2020  ory where the.  
│ │ │ │ +000169a0: 2020 2020 2020 6d61 696e 5f64 6174 6120        main_data 
│ │ │ │ +000169b0: 6669 6c65 2069 7320 6c6f 6361 7465 642e  file is located.
│ │ │ │ +000169c0: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
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│ │ │ │ +000169e0: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ +000169f0: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
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│ │ │ │ +00016a10: 0a20 2020 2020 202a 204d 3250 7265 6369  .      * M2Preci
│ │ │ │ +00016a20: 7369 6f6e 2028 6d69 7373 696e 6720 646f  sion (missing do
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│ │ │ │ +00016a40: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00016a50: 7565 2035 332c 200a 2020 2020 2020 2a20  ue 53, .      * 
│ │ │ │ +00016a60: 4e61 6d65 4d61 696e 4461 7461 4669 6c65  NameMainDataFile
│ │ │ │ +00016a70: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
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│ │ │ │ +00016a90: 2064 6566 6175 6c74 2076 616c 7565 0a20   default value. 
│ │ │ │ +00016aa0: 2020 2020 2020 2022 6d61 696e 5f64 6174         "main_dat
│ │ │ │ +00016ab0: 6122 2c0a 2020 2020 2020 2a20 5061 7468  a",.      * Path
│ │ │ │ +00016ac0: 4c69 7374 2028 6d69 7373 696e 6720 646f  List (missing do
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│ │ │ │ +00016ae0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00016af0: 7565 206e 756c 6c2c 200a 2020 2020 2020  ue null, .      
│ │ │ │ +00016b00: 2a20 5370 6563 6966 7944 696d 2028 6d69  * SpecifyDim (mi
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│ │ │ │ +00016b30: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ +00016b40: 2c20 0a20 2020 2020 202a 202a 6e6f 7465  , .      * *note
│ │ │ │ +00016b50: 2056 6572 626f 7365 3a20 6265 7274 696e   Verbose: bertin
│ │ │ │ +00016b60: 6954 7261 636b 486f 6d6f 746f 7079 5f6c  iTrackHomotopy_l
│ │ │ │ +00016b70: 705f 7064 5f70 645f 7064 5f63 6d56 6572  p_pd_pd_pd_cmVer
│ │ │ │ +00016b80: 626f 7365 3d3e 5f70 645f 7064 5f70 645f  bose=>_pd_pd_pd_
│ │ │ │ +00016b90: 7270 0a20 2020 2020 2020 202c 203d 3e20  rp.        , => 
│ │ │ │ +00016ba0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
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│ │ │ │ +00016bf0: 3d3d 3d3d 3d3d 0a0a 5468 6973 2066 756e  ======..This fun
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│ │ │ │ +00016c20: 636f 6f72 6469 6e61 7465 732e 2049 6e73  coordinates. Ins
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│ │ │ │ +00016c40: 706f 696e 7473 0a66 726f 6d20 6120 6d61  points.from a ma
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│ │ │ │ +00016c80: 636f 6e64 6974 696f 6e20 6e75 6d62 6572  condition number
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│ │ │ │ +00016cb0: 706f 696e 7473 2063 6f6e 7461 696e 2064  points contain d
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│ │ │ │ +000173e0: 2020 7c0a 7c31 2020 2020 2020 2020 2020    |.|1          
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│ │ │ │ -00018600: 7465 2073 7472 696e 673a 2028 4d61 6361  te string: (Maca
│ │ │ │ -00018610: 756c 6179 3244 6f63 2953 7472 696e 672c  ulay2Doc)String,
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│ │ │ │ -00018630: 7768 6572 6520 7468 6520 6669 6c65 0a20  where the file. 
│ │ │ │ -00018640: 2020 2020 2020 2069 7320 7374 6f72 6564         is stored
│ │ │ │ -00018650: 2e0a 2020 2a20 2a6e 6f74 6520 4f70 7469  ..  * *note Opti
│ │ │ │ -00018660: 6f6e 616c 2069 6e70 7574 733a 2028 4d61  onal inputs: (Ma
│ │ │ │ -00018670: 6361 756c 6179 3244 6f63 2975 7369 6e67  caulay2Doc)using
│ │ │ │ -00018680: 2066 756e 6374 696f 6e73 2077 6974 6820   functions with 
│ │ │ │ -00018690: 6f70 7469 6f6e 616c 2069 6e70 7574 732c  optional inputs,
│ │ │ │ -000186a0: 3a0a 2020 2020 2020 2a20 4d32 5072 6563  :.      * M2Prec
│ │ │ │ -000186b0: 6973 696f 6e20 286d 6973 7369 6e67 2064  ision (missing d
│ │ │ │ -000186c0: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e  ocumentation) =>
│ │ │ │ -000186d0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -000186e0: 6c75 6520 3533 2c20 0a20 2020 2020 202a  lue 53, .      *
│ │ │ │ -000186f0: 204e 616d 6553 6f6c 7574 696f 6e73 4669   NameSolutionsFi
│ │ │ │ -00018700: 6c65 2028 6d69 7373 696e 6720 646f 6375  le (missing docu
│ │ │ │ -00018710: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e  mentation) => ..
│ │ │ │ -00018720: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -00018730: 0a20 2020 2020 2020 2022 7261 775f 736f  .        "raw_so
│ │ │ │ -00018740: 6c75 7469 6f6e 7322 2c0a 2020 2020 2020  lutions",.      
│ │ │ │ -00018750: 2a20 4f72 6465 7250 6174 6873 2028 6d69  * OrderPaths (mi
│ │ │ │ -00018760: 7373 696e 6720 646f 6375 6d65 6e74 6174  ssing documentat
│ │ │ │ -00018770: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566  ion) => ..., def
│ │ │ │ -00018780: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ -00018790: 2c20 0a20 2020 2020 202a 2053 746f 7261  , .      * Stora
│ │ │ │ -000187a0: 6765 466f 6c64 6572 2028 6d69 7373 696e  geFolder (missin
│ │ │ │ -000187b0: 6720 646f 6375 6d65 6e74 6174 696f 6e29  g documentation)
│ │ │ │ -000187c0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -000187d0: 2076 616c 7565 206e 756c 6c2c 200a 2020   value null, .  
│ │ │ │ -000187e0: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
│ │ │ │ -000187f0: 6f73 653a 2062 6572 7469 6e69 5472 6163  ose: bertiniTrac
│ │ │ │ -00018800: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
│ │ │ │ -00018810: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ -00018820: 3e5f 7064 5f70 645f 7064 5f72 700a 2020  >_pd_pd_pd_rp.  
│ │ │ │ -00018830: 2020 2020 2020 2c20 3d3e 202e 2e2e 2c20        , => ..., 
│ │ │ │ -00018840: 6465 6661 756c 7420 7661 6c75 6520 6661  default value fa
│ │ │ │ -00018850: 6c73 652c 204f 7074 696f 6e20 746f 2073  lse, Option to s
│ │ │ │ -00018860: 696c 656e 6365 2061 6464 6974 696f 6e61  ilence additiona
│ │ │ │ -00018870: 6c20 6f75 7470 7574 0a0a 4465 7363 7269  l output..Descri
│ │ │ │ -00018880: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -00018890: 3d0a 0a41 6674 6572 2042 6572 7469 6e69  =..After Bertini
│ │ │ │ -000188a0: 2064 6f65 7320 6120 7275 6e20 6d61 6e79   does a run many
│ │ │ │ -000188b0: 2066 696c 6573 2061 7265 2063 7265 6174   files are creat
│ │ │ │ -000188c0: 6564 2e20 5468 6973 2066 756e 6374 696f  ed. This functio
│ │ │ │ -000188d0: 6e20 696d 706f 7274 7320 7468 650a 636f  n imports the.co
│ │ │ │ -000188e0: 6f72 6469 6e61 7465 7320 6f66 2073 6f6c  ordinates of sol
│ │ │ │ -000188f0: 7574 696f 6e73 2066 726f 6d20 7468 6520  utions from the 
│ │ │ │ -00018900: 7369 6d70 6c65 2022 7261 775f 736f 6c75  simple "raw_solu
│ │ │ │ -00018910: 7469 6f6e 7322 2066 696c 652e 2042 7920  tions" file. By 
│ │ │ │ -00018920: 7573 696e 6720 7468 650a 6f70 7469 6f6e  using the.option
│ │ │ │ -00018930: 204e 616d 6553 6f6c 7574 696f 6e73 4669   NameSolutionsFi
│ │ │ │ -00018940: 6c65 3d3e 2272 6561 6c5f 6669 6e69 7465  le=>"real_finite
│ │ │ │ -00018950: 5f73 6f6c 7574 696f 6e73 2220 7765 2077  _solutions" we w
│ │ │ │ -00018960: 6f75 6c64 2069 6d70 6f72 7420 736f 6c75  ould import solu
│ │ │ │ -00018970: 7469 6f6e 730a 6672 6f6d 2072 6561 6c20  tions.from real 
│ │ │ │ -00018980: 6669 6e69 7465 2073 6f6c 7574 696f 6e73  finite solutions
│ │ │ │ -00018990: 2e20 4f74 6865 7220 636f 6d6d 6f6e 2066  . Other common f
│ │ │ │ -000189a0: 696c 6520 6e61 6d65 7320 6172 650a 226e  ile names are."n
│ │ │ │ -000189b0: 6f6e 7369 6e67 756c 6172 5f73 6f6c 7574  onsingular_solut
│ │ │ │ -000189c0: 696f 6e73 222c 2022 6669 6e69 7465 5f73  ions", "finite_s
│ │ │ │ -000189d0: 6f6c 7574 696f 6e73 222c 2022 696e 6669  olutions", "infi
│ │ │ │ -000189e0: 6e69 7465 5f73 6f6c 7574 696f 6e73 222c  nite_solutions",
│ │ │ │ -000189f0: 2061 6e64 0a22 7369 6e67 756c 6172 5f73   and."singular_s
│ │ │ │ -00018a00: 6f6c 7574 696f 6e73 222e 0a0a 4966 2074  olutions"...If t
│ │ │ │ -00018a10: 6865 204e 616d 6553 6f6c 7574 696f 6e73  he NameSolutions
│ │ │ │ -00018a20: 4669 6c65 206f 7074 696f 6e20 6973 2073  File option is s
│ │ │ │ -00018a30: 6574 2074 6f20 3020 7468 656e 2022 6e6f  et to 0 then "no
│ │ │ │ -00018a40: 6e73 696e 6775 6c61 725f 736f 6c75 7469  nsingular_soluti
│ │ │ │ -00018a50: 6f6e 7322 2069 730a 696d 706f 7274 6564  ons" is.imported
│ │ │ │ -00018a60: 2c20 6973 2073 6574 2074 6f20 3120 7468  , is set to 1 th
│ │ │ │ -00018a70: 656e 2022 7265 616c 5f66 696e 6974 655f  en "real_finite_
│ │ │ │ -00018a80: 736f 6c75 7469 6f6e 7322 2069 7320 696d  solutions" is im
│ │ │ │ -00018a90: 706f 7274 6564 2c20 6973 2073 6574 2074  ported, is set t
│ │ │ │ -00018aa0: 6f20 320a 7468 656e 2022 696e 6669 6e69  o 2.then "infini
│ │ │ │ -00018ab0: 7465 5f73 6f6c 7574 696f 6e73 2220 6973  te_solutions" is
│ │ │ │ -00018ac0: 2069 6d70 6f72 7465 642c 2069 7320 7365   imported, is se
│ │ │ │ -00018ad0: 7420 746f 2033 2074 6865 6e20 2266 696e  t to 3 then "fin
│ │ │ │ -00018ae0: 6974 655f 736f 6c75 7469 6f6e 7322 2069  ite_solutions" i
│ │ │ │ -00018af0: 730a 696d 706f 7274 6564 2c20 6973 2073  s.imported, is s
│ │ │ │ -00018b00: 6574 2074 6f20 3420 7468 656e 2022 7374  et to 4 then "st
│ │ │ │ -00018b10: 6172 7422 2069 7320 696d 706f 7274 6564  art" is imported
│ │ │ │ -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                 |
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│ │ │ │ +00018dd0: 2020 7c0a 7c20 2020 2020 2020 2020 4166    |.|         Af
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│ │ │ │  000198e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000198f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573  ***********..Des
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│ │ │ │ -00019930: 6f20 312c 2074 6869 7320 6f70 7469 6f6e  o 1, this option
│ │ │ │ -00019940: 2069 6e64 6963 6174 6573 2074 6861 7420   indicates that 
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│ │ │ │  000199c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00019a10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │ +000199e0: 3120 3a20 5220 3d20 4343 5b78 2c79 2c7a  1 : R = CC[x,y,z
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│ │ │ │  00019aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00019b80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00019b90: 2a20 6265 7274 696e 695a 6572 6f44 696d  * bertiniZeroDim
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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
│ │ │ │ +00019c30: 6374 6976 653d 3e2e 2e2e 2922 0a20 202a  ctive=>...)".  *
│ │ │ │ +00019c40: 2022 6265 7274 696e 6952 6566 696e 6553   "bertiniRefineS
│ │ │ │ +00019c50: 6f6c 7328 2e2e 2e2c 4973 5072 6f6a 6563  ols(...,IsProjec
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│ │ │ │ +00019c70: 2262 6572 7469 6e69 5361 6d70 6c65 282e  "bertiniSample(.
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│ │ │ │ +00019c90: 3e2e 2e2e 2922 0a20 202a 2022 6265 7274  >...)".  * "bert
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│ │ │ │ +00019d10: 4973 5072 6f6a 6563 7469 7665 2c20 6973  IsProjective, is
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│ │ │ │ +00019d30: 0a28 4d61 6361 756c 6179 3244 6f63 2953  .(Macaulay2Doc)S
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│ │ │ │ +00019d50: 4265 7274 696e 692e 696e 666f 2c20 4e6f  Bertini.info, No
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│ │ │ │ +00019da0: 2c20 5570 3a20 546f 700a 0a4d 6169 6e44  , Up: Top..MainD
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│ │ │ │ +00019dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00019dd0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +00019de0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +00019df0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +00019e00: 7420 4d61 696e 4461 7461 4469 7265 6374  t MainDataDirect
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│ │ │ │ +00019e20: 756d 656e 7461 7469 6f6e 2920 6973 2061  umentation) is a
│ │ │ │ +00019e30: 202a 6e6f 7465 2073 796d 626f 6c3a 0a28   *note symbol:.(
│ │ │ │ +00019e40: 4d61 6361 756c 6179 3244 6f63 2953 796d  Macaulay2Doc)Sym
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│ │ │ │ +00019e80: 652c 204e 6578 743a 206d 616b 6542 2753  e, Next: makeB'S
│ │ │ │ +00019e90: 6563 7469 6f6e 2c20 5072 6576 3a20 4d61  ection, Prev: Ma
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│ │ │ │ +00019ec0: 496e 7075 7446 696c 6520 2d2d 2077 7269  InputFile -- wri
│ │ │ │ +00019ed0: 7465 2061 2042 6572 7469 6e69 2069 6e70  te a Bertini inp
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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: 
│ │ │ │ -00019f50: 0a20 2020 2020 2020 206d 616b 6542 2749  .        makeB'I
│ │ │ │ -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
│ │ │ │ -00019fa0: 2953 7472 696e 672c 2c20 6120 6469 7265  )String,, a dire
│ │ │ │ -00019fb0: 6374 6f72 7920 7768 6572 6520 7468 6520  ctory where the 
│ │ │ │ -00019fc0: 696e 7075 740a 2020 2020 2020 2020 6669  input.        fi
│ │ │ │ -00019fd0: 6c65 2077 696c 6c20 6265 2077 7269 7474  le will be writt
│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -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 
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│ │ │ │ -0001a530: 735f 636f 2072 616e 646f 6d20 6e75 6d62  s_co random numb
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│ │ │ │ -0001a550: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
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│ │ │ │ -0001a590: 732f 7374 7269 6e67 732f 7661 7269 6162  s/strings/variab
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│ │ │ │ -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.
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│ │ │ │ -0001a5e0: 6d20 636f 6d70 6c65 7820 6e75 6d62 6572  m complex number
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│ │ │ │ -0001a640: 2020 2020 2a20 5374 6f72 6167 6546 6f6c      * StorageFol
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│ │ │ │ -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
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│ │ │ │ -0001a6f0: 7261 636b 486f 6d6f 746f 7079 5f6c 705f  rackHomotopy_lp_
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -0001a810: 7368 6f75 6c64 2073 7065 6369 6679 0a76  should specify.v
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│ │ │ │ -0001a940: 6572 2073 686f 756c 6420 7573 6520 7468  er should use th
│ │ │ │ -0001a950: 650a 4e61 6d65 506f 6c79 6e6f 6d69 616c  e.NamePolynomial
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│ │ │ │ +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
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│ │ │ │ +00019fa0: 446f 6329 5374 7269 6e67 2c2c 2061 2064  Doc)String,, a d
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│ │ │ │ +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
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│ │ │ │ +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 {},
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│ │ │ │ +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
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│ │ │ │ +0001a100: 7420 7661 6c75 6520 7b7d 2c20 616e 206f  t value {}, an o
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│ │ │ │ +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
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│ │ │ │ +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
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│ │ │ │ +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
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│ │ │ │ +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
│ │ │ │ +0001a310: 7420 7661 6c75 6520 2269 6e70 7574 222c  t value "input",
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│ │ │ │ +0001a330: 6c79 6e6f 6d69 616c 7320 286d 6973 7369  lynomials (missi
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│ │ │ │ +0001a350: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c  ) => ..., defaul
│ │ │ │ +0001a360: 7420 7661 6c75 6520 7b7d 2c20 0a20 2020  t value {}, .   
│ │ │ │ +0001a370: 2020 202a 2050 6172 616d 6574 6572 4772     * ParameterGr
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│ │ │ │ +0001a3a0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
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│ │ │ │ +0001a400: 2020 2020 202a 202a 6e6f 7465 2052 616e       * *note Ran
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│ │ │ │ +0001a4e0: 2020 6f72 2072 616e 646f 6d20 636f 6d70    or random comp
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│ │ │ │ +0001a5c0: 7261 6e64 6f6d 2072 6561 6c20 6e75 6d62  random real numb
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│ │ │ │ +0001a690: 2020 2a20 5661 7269 6162 6c65 4c69 7374    * VariableList
│ │ │ │ +0001a6a0: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
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│ │ │ │ +0001a6e0: 6520 5665 7262 6f73 653a 2062 6572 7469  e Verbose: berti
│ │ │ │ +0001a6f0: 6e69 5472 6163 6b48 6f6d 6f74 6f70 795f  niTrackHomotopy_
│ │ │ │ +0001a700: 6c70 5f70 645f 7064 5f70 645f 636d 5665  lp_pd_pd_pd_cmVe
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│ │ │ │ +0001a7a0: 4265 7274 696e 6920 696e 7075 7420 6669  Bertini input fi
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│ │ │ │ +0001a850: 696e 6520 7661 7269 6162 6c65 2067 726f  ine variable gro
│ │ │ │ +0001a860: 7570 2920 6f70 7469 6f6e 206f 720a 486f  up) option or.Ho
│ │ │ │ +0001a870: 6d56 6172 6961 626c 6547 726f 7570 2028  mVariableGroup (
│ │ │ │ +0001a880: 686f 6d6f 6765 6e65 6f75 7320 7661 7269  homogeneous vari
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│ │ │ │ +0001a8c0: 706f 6c79 6e6f 6d69 616c 2073 7973 7465  polynomial syste
│ │ │ │ +0001a8d0: 6d20 7468 6579 2077 616e 7420 746f 2073  m they want to s
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│ │ │ │ +0001a910: 6e73 206f 7074 696f 6e2e 2049 6620 4227  ns option. If B'
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│ │ │ │ +0001a940: 2075 7365 7220 7368 6f75 6c64 2075 7365   user should use
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│ │ │ │ +0001a960: 6961 6c73 206f 7074 696f 6e2e 0a0a 2b2d  ials option...+-
│ │ │ │  0001a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a9a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2052  -------+.|i1 : R
│ │ │ │ -0001a9b0: 3d51 515b 7831 2c78 322c 795d 2020 2020  =QQ[x1,x2,y]    
│ │ │ │ +0001a9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +0001a9b0: 3a20 523d 5151 5b78 312c 7832 2c79 5d20  : R=QQ[x1,x2,y] 
│ │ │ │  0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a9e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa20: 2020 207c 0a7c 6f31 203d 2052 2020 2020     |.|o1 = R    
│ │ │ │ +0001aa20: 2020 2020 2020 7c0a 7c6f 3120 3d20 5220        |.|o1 = R 
│ │ │ │  0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001aa60: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 207c                 |
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│ │ │ │ +0001aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aaa0: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f    |.|o1 : Polyno
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│ │ │ │  0001aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aad0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -0001aae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aae0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
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│ │ │ │ -0001af10: 2042 6572 7469 6e69 496e 7075 7443 6f6e   BertiniInputCon
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│ │ │ │ +0001af10: 2020 2020 4265 7274 696e 6949 6e70 7574      BertiniInput
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│ │ │ │ +0001af80: 2020 4e61 6d65 506f 6c79 6e6f 6d69 616c    NamePolynomial
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│ │ │ │ -0001b050: 2020 2020 2020 2020 2020 7b66 322c 7831            {f2,x1
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│ │ │ │ +0001b060: 2c78 312d 7832 2b31 7d2c 2020 2020 2020  ,x1-x2+1},      
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│ │ │ │ -0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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  ]               
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│ │ │ │ -0001b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +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                  
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│ │ │ │ +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
│ │ │ │ -0001b210: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ +0001b200: 2020 2020 2020 7c0a 7c6f 3720 3a20 506f        |.|o7 : Po
│ │ │ │ +0001b210: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
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│ │ │ │  0001b260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001b280: 6e70 7574 4669 6c65 2874 6865 4469 722c  nputFile(theDir,
│ │ │ │ -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
│ │ │ │ -0001b2c0: 496e 7075 7443 6f6e 6669 6775 7261 7469  InputConfigurati
│ │ │ │ -0001b2d0: 6f6e 3d3e 7b4d 5079 7065 3d3e 327d 2c7c  on=>{MPype=>2},|
│ │ │ │ -0001b2e0: 0a7c 2020 2020 2020 2020 2020 4166 6656  .|          AffV
│ │ │ │ -0001b2f0: 6172 6961 626c 6547 726f 7570 3d3e 7b7b  ariableGroup=>{{
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│ │ │ │ -0001b310: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b320: 2020 2020 2020 2020 4227 506f 6c79 6e6f          B'Polyno
│ │ │ │ -0001b330: 6d69 616c 733d 3e7b 792a 585e 322b 312c  mials=>{y*X^2+1,
│ │ │ │ -0001b340: 7831 2d78 322b 312c 792d 327d 2c7c 0a7c  x1-x2+1,y-2},|.|
│ │ │ │ -0001b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b360: 4227 4675 6e63 7469 6f6e 733d 3e7b 2020  B'Functions=>{  
│ │ │ │ -0001b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b380: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001b390: 2020 2020 2020 2020 2020 7b58 2c78 312b            {X,x1+
│ │ │ │ -0001b3a0: 7832 2b31 7d7d 293b 2020 2020 2020 2020  x2+1}});        
│ │ │ │ -0001b3b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0001b3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b270: 2d2d 2d2d 2b0a 7c69 3820 3a20 6d61 6b65  ----+.|i8 : make
│ │ │ │ +0001b280: 4227 496e 7075 7446 696c 6528 7468 6544  B'InputFile(theD
│ │ │ │ +0001b290: 6972 2c20 2020 2020 2020 2020 2020 2020  ir,             
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│ │ │ │ +0001b2b0: 2020 2020 2020 2020 2020 2020 4265 7274              Bert
│ │ │ │ +0001b2c0: 696e 6949 6e70 7574 436f 6e66 6967 7572  iniInputConfigur
│ │ │ │ +0001b2d0: 6174 696f 6e3d 3e7b 4d50 7970 653d 3e32  ation=>{MPype=>2
│ │ │ │ +0001b2e0: 7d2c 7c0a 7c20 2020 2020 2020 2020 2041  },|.|          A
│ │ │ │ +0001b2f0: 6666 5661 7269 6162 6c65 4772 6f75 703d  ffVariableGroup=
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│ │ │ │ +0001b310: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001b320: 2020 2020 2020 2020 2020 2042 2750 6f6c             B'Pol
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│ │ │ │ +0001b390: 2020 2020 2020 2020 2020 2020 207b 582c               {X,
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│ │ │ │  0001b6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -0001b880: 7420 7661 6c75 6520 6e75 6c6c 0a20 2020  t value null.   
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│ │ │ │ -0001bbf0: 486f 6d6f 6765 6e69 7a61 7469 6f6e 206f  Homogenization o
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│ │ │ │ +0001b6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001b6f0: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
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│ │ │ │ +0001b7d0: 2042 2748 6f6d 6f67 656e 697a 6174 696f   B'Homogenizatio
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│ │ │ │ +0001b820: 6566 6175 6c74 2076 616c 7565 207b 7d0a  efault value {}.
│ │ │ │ +0001b830: 2020 2020 2020 2a20 436f 6e74 6169 6e73        * Contains
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│ │ │ │ +0001b850: 6661 756c 7420 7661 6c75 6520 7b7d 0a20  fault value {}. 
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│ │ │ │  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                  
│ │ │ │ -0001d410: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001d410: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0001d420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d460: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a  --------+.|i13 :
│ │ │ │ -0001d470: 2072 3d20 6166 6669 6e65 436f 6e74 6169   r= affineContai
│ │ │ │ -0001d480: 6e69 6e67 506f 696e 7423 4227 5365 6374  ningPoint#B'Sect
│ │ │ │ -0001d490: 696f 6e53 7472 696e 6720 2020 2020 2020  ionString       
│ │ │ │ +0001d460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +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    
│ │ │ │  0001d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d4b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d4b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d500: 2020 2020 2020 2020 7c0a 7c6f 3133 203d          |.|o13 =
│ │ │ │ -0001d510: 2028 312e 3138 3932 312b 2e38 3439 3533   (1.18921+.84953
│ │ │ │ -0001d520: 392a 6969 292a 2878 2d28 3129 2a28 3129  9*ii)*(x-(1)*(1)
│ │ │ │ -0001d530: 292b 282d 2e35 3432 3337 312b 2e33 3037  )+(-.542371+.307
│ │ │ │ -0001d540: 3133 372a 6969 292a 2879 2d28 3129 2a28  137*ii)*(y-(1)*(
│ │ │ │ -0001d550: 3229 292b 2820 2020 7c0a 7c20 2020 2020  2))+(   |.|     
│ │ │ │ -0001d560: 2031 2e33 3639 3435 2b2e 3031 3536 3333   1.36945+.015633
│ │ │ │ -0001d570: 2a69 6929 2a28 7a2d 2831 292a 2833 2929  *ii)*(z-(1)*(3))
│ │ │ │ -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)*(
│ │ │ │ +0001d580: 3329 2920 2020 2020 2020 2020 2020 2020  3))             
│ │ │ │  0001d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d5a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001d5a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0001d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d5f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a  --------+.|i14 :
│ │ │ │ -0001d600: 2070 7269 6e74 2072 2020 2020 2020 2020   print r        
│ │ │ │ +0001d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001d600: 3420 3a20 7072 696e 7420 7220 2020 2020  4 : print r     
│ │ │ │  0001d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d640: 2020 2020 2020 2020 7c0a 7c28 312e 3138          |.|(1.18
│ │ │ │ -0001d650: 3932 312b 2e38 3439 3533 392a 6969 292a  921+.849539*ii)*
│ │ │ │ -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
│ │ │ │ -0001d680: 292a 2879 2d28 3129 2a28 3229 292b 2831  )*(y-(1)*(2))+(1
│ │ │ │ -0001d690: 2e33 3639 3435 2020 7c0a 7c2d 2d2d 2d2d  .36945  |.|-----
│ │ │ │ +0001d640: 2020 2020 2020 2020 2020 207c 0a7c 2831             |.|(1
│ │ │ │ +0001d650: 2e31 3839 3231 2b2e 3834 3935 3339 2a69  .18921+.849539*i
│ │ │ │ +0001d660: 6929 2a28 782d 2831 292a 2831 2929 2b28  i)*(x-(1)*(1))+(
│ │ │ │ +0001d670: 2d2e 3534 3233 3731 2b2e 3330 3731 3337  -.542371+.307137
│ │ │ │ +0001d680: 2a69 6929 2a28 792d 2831 292a 2832 2929  *ii)*(y-(1)*(2))
│ │ │ │ +0001d690: 2b28 312e 3336 3934 3520 207c 0a7c 2d2d  +(1.36945  |.|--
│ │ │ │  0001d6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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)*(
│ │ │ │ -0001d700: 3329 2920 2020 2020 2020 2020 2020 2020  3))             
│ │ │ │ +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))          
│ │ │ │  0001d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d730: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001d730: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0001d740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d780: 2d2d 2d2d 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d  --------+.+-----
│ │ │ │ +0001d780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b 2d2d  -----------+.+--
│ │ │ │  0001d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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
│ │ │ │ -0001d810: 6f69 6e74 3d3e 502c 4227 486f 6d6f 6765  oint=>P,B'Homoge
│ │ │ │ -0001d820: 6e69 7a61 7469 6f6e 7c0a 7c20 2020 2020  nization|.|     
│ │ │ │ +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  ..}             
│ │ │ │ +0001d870: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001d880: 3520 3d20 4227 5365 6374 696f 6e7b 2e2e  5 = B'Section{..
│ │ │ │ +0001d890: 2e33 2e2e 2e7d 2020 2020 2020 2020 2020  .3...}          
│ │ │ │  0001d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d8c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001d8c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d910: 2020 2020 2020 2020 7c0a 7c6f 3135 203a          |.|o15 :
│ │ │ │ -0001d920: 2042 2753 6563 7469 6f6e 2020 2020 2020   B'Section      
│ │ │ │ +0001d910: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001d920: 3520 3a20 4227 5365 6374 696f 6e20 2020  5 : B'Section   
│ │ │ │  0001d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d960: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +0001d960: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  0001d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d9b0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c3d 3e22 782b  --------|.|=>"x+
│ │ │ │ -0001d9c0: 792b 7a22 2920 2020 2020 2020 2020 2020  y+z")           
│ │ │ │ +0001d9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3d3e  -----------|.|=>
│ │ │ │ +0001d9c0: 2278 2b79 2b7a 2229 2020 2020 2020 2020  "x+y+z")        
│ │ │ │  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                  
│ │ │ │ -0001da00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001da00: 2020 2020 2020 2020 2020 207c 0a2b 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 2d2d  ----------------
│ │ │ │  0001da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001da50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a  --------+.|i16 :
│ │ │ │ -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                  
│ │ │ │ -0001daf0: 2020 2020 2020 2020 7c0a 7c6f 3136 203d          |.|o16 =
│ │ │ │ -0001db00: 2042 2753 6563 7469 6f6e 7b42 2748 6f6d   B'Section{B'Hom
│ │ │ │ -0001db10: 6f67 656e 697a 6174 696f 6e20 3d3e 2078  ogenization => x
│ │ │ │ -0001db20: 2b79 2b7a 2020 2020 2020 2020 2020 2020  +y+z            
│ │ │ │ +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 =
│ │ │ │ +0001db20: 3e20 782b 792b 7a20 2020 2020 2020 2020  > x+y+z         
│ │ │ │  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
│ │ │ │ -0001db60: 6265 7243 6f65 6666 6963 6965 6e74 7320  berCoefficients 
│ │ │ │ -0001db70: 3d3e 207b 2e35 3334 3631 342d 2e31 3735  => {.534614-.175
│ │ │ │ -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 => (.
│ │ │ │ -0001dbc0: 3533 3436 3134 2d2e 3137 3539 3435 2a69  534614-.175945*i
│ │ │ │ -0001dbd0: 6929 2a28 782d 2878 2b79 2b7a 292a 2820  i)*(x-(x+y+z)*( 
│ │ │ │ -0001dbe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001dbf0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ +0001db40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001db50: 2020 2020 2020 2020 2020 2020 2020 4227                B'
│ │ │ │ +0001db60: 4e75 6d62 6572 436f 6566 6669 6369 656e  NumberCoefficien
│ │ │ │ +0001db70: 7473 203d 3e20 7b2e 3533 3436 3134 2d2e  ts => {.534614-.
│ │ │ │ +0001db80: 3137 3539 3435 2a69 692c 202e 3432 3637  175945*ii, .4267
│ │ │ │ +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 =>
│ │ │ │ +0001dbc0: 2028 2e35 3334 3631 342d 2e31 3735 3934   (.534614-.17594
│ │ │ │ +0001dbd0: 352a 6969 292a 2878 2d28 782b 792b 7a29  5*ii)*(x-(x+y+z)
│ │ │ │ +0001dbe0: 2a28 2020 2020 2020 2020 207c 0a7c 2020  *(         |.|  
│ │ │ │ +0001dbf0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │  0001dc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dc30: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0001dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │  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 2020                  
│ │ │ │  0001dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001dc90: 202d 2e39 3735 3339 2a69 692c 202d 2e34   -.97539*ii, -.4
│ │ │ │ -0001dca0: 3738 3830 332b 2e30 3431 3630 3038 2a69  78803+.0416008*i
│ │ │ │ -0001dcb0: 697d 2020 2020 2020 2020 2020 2020 2020  i}              
│ │ │ │ +0001dc80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001dc90: 2020 2020 2d2e 3937 3533 392a 6969 2c20      -.97539*ii, 
│ │ │ │ +0001dca0: 2d2e 3437 3838 3033 2b2e 3034 3136 3030  -.478803+.041600
│ │ │ │ +0001dcb0: 382a 6969 7d20 2020 2020 2020 2020 2020  8*ii}           
│ │ │ │  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
│ │ │ │ -0001dd10: 3033 2b2e 3034 3136 3030 382a 6969 292a  03+.0416008*ii)*
│ │ │ │ -0001dd20: 287a 2d28 782b 792b 7c0a 7c20 2020 2020  (z-(x+y+|.|     
│ │ │ │ -0001dd30: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ +0001dcd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001dce0: 2020 2020 3129 292b 282e 3432 3637 3034      1))+(.426704
│ │ │ │ +0001dcf0: 2d2e 3937 3533 392a 6969 292a 2879 2d28  -.97539*ii)*(y-(
│ │ │ │ +0001dd00: 782b 792b 7a29 2a28 3229 292b 282d 2e34  x+y+z)*(2))+(-.4
│ │ │ │ +0001dd10: 3738 3830 332b 2e30 3431 3630 3038 2a69  78803+.0416008*i
│ │ │ │ +0001dd20: 6929 2a28 7a2d 2878 2b79 2b7c 0a7c 2020  i)*(z-(x+y+|.|  
│ │ │ │ +0001dd30: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 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 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -0001dd80: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │ +0001dd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0001dd80: 2020 2020 2020 2020 2020 207d 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 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001ddc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001de20: 207a 292a 2833 2929 2020 2020 2020 2020   z)*(3))        
│ │ │ │ +0001de10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001de20: 2020 2020 7a29 2a28 3329 2920 2020 2020      z)*(3))     
│ │ │ │  0001de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001de60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0001de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001deb0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a  --------+.|i17 :
│ │ │ │ -0001dec0: 2070 7269 6e74 2072 486f 6d6f 6765 5365   print rHomogeSe
│ │ │ │ -0001ded0: 6374 696f 6e23 4227 5365 6374 696f 6e53  ction#B'SectionS
│ │ │ │ -0001dee0: 7472 696e 6720 2020 2020 2020 2020 2020  tring           
│ │ │ │ +0001deb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001dec0: 3720 3a20 7072 696e 7420 7248 6f6d 6f67  7 : print rHomog
│ │ │ │ +0001ded0: 6553 6563 7469 6f6e 2342 2753 6563 7469  eSection#B'Secti
│ │ │ │ +0001dee0: 6f6e 5374 7269 6e67 2020 2020 2020 2020  onString        
│ │ │ │  0001def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df00: 2020 2020 2020 2020 7c0a 7c28 2e35 3334          |.|(.534
│ │ │ │ -0001df10: 3631 342d 2e31 3735 3934 352a 6969 292a  614-.175945*ii)*
│ │ │ │ -0001df20: 2878 2d28 782b 792b 7a29 2a28 3129 292b  (x-(x+y+z)*(1))+
│ │ │ │ -0001df30: 282e 3432 3637 3034 2d2e 3937 3533 392a  (.426704-.97539*
│ │ │ │ -0001df40: 6969 292a 2879 2d28 782b 792b 7a29 2a28  ii)*(y-(x+y+z)*(
│ │ │ │ -0001df50: 3220 2020 2020 2020 7c0a 7c2d 2d2d 2d2d  2       |.|-----
│ │ │ │ +0001df00: 2020 2020 2020 2020 2020 207c 0a7c 282e             |.|(.
│ │ │ │ +0001df10: 3533 3436 3134 2d2e 3137 3539 3435 2a69  534614-.175945*i
│ │ │ │ +0001df20: 6929 2a28 782d 2878 2b79 2b7a 292a 2831  i)*(x-(x+y+z)*(1
│ │ │ │ +0001df30: 2929 2b28 2e34 3236 3730 342d 2e39 3735  ))+(.426704-.975
│ │ │ │ +0001df40: 3339 2a69 6929 2a28 792d 2878 2b79 2b7a  39*ii)*(y-(x+y+z
│ │ │ │ +0001df50: 292a 2832 2020 2020 2020 207c 0a7c 2d2d  )*(2       |.|--
│ │ │ │  0001df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dfa0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c29 292b 282d  --------|.|))+(-
│ │ │ │ -0001dfb0: 2e34 3738 3830 332b 2e30 3431 3630 3038  .478803+.0416008
│ │ │ │ -0001dfc0: 2a69 6929 2a28 7a2d 2878 2b79 2b7a 292a  *ii)*(z-(x+y+z)*
│ │ │ │ -0001dfd0: 2833 2929 2020 2020 2020 2020 2020 2020  (3))            
│ │ │ │ +0001dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2929  -----------|.|))
│ │ │ │ +0001dfb0: 2b28 2d2e 3437 3838 3033 2b2e 3034 3136  +(-.478803+.0416
│ │ │ │ +0001dfc0: 3030 382a 6969 292a 287a 2d28 782b 792b  008*ii)*(z-(x+y+
│ │ │ │ +0001dfd0: 7a29 2a28 3329 2920 2020 2020 2020 2020  z)*(3))         
│ │ │ │  0001dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dff0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001dff0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0001e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e040: 2d2d 2d2d 2d2d 2d2d 2b0a 2b2d 2d2d 2d2d  --------+.+-----
│ │ │ │ +0001e040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b 2d2d  -----------+.+--
│ │ │ │  0001e050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e070: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20  -------+.|i18 : 
│ │ │ │ -0001e080: 663d 2279 5e33 2d78 2a79 2b31 2220 2020  f="y^3-x*y+1"   
│ │ │ │ +0001e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ +0001e080: 203a 2066 3d22 795e 332d 782a 792b 3122   : f="y^3-x*y+1"
│ │ │ │  0001e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e0a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0d0: 2020 2020 207c 0a7c 6f31 3820 3d20 795e       |.|o18 = y^
│ │ │ │ -0001e0e0: 332d 782a 792b 3120 2020 2020 2020 2020  3-x*y+1         
│ │ │ │ +0001e0d0: 2020 2020 2020 2020 7c0a 7c6f 3138 203d          |.|o18 =
│ │ │ │ +0001e0e0: 2079 5e33 2d78 2a79 2b31 2020 2020 2020   y^3-x*y+1      
│ │ │ │  0001e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e100: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001e100: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0001e110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e130: 2d2d 2d2b 0a7c 6931 3920 3a20 7331 3d6d  ---+.|i19 : s1=m
│ │ │ │ -0001e140: 616b 6542 2753 6563 7469 6f6e 287b 782c  akeB'Section({x,
│ │ │ │ -0001e150: 792c 317d 2920 2020 2020 2020 2020 2020  y,1})           
│ │ │ │ -0001e160: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001e130: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2073  ------+.|i19 : s
│ │ │ │ +0001e140: 313d 6d61 6b65 4227 5365 6374 696f 6e28  1=makeB'Section(
│ │ │ │ +0001e150: 7b78 2c79 2c31 7d29 2020 2020 2020 2020  {x,y,1})        
│ │ │ │ +0001e160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e190: 207c 0a7c 6f31 3920 3d20 4227 5365 6374   |.|o19 = B'Sect
│ │ │ │ -0001e1a0: 696f 6e7b 2e2e 2e32 2e2e 2e7d 2020 2020  ion{...2...}    
│ │ │ │ +0001e190: 2020 2020 7c0a 7c6f 3139 203d 2042 2753      |.|o19 = B'S
│ │ │ │ +0001e1a0: 6563 7469 6f6e 7b2e 2e2e 322e 2e2e 7d20  ection{...2...} 
│ │ │ │  0001e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001e1c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001e1f0: 0a7c 6f31 3920 3a20 4227 5365 6374 696f  .|o19 : B'Sectio
│ │ │ │ -0001e200: 6e20 2020 2020 2020 2020 2020 2020 2020  n               
│ │ │ │ -0001e210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e220: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1f0: 2020 7c0a 7c6f 3139 203a 2042 2753 6563    |.|o19 : B'Sec
│ │ │ │ +0001e200: 7469 6f6e 2020 2020 2020 2020 2020 2020  tion            
│ │ │ │ +0001e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e220: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0001e250: 6932 3020 3a20 6d61 6b65 4227 496e 7075  i20 : makeB'Inpu
│ │ │ │ -0001e260: 7446 696c 6528 7374 6f72 6542 4d32 4669  tFile(storeBM2Fi
│ │ │ │ -0001e270: 6c65 732c 2020 2020 2020 2020 7c0a 7c20  les,        |.| 
│ │ │ │ -0001e280: 2020 2020 2020 2041 6666 5661 7269 6162         AffVariab
│ │ │ │ -0001e290: 6c65 4772 6f75 703d 3e7b 782c 797d 2c20  leGroup=>{x,y}, 
│ │ │ │ -0001e2a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001e2b0: 2020 2020 2020 4227 506f 6c79 6e6f 6d69        B'Polynomi
│ │ │ │ -0001e2c0: 616c 733d 3e7b 662c 7331 7d29 3b20 2020  als=>{f,s1});   
│ │ │ │ -0001e2d0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e250: 2b0a 7c69 3230 203a 206d 616b 6542 2749  +.|i20 : makeB'I
│ │ │ │ +0001e260: 6e70 7574 4669 6c65 2873 746f 7265 424d  nputFile(storeBM
│ │ │ │ +0001e270: 3246 696c 6573 2c20 2020 2020 2020 207c  2Files,        |
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│ │ │ │  0001e5f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001e600: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -0001e630: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
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│ │ │ │ -0001e860: 2020 2020 2a20 436f 6e74 6169 6e73 4d75      * ContainsMu
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│ │ │ │ +0001e840: 6963 6965 6e74 7320 3d3e 202e 2e2e 2c20  icients => ..., 
│ │ │ │ +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
│ │ │ │ +0001e8b0: 6f69 6e74 203d 3e20 2e2e 2e2c 2064 6566  oint => ..., def
│ │ │ │ +0001e8c0: 6175 6c74 2076 616c 7565 207b 7d0a 2020  ault value {}.  
│ │ │ │ +0001e8d0: 2020 2020 2a20 4e61 6d65 4227 536c 6963      * NameB'Slic
│ │ │ │ +0001e8e0: 6520 3d3e 202e 2e2e 2c20 6465 6661 756c  e => ..., defaul
│ │ │ │ +0001e8f0: 7420 7661 6c75 6520 6e75 6c6c 0a20 2020  t value null.   
│ │ │ │ +0001e900: 2020 202a 2052 616e 646f 6d43 6f65 6666     * RandomCoeff
│ │ │ │ +0001e910: 6963 6965 6e74 4765 6e65 7261 746f 7220  icientGenerator 
│ │ │ │ +0001e920: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +0001e930: 7661 6c75 650a 2020 2020 2020 2020 4675  value.        Fu
│ │ │ │ +0001e940: 6e63 7469 6f6e 436c 6f73 7572 655b 2e2e  nctionClosure[..
│ │ │ │ +0001e950: 2f42 6572 7469 6e69 2e6d 323a 3233 3536  /Bertini.m2:2356
│ │ │ │ +0001e960: 3a33 372d 3233 3536 3a36 365d 0a0a 4465  :37-2356:66]..De
│ │ │ │ +0001e970: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0001e980: 3d3d 3d3d 3d0a 0a6d 616b 6542 2753 6c69  =====..makeB'Sli
│ │ │ │ +0001e990: 6365 2061 6c6c 6f77 7320 666f 7220 6561  ce allows for ea
│ │ │ │ +0001e9a0: 7379 2063 7265 6174 696f 6e20 6f66 2065  sy creation of e
│ │ │ │ +0001e9b0: 7175 6174 696f 6e73 2074 6861 7420 6465  quations that de
│ │ │ │ +0001e9c0: 6669 6e65 206c 696e 6561 7220 7370 6163  fine linear spac
│ │ │ │ +0001e9d0: 6573 2c0a 692e 652e 2073 6c69 6365 732e  es,.i.e. slices.
│ │ │ │ +0001e9e0: 2054 6865 2064 6566 6175 6c74 2063 7265   The default cre
│ │ │ │ +0001e9f0: 6174 6573 2061 2068 6173 6820 7461 626c  ates a hash tabl
│ │ │ │ +0001ea00: 6520 7769 7468 2074 776f 206b 6579 733a  e with two keys:
│ │ │ │ +0001ea10: 0a42 274e 756d 6265 7243 6f65 6666 6963  .B'NumberCoeffic
│ │ │ │ +0001ea20: 6965 6e74 7320 616e 6420 4227 5365 6374  ients and B'Sect
│ │ │ │ +0001ea30: 696f 6e53 7472 696e 672e 2057 6865 6e20  ionString. When 
│ │ │ │ +0001ea40: 7765 2068 6176 6520 6120 6d75 6c74 6970  we have a multip
│ │ │ │ +0001ea50: 726f 6a65 6374 6976 650a 7661 7269 6574  rojective.variet
│ │ │ │ +0001ea60: 7920 7765 2063 616e 2064 6966 6665 7265  y we can differe
│ │ │ │ +0001ea70: 6e74 2074 7970 6573 206f 6620 736c 6963  nt types of slic
│ │ │ │ +0001ea80: 6573 2e20 546f 206d 616b 6520 6120 736c  es. To make a sl
│ │ │ │ +0001ea90: 6963 6520 7765 206e 6565 6420 746f 2073  ice we need to s
│ │ │ │ +0001eaa0: 7065 6369 6679 0a74 6865 2074 7970 6520  pecify.the type 
│ │ │ │ +0001eab0: 6f66 2073 6c69 6365 2077 6520 7761 6e74  of slice we want
│ │ │ │ +0001eac0: 2066 6f6c 6c6f 7765 6420 6279 2076 6172   followed by var
│ │ │ │ +0001ead0: 6961 626c 6520 6772 6f75 7073 2e0a 0a2b  iable groups...+
│ │ │ │  0001eae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ -0001eb30: 3a20 736c 6963 6554 7970 653d 7b31 2c31  : sliceType={1,1
│ │ │ │ -0001eb40: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0001eb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001eb30: 6931 203a 2073 6c69 6365 5479 7065 3d7b  i1 : sliceType={
│ │ │ │ +0001eb40: 312c 317d 2020 2020 2020 2020 2020 2020  1,1}            
│ │ │ │  0001eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001eb70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001eb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ebc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -0001ebd0: 3d20 7b31 2c20 317d 2020 2020 2020 2020  = {1, 1}        
│ │ │ │ +0001ebc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ebd0: 6f31 203d 207b 312c 2031 7d20 2020 2020  o1 = {1, 1}     
│ │ │ │  0001ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ec10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001ec10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ec60: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -0001ec70: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +0001ec60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ec70: 6f31 203a 204c 6973 7420 2020 2020 2020  o1 : List       
│ │ │ │  0001ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ecb0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001ecb0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ecf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -0001ed10: 3a20 7661 7269 6162 6c65 4772 6f75 7073  : variableGroups
│ │ │ │ -0001ed20: 3d7b 7b78 302c 7831 7d2c 7b79 302c 7931  ={{x0,x1},{y0,y1
│ │ │ │ -0001ed30: 2c79 327d 7d20 2020 2020 2020 2020 2020  ,y2}}           
│ │ │ │ +0001ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001ed10: 6932 203a 2076 6172 6961 626c 6547 726f  i2 : variableGro
│ │ │ │ +0001ed20: 7570 733d 7b7b 7830 2c78 317d 2c7b 7930  ups={{x0,x1},{y0
│ │ │ │ +0001ed30: 2c79 312c 7932 7d7d 2020 2020 2020 2020  ,y1,y2}}        
│ │ │ │  0001ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001ed50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eda0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -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}, 
│ │ │ │ +0001edc0: 7b79 302c 2079 312c 2079 327d 7d20 2020  {y0, y1, y2}}   
│ │ │ │  0001edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001edf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001edf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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          
│ │ │ │ +0001ee40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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                  
│ │ │ │  0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001ee90: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -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                  
│ │ │ │ -0001ef30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +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
│ │ │ │ +0001ef20: 7329 2020 2020 2020 2020 2020 2020 2020  s)              
│ │ │ │ +0001ef30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef80: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -0001ef90: 3d20 4227 536c 6963 657b 2e2e 2e34 2e2e  = B'Slice{...4..
│ │ │ │ -0001efa0: 2e7d 2020 2020 2020 2020 2020 2020 2020  .}              
│ │ │ │ +0001ef80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ef90: 6f33 203d 2042 2753 6c69 6365 7b2e 2e2e  o3 = B'Slice{...
│ │ │ │ +0001efa0: 342e 2e2e 7d20 2020 2020 2020 2020 2020  4...}           
│ │ │ │  0001efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001efd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f020: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -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    
│ │ │ │  0001f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f070: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001f070: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0001f080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -0001f0d0: 3a20 7065 656b 2078 7953 6c69 6365 2020  : peek xySlice  
│ │ │ │ -0001f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001f0d0: 6934 203a 2070 6565 6b20 7879 536c 6963  i4 : peek xySlic
│ │ │ │ +0001f0e0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  0001f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f110: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f160: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -0001f170: 3d20 4227 536c 6963 657b 4227 4e75 6d62  = B'Slice{B'Numb
│ │ │ │ -0001f180: 6572 436f 6566 6669 6369 656e 7473 203d  erCoefficients =
│ │ │ │ -0001f190: 3e20 7b7b 312e 3439 3134 342b 2e37 3133  > {{1.49144+.713
│ │ │ │ -0001f1a0: 3834 362a 6969 2c20 2d2e 3834 3031 3133  846*ii, -.840113
│ │ │ │ -0001f1b0: 2b31 2e31 3938 362a 2020 7c0a 7c20 2020  +1.1986*  |.|   
│ │ │ │ -0001f1c0: 2020 2020 2020 2020 2020 4227 5365 6374            B'Sect
│ │ │ │ -0001f1d0: 696f 6e53 7472 696e 6720 3d3e 207b 2831  ionString => {(1
│ │ │ │ -0001f1e0: 2e34 3931 3434 2b2e 3731 3338 3436 2a69  .49144+.713846*i
│ │ │ │ -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
│ │ │ │ -0001f230: 6563 7469 6f6e 7b2e 2e2e 322e 2e2e 7d2c  ection{...2...},
│ │ │ │ -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   
│ │ │ │ +0001f160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f170: 6f34 203d 2042 2753 6c69 6365 7b42 274e  o4 = B'Slice{B'N
│ │ │ │ +0001f180: 756d 6265 7243 6f65 6666 6963 6965 6e74  umberCoefficient
│ │ │ │ +0001f190: 7320 3d3e 207b 7b31 2e34 3931 3434 2b2e  s => {{1.49144+.
│ │ │ │ +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
│ │ │ │ +0001f1d0: 6563 7469 6f6e 5374 7269 6e67 203d 3e20  ectionString => 
│ │ │ │ +0001f1e0: 7b28 312e 3439 3134 342b 2e37 3133 3834  {(1.49144+.71384
│ │ │ │ +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 => {
│ │ │ │ +0001f230: 4227 5365 6374 696f 6e7b 2e2e 2e32 2e2e  B'Section{...2..
│ │ │ │ +0001f240: 2e7d 2c20 4227 5365 6374 696f 6e7b 2e2e  .}, B'Section{..
│ │ │ │ +0001f250: 2e32 2e2e 2e7d 7d20 2020 2020 207c 0a7c  .2...}}      |.|
│ │ │ │ +0001f260: 2020 2020 2020 2020 2020 2020 204e 616d               Nam
│ │ │ │ +0001f270: 6542 2753 6c69 6365 203d 3e20 6e75 6c6c  eB'Slice => null
│ │ │ │  0001f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +0001f2a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 7c0a 7c69 697d  ----------|.|ii}
│ │ │ │ -0001f300: 2c20 7b2e 3031 3438 3432 2b31 2e32 3335  , {.014842+1.235
│ │ │ │ -0001f310: 3438 2a69 692c 202d 2e32 3134 3436 382b  48*ii, -.214468+
│ │ │ │ -0001f320: 2e39 3131 3239 332a 6969 2c20 2d2e 3438  .911293*ii, -.48
│ │ │ │ -0001f330: 3631 3736 2b2e 3430 3035 3737 2a69 697d  6176+.400577*ii}
│ │ │ │ -0001f340: 7d20 2020 2020 2020 2020 7c0a 7c2a 6969  }         |.|*ii
│ │ │ │ -0001f350: 292a 2878 3129 2c20 282e 3031 3438 3432  )*(x1), (.014842
│ │ │ │ -0001f360: 2b31 2e32 3335 3438 2a69 6929 2a28 7930  +1.23548*ii)*(y0
│ │ │ │ -0001f370: 292b 282d 2e32 3134 3436 382b 2e39 3131  )+(-.214468+.911
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│ │ │ │ +0001f8f0: 6f37 203a 204c 6973 7420 2020 2020 2020  o7 : List       
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│ │ │ │ +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      
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│ │ │ │ +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)     
│ │ │ │ +0001fff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020020: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020030: 7c6f 3133 203d 2042 2753 6c69 6365 7b2e  |o13 = B'Slice{.
│ │ │ │ -00020040: 2e2e 342e 2e2e 7d20 2020 2020 2020 2020  ..4...}         
│ │ │ │ +00020020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020030: 207c 0a7c 6f31 3320 3d20 4227 536c 6963   |.|o13 = B'Slic
│ │ │ │ +00020040: 657b 2e2e 2e34 2e2e 2e7d 2020 2020 2020  e{...4...}      
│ │ │ │  00020050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020060: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00020070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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       
│ │ │ │  000200c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000200d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000200e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  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  --------------+.
│ │ │ │ -00020120: 7c69 3134 203a 2079 7953 6c69 6365 3d6d  |i14 : yySlice=m
│ │ │ │ -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                  
│ │ │ │ -00020200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -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               
│ │ │ │  00020230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020240: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00020240: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  00020250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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
│ │ │ │ -000202a0: 746f 7265 424d 3246 696c 6573 2c20 2020  toreBM2Files,   
│ │ │ │ +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  ----------------
│ │ │ │  00020350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020370: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2072  ------+.|i16 : r
│ │ │ │ -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
│ │ │ │ +00020390: 7265 424d 3246 696c 6573 2920 2020 2020  reBM2Files)     
│ │ │ │  000203a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +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
│ │ │ │ -00020410: 4669 6c65 2873 746f 7265 424d 3246 696c  File(storeBM2Fil
│ │ │ │ -00020420: 6573 2920 2020 2020 2020 7c0a 7c20 2020  es)       |.|   
│ │ │ │ +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
│ │ │ │ -00020590: 293b 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  );|.+-----------
│ │ │ │ +000204d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000204e0: 2d2b 0a7c 6931 3820 3a20 6d61 6b65 4227  -+.|i18 : makeB'
│ │ │ │ +000204f0: 496e 7075 7446 696c 6528 7374 6f72 6542  InputFile(storeB
│ │ │ │ +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
│ │ │ │ +00020530: 6961 626c 6547 726f 7570 3d3e 7661 7269  iableGroup=>vari
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│ │ │ │  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
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│ │ │ │ -00020670: 7265 424d 3246 696c 6573 2920 2020 2020  reBM2Files)     
│ │ │ │ -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
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│ │ │ │ +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            |.+---
│ │ │ │ +000206f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  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
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│ │ │ │ -00020770: 2020 7c0a 7c20 2020 2020 2020 2020 2048    |.|          H
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│ │ │ │ +00020770: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020780: 2020 486f 6d56 6172 6961 626c 6547 726f    HomVariableGro
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│ │ │ │  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 
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│ │ │ │  00020850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020860: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00020860: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00020870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000208a0: 7c69 3233 203a 2079 7944 6567 7265 653d  |i23 : yyDegree=
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│ │ │ │ +00020890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000208a0: 2d2b 0a7c 6932 3320 3a20 7979 4465 6772  -+.|i23 : yyDegr
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│ │ │ │  000208e0: 2020 2020 2020 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 7c0a 7c6f 3233 203d 2031        |.|o23 = 1
│ │ │ │ -00020920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020910: 2020 2020 2020 2020 207c 0a7c 6f32 3320           |.|o23 
│ │ │ │ +00020920: 3d20 3120 2020 2020 2020 2020 2020 2020  = 1             
│ │ │ │  00020930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020950: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00020950: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00020960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00020990: 0a57 6179 7320 746f 2075 7365 206d 616b  .Ways to use mak
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│ │ │ │ +00020980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020990: 2d2b 0a0a 5761 7973 2074 6f20 7573 6520  -+..Ways to use 
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│ │ │ │  000209b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -000209d0: 6c69 6365 2854 6869 6e67 2c4c 6973 7429  lice(Thing,List)
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│ │ │ │ -00020ab0: 4e65 7874 3a20 4e75 6d62 6572 546f 4227  Next: NumberToB'
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│ │ │ │ -00020af0: 2d20 4d6f 7665 206f 7220 636f 7079 2066  - Move or copy f
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│ │ │ │ +000209c0: 3d3d 3d3d 3d0a 0a20 202a 2022 6d61 6b65  =====..  * "make
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│ │ │ │ +00020a00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
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│ │ │ │ +00020a60: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
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│ │ │ │ +00020a90: 653a 2042 6572 7469 6e69 2e69 6e66 6f2c  e: Bertini.info,
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│ │ │ │ +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
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│ │ │ │ -00020b40: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
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│ │ │ │ -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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│ │ │ │ -00020c70: 3244 6f63 2975 7369 6e67 2066 756e 6374  2Doc)using funct
│ │ │ │ -00020c80: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
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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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│ │ │ │ -00020d50: 7661 6c75 6520 6e75 6c6c 0a20 2020 2020  value null.     
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│ │ │ │ +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
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│ │ │ │ -00021620: 2d2d 2b0a 0a54 6865 206f 7074 696f 6e73  --+..The options
│ │ │ │ -00021630: 204d 6f76 6554 6f44 6972 6563 746f 7279   MoveToDirectory
│ │ │ │ -00021640: 2061 6e64 2053 7562 466f 6c64 6572 2067   and SubFolder g
│ │ │ │ -00021650: 6976 6520 6772 6561 7465 7220 636f 6e74  ive greater cont
│ │ │ │ -00021660: 726f 6c20 666f 7220 7768 6572 6520 746f  rol for where to
│ │ │ │ -00021670: 0a6d 6f76 6520 7468 6520 6669 6c65 2e0a  .move the file..
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│ │ │ │ +00021620: 2d2d 2d2d 2d2b 0a0a 5468 6520 6f70 7469  -----+..The opti
│ │ │ │ +00021630: 6f6e 7320 4d6f 7665 546f 4469 7265 6374  ons MoveToDirect
│ │ │ │ +00021640: 6f72 7920 616e 6420 5375 6246 6f6c 6465  ory and SubFolde
│ │ │ │ +00021650: 7220 6769 7665 2067 7265 6174 6572 2063  r give greater c
│ │ │ │ +00021660: 6f6e 7472 6f6c 2066 6f72 2077 6865 7265  ontrol for where
│ │ │ │ +00021670: 2074 6f0a 6d6f 7665 2074 6865 2066 696c   to.move the fil
│ │ │ │ +00021680: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  e...+-----------
│ │ │ │  00021690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000216a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000216b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000216c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000216d0: 0a7c 6939 203a 2044 6972 3120 3d20 7465  .|i9 : Dir1 = te
│ │ │ │ -000216e0: 6d70 6f72 6172 7946 696c 654e 616d 6528  mporaryFileName(
│ │ │ │ -000216f0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +000216c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000216d0: 2d2d 2b0a 7c69 3920 3a20 4469 7231 203d  --+.|i9 : Dir1 =
│ │ │ │ +000216e0: 2074 656d 706f 7261 7279 4669 6c65 4e61   temporaryFileNa
│ │ │ │ +000216f0: 6d65 2829 3b20 2020 2020 2020 2020 2020  me();           
│ │ │ │  00021700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021720: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00021710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021720: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00021730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -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                  
│ │ │ │ -000217b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000217c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000217b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000217d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000217e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000217f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00021810: 0a7c 6931 3120 3a20 7772 6974 6550 6172  .|i11 : writePar
│ │ │ │ -00021820: 616d 6574 6572 4669 6c65 2873 746f 7265  ameterFile(store
│ │ │ │ -00021830: 424d 3246 696c 6573 2c7b 322c 332c 352c  BM2Files,{2,3,5,
│ │ │ │ -00021840: 377d 293b 2020 2020 2020 2020 2020 2020  7});            
│ │ │ │ -00021850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021860: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00021800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +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    |.+-----------
│ │ │ │  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 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
│ │ │ │ -000218d0: 2c22 6669 6e61 6c5f 7061 7261 6d65 7465  ,"final_paramete
│ │ │ │ -000218e0: 7273 222c 2273 7461 7274 5f70 6172 616d  rs","start_param
│ │ │ │ -000218f0: 6574 6572 7322 2c20 2020 2020 2020 207c  eters",        |
│ │ │ │ -00021900: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000218a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000218b0: 2d2d 2b0a 7c69 3132 203a 206d 6f76 6542  --+.|i12 : moveB
│ │ │ │ +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",      
│ │ │ │ +00021900: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  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 2d7c  ---------------|
│ │ │ │ -00021950: 0a7c 4d6f 7665 546f 4469 7265 6374 6f72  .|MoveToDirector
│ │ │ │ -00021960: 793d 3e44 6972 3129 2020 2020 2020 2020  y=>Dir1)        
│ │ │ │ +00021940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021950: 2d2d 7c0a 7c4d 6f76 6554 6f44 6972 6563  --|.|MoveToDirec
│ │ │ │ +00021960: 746f 7279 3d3e 4469 7231 2920 2020 2020  tory=>Dir1)     
│ │ │ │  00021970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000219a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00021990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000219a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000219b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000219c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000219d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000219e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -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  ----------------
│ │ │ │ +000219f0: 2d2d 2b0a 7c69 3133 203a 2066 696c 6545  --+.|i13 : fileE
│ │ │ │ +00021a00: 7869 7374 7328 4469 7231 7c22 2f73 7461  xists(Dir1|"/sta
│ │ │ │ +00021a10: 7274 5f70 6172 616d 6574 6572 7322 2920  rt_parameters") 
│ │ │ │  00021a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021a40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  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 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                  
│ │ │ │  00021ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021ae0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00021ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021ae0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  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 2d2b  ---------------+
│ │ │ │ -00021b30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00021b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021b30: 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  --+.+-----------
│ │ │ │  00021b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -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  .+--------------
│ │ │ │ +00021da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021db0: 2020 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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  .|              
│ │ │ │ +00021df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00021e10: 7869 7374 7328 7374 6f72 6542 4d32 4669  xists(storeBM2Fi
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│ │ │ │  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
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│ │ │ │  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                  
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│ │ │ │ -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
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│ │ │ │ -000224a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000224b0: 5761 726e 696e 673a 2072 6174 696f 6e61  Warning: rationa
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│ │ │ │ -000224e0: 6c6f 6174 696e 6720 706f 696e 742e 7c0a  loating point.|.
│ │ │ │ -000224f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +00022470: 0a7c 6932 203a 204e 756d 6265 7254 6f42  .|i2 : NumberToB
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│ │ │ │ +000224a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224b0: 7c0a 7c57 6172 6e69 6e67 3a20 7261 7469  |.|Warning: rati
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│ │ │ │ +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
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│ │ │ │ -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
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│ │ │ │ +000225b0: 2d2d 2d2d 2b0a 7c69 3320 3a20 4e75 6d62  ----+.|i3 : Numb
│ │ │ │ +000225c0: 6572 546f 4227 5374 7269 6e67 2831 2f33  erToB'String(1/3
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│ │ │ │ +000225e0: 3829 2020 2020 2020 2020 2020 2020 2020  8)              
│ │ │ │ +000225f0: 2020 2020 207c 0a7c 5761 726e 696e 673a       |.|Warning:
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│ │ │ │ +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
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│ │ │ │  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
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│ │ │ │ -000227a0: 756d 6265 7254 6f42 2753 7472 696e 672c  umberToB'String,
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│ │ │ │ +00022740: 6d62 6572 546f 4227 5374 7269 6e67 2854  mberToB'String(T
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│ │ │ │ +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
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│ │ │ │ +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.
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│ │ │ │ +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  ================
│ │ │ │ -000228d0: 3d3d 0a0a 2020 2a20 696d 706f 7274 4d61  ==..  * importMa
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│ │ │ │ -00022930: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -00022940: 6520 6f62 6a65 6374 2050 6174 684c 6973  e object PathLis
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│ │ │ │ -00022980: 6361 756c 6179 3244 6f63 2953 796d 626f  caulay2Doc)Symbo
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│ │ │ │ -000229b0: 7261 6469 6361 6c4c 6973 742c 204e 6578  radicalList, Nex
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│ │ │ │ -00022a50: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
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│ │ │ │ +000228e0: 744d 6169 6e44 6174 6146 696c 6528 2e2e  tMainDataFile(..
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│ │ │ │ +00022900: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
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│ │ │ │ +00022930: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00022940: 0a54 6865 206f 626a 6563 7420 5061 7468  .The object Path
│ │ │ │ +00022950: 4c69 7374 2028 6d69 7373 696e 6720 646f  List (missing do
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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
│ │ │ │ +000229d0: 6c65 732c 2050 7265 763a 2050 6174 684c  les, Prev: PathL
│ │ │ │ +000229e0: 6973 742c 2055 703a 2054 6f70 0a0a 7261  ist, Up: Top..ra
│ │ │ │ +000229f0: 6469 6361 6c4c 6973 7420 2d2d 2041 2073  dicalList -- A s
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│ │ │ │ +00022a40: 2075 7020 746f 2061 2074 6f6c 6572 616e   up to a toleran
│ │ │ │ +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
│ │ │ │ -00022b10: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ -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
│ │ │ │ -00022be0: 7075 7473 2061 2073 7562 6c69 7374 206f  puts a sublist o
│ │ │ │ -00022bf0: 6620 636f 6d70 6c65 7820 6f72 2072 6561  f complex or rea
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│ │ │ │ -00022c10: 6c6c 2068 6176 6520 6469 7374 696e 6374  ll have distinct
│ │ │ │ -00022c20: 206e 6f72 6d73 0a75 7020 746f 2074 6865   norms.up to the
│ │ │ │ -00022c30: 2074 6f6c 6572 616e 6365 204e 2028 6465   tolerance N (de
│ │ │ │ -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
│ │ │ │ +00022af0: 7374 2c4e 756d 6265 7229 0a20 2020 2020  st,Number).     
│ │ │ │ +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
│ │ │ │ +00022b60: 206f 7220 7265 616c 0a20 2020 2020 2020   or real.       
│ │ │ │ +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
│ │ │ │ +00022bf0: 7420 6f66 2063 6f6d 706c 6578 206f 7220  t of complex or 
│ │ │ │ +00022c00: 7265 616c 206e 756d 6265 7273 2074 6861  real numbers tha
│ │ │ │ +00022c10: 7420 616c 6c20 6861 7665 2064 6973 7469  t all have disti
│ │ │ │ +00022c20: 6e63 7420 6e6f 726d 730a 7570 2074 6f20  nct norms.up to 
│ │ │ │ +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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│ │ │ │ -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
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│ │ │ │ -00023060: 653a 2042 6572 7469 6e69 2e69 6e66 6f2c  e: Bertini.info,
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│ │ │ │ -000230c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a46  *************..F
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│ │ │ │ -00023130: 6520 7374 7269 6e67 3a0a 284d 6163 6175  e string:.(Macau
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│ │ │ │ -000231a0: 5570 3a20 546f 700a 0a73 7562 506f 696e  Up: Top..subPoin
│ │ │ │ -000231b0: 7420 2d2d 2054 6869 7320 6675 6e63 7469  t -- This functi
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│ │ │ │ -000231f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00022f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761  -----------+..Wa
│ │ │ │ +00022f50: 7973 2074 6f20 7573 6520 7261 6469 6361  ys to use radica
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│ │ │ │ +00022f70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
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│ │ │ │ +00022f90: 7428 4c69 7374 2922 0a20 202a 2022 7261  t(List)".  * "ra
│ │ │ │ +00022fa0: 6469 6361 6c4c 6973 7428 4c69 7374 2c4e  dicalList(List,N
│ │ │ │ +00022fb0: 756d 6265 7229 220a 0a46 6f72 2074 6865  umber)"..For the
│ │ │ │ +00022fc0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +00022fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00022fe0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +00022ff0: 2072 6164 6963 616c 4c69 7374 3a20 7261   radicalList: ra
│ │ │ │ +00023000: 6469 6361 6c4c 6973 742c 2069 7320 6120  dicalList, is a 
│ │ │ │ +00023010: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +00023020: 6374 696f 6e20 7769 7468 0a6f 7074 696f  ction with.optio
│ │ │ │ +00023030: 6e73 3a20 284d 6163 6175 6c61 7932 446f  ns: (Macaulay2Do
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│ │ │ │ +00023050: 5769 7468 4f70 7469 6f6e 732c 2e0a 1f0a  WithOptions,....
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│ │ │ │ +00023090: 7562 506f 696e 742c 2050 7265 763a 2072  ubPoint, Prev: r
│ │ │ │ +000230a0: 6164 6963 616c 4c69 7374 2c20 5570 3a20  adicalList, Up: 
│ │ │ │ +000230b0: 546f 700a 0a73 746f 7265 424d 3246 696c  Top..storeBM2Fil
│ │ │ │ +000230c0: 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  es.*************
│ │ │ │ +000230d0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +000230e0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +000230f0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +00023100: 6563 7420 7374 6f72 6542 4d32 4669 6c65  ect storeBM2File
│ │ │ │ +00023110: 7320 286d 6973 7369 6e67 2064 6f63 756d  s (missing docum
│ │ │ │ +00023120: 656e 7461 7469 6f6e 2920 6973 2061 202a  entation) is a *
│ │ │ │ +00023130: 6e6f 7465 2073 7472 696e 673a 0a28 4d61  note string:.(Ma
│ │ │ │ +00023140: 6361 756c 6179 3244 6f63 2953 7472 696e  caulay2Doc)Strin
│ │ │ │ +00023150: 672c 2e0a 1f0a 4669 6c65 3a20 4265 7274  g,....File: Bert
│ │ │ │ +00023160: 696e 692e 696e 666f 2c20 4e6f 6465 3a20  ini.info, Node: 
│ │ │ │ +00023170: 7375 6250 6f69 6e74 2c20 4e65 7874 3a20  subPoint, Next: 
│ │ │ │ +00023180: 546f 7044 6972 6563 746f 7279 2c20 5072  TopDirectory, Pr
│ │ │ │ +00023190: 6576 3a20 7374 6f72 6542 4d32 4669 6c65  ev: storeBM2File
│ │ │ │ +000231a0: 732c 2055 703a 2054 6f70 0a0a 7375 6250  s, Up: Top..subP
│ │ │ │ +000231b0: 6f69 6e74 202d 2d20 5468 6973 2066 756e  oint -- This fun
│ │ │ │ +000231c0: 6374 696f 6e20 6576 616c 7561 7465 7320  ction evaluates 
│ │ │ │ +000231d0: 6120 706f 6c79 6e6f 6d69 616c 206f 7220  a polynomial or 
│ │ │ │ +000231e0: 6d61 7472 6978 2061 7420 6120 706f 696e  matrix at a poin
│ │ │ │ +000231f0: 742e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  t..*************
│ │ │ │  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
│ │ │ │ -00023450: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ -00023460: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175  n) => ..., defau
│ │ │ │ -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) => ...
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│ │ │ │ +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
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│ │ │ │ +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
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│ │ │ │ +00024790: 5265 6765 6e65 7261 7469 6f6e 202d 2d20  Regeneration -- 
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│ │ │ │ -00024820: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00024830: 2020 2020 6265 7274 696e 6950 6172 616d      bertiniParam
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│ │ │ │ -00024860: 7472 696e 6729 0a20 2020 2020 2020 2062  tring).        b
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│ │ │ │ -000248a0: 2020 2020 2062 6572 7469 6e69 5573 6572       bertiniUser
│ │ │ │ -000248b0: 486f 6d6f 746f 7079 282e 2e2e 2c54 6f70  Homotopy(...,Top
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│ │ │ │ +00024870: 2020 6265 7274 696e 695a 6572 6f44 696d    bertiniZeroDim
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│ │ │ │  00024990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  000249a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000249b0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
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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  ****************
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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
│ │ │ │ +00024ca0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +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
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│ │ │ │ +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
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│ │ │ │ +00025c10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00025c90: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
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│ │ │ │ +00025ce0: 653a 2042 6572 7469 6e69 2e69 6e66 6f2c  e: Bertini.info,
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│ │ │ │ +00025d60: 6961 626c 6573 2061 6e64 2075 7365 206d  iables and use m
│ │ │ │ +00025d70: 756c 7469 686f 6d6f 6765 6e65 6f75 7320  ultihomogeneous 
│ │ │ │ +00025d80: 686f 6d6f 746f 7069 6573 0a2a 2a2a 2a2a  homotopies.*****
│ │ │ │  00025d90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025da0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025db0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00025dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a47 726f  ***********..Gro
│ │ │ │ -00025de0: 7570 696e 6720 7468 6520 7661 7269 6162  uping the variab
│ │ │ │ -00025df0: 6c65 7320 6861 7320 4265 7274 696e 6920  les has Bertini 
│ │ │ │ -00025e00: 736f 6c76 6520 7a65 726f 2064 696d 656e  solve zero dimen
│ │ │ │ -00025e10: 7369 6f6e 616c 2073 7973 7465 6d73 2075  sional systems u
│ │ │ │ -00025e20: 7369 6e67 0a6d 756c 7469 686f 6d6f 6765  sing.multihomoge
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│ │ │ │ -00025e40: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
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│ │ │ │  00025e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00025e80: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ -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                  
│ │ │ │ -00025ec0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00025ec0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
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│ │ │ │  00025ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00025f10: 6932 203a 2046 3120 3d20 7b78 2a79 2b31  i2 : F1 = {x*y+1
│ │ │ │ -00025f20: 2c32 2a78 2a79 2b33 2a78 2b34 2a79 2b35  ,2*x*y+3*x+4*y+5
│ │ │ │ -00025f30: 7d3b 2020 2020 2020 2020 2020 2020 2020  };              
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│ │ │ │ +00025f20: 792b 312c 322a 782a 792b 332a 782b 342a  y+1,2*x*y+3*x+4*
│ │ │ │ +00025f30: 792b 357d 3b20 2020 2020 2020 2020 2020  y+5};           
│ │ │ │  00025f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00025fa0: 7469 6e69 5a65 726f 4469 6d53 6f6c 7665  tiniZeroDimSolve
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│ │ │ │ -00025fc0: 4772 6f75 703d 3e7b 7b78 7d2c 7b79 7d7d  Group=>{{x},{y}}
│ │ │ │ -00025fd0: 293b 2020 2020 2020 207c 0a2b 2d2d 2d2d  );       |.+----
│ │ │ │ +00025f90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +00025fa0: 6265 7274 696e 695a 6572 6f44 696d 536f  bertiniZeroDimSo
│ │ │ │ +00025fb0: 6c76 6528 4631 2c20 4166 6656 6172 6961  lve(F1, AffVaria
│ │ │ │ +00025fc0: 626c 6547 726f 7570 3d3e 7b7b 787d 2c7b  bleGroup=>{{x},{
│ │ │ │ +00025fd0: 797d 7d29 3b20 2020 2020 2020 7c0a 2b2d  y}});       |.+-
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│ │ │ │  00025ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -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  ----------------
│ │ │ │ +00026020: 2b0a 7c69 3420 3a20 6852 203d 4343 5b78  +.|i4 : hR =CC[x
│ │ │ │ +00026030: 302c 7831 2c79 302c 7931 5d20 2020 2020  0,x1,y0,y1]     
│ │ │ │  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   |.|            
│ │ │ │ +00026060: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00026070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000260a0: 2020 2020 207c 0a7c 6f34 203d 2068 5220       |.|o4 = hR 
│ │ │ │ -000260b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000260a0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ +000260b0: 6852 2020 2020 2020 2020 2020 2020 2020  hR              
│ │ │ │  000260c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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              |.| 
│ │ │ │  000260f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026120: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00026130: 6f34 203a 2050 6f6c 796e 6f6d 6961 6c52  o4 : PolynomialR
│ │ │ │ -00026140: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00026120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026130: 7c0a 7c6f 3420 3a20 506f 6c79 6e6f 6d69  |.|o4 : Polynomi
│ │ │ │ +00026140: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │  00026150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026170: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
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│ │ │ │  00026380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -000263c0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
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│ │ │ │ -000265d0: 2c20 0a20 2020 2020 202a 2053 746f 7261  , .      * Stora
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│ │ │ │ +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 2e34 3130 3631 3173 2065 6c61   -- .410611s 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 2e34 3639 3731 3873   |.| -- .469718s
│ │ │ │  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 2e37 3239 3839     |.| -- .72989
│ │ │ │ +0003ab80: 3973 2065 6c61 7073 6564 2020 2020 2020  9s 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 3239 2e32 3633 3173 2065  .| -- 29.2631s 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 2031 332e 3332 3838  1)|.| -- 13.3288
│ │ │ │  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 2e31 3931 3935   -- used 0.19195
│ │ │ │ +00004020: 3573 2028 6370 7529 3b20 302e 3139 3232  5s (cpu); 0.1922
│ │ │ │ +00004030: 3138 7320 2874 6872 6561 6429 3b20 3073  18s (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
│ │ │ │ @@ -2790,33 +2790,33 @@
│ │ │ │  0000ae50: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020                  
│ │ │ │  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
│ │ │ │ @@ -4516,18 +4516,18 @@
│ │ │ │  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); 
│ │ │ │ -00011ad0: 3073 2028 6763 297c 0a2b 2d2d 2d2d 2d2d  0s (gc)|.+------
│ │ │ │ +00011aa0: 0a7c 202d 2d20 7573 6564 2030 2e33 3238  .| -- used 0.328
│ │ │ │ +00011ab0: 3331 7320 2863 7075 293b 2030 2e32 3538  31s (cpu); 0.258
│ │ │ │ +00011ac0: 3039 7320 2874 6872 6561 6429 3b20 3073  09s (thread); 0s
│ │ │ │ +00011ad0: 2028 6763 2920 207c 0a2b 2d2d 2d2d 2d2d   (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                  
│ │ │ │  00011b40: 2020 2020 2020 207c 0a7c 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 3131 3137 3532 7320 2863 7075  d 0.111752s (cpu
│ │ │ │ +000183b0: 293b 2030 2e31 3039 3932 3573 2028 7468  ); 0.109925s (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 3232 3737 3934 7320 2863 7075  d 0.227794s (cpu
│ │ │ │ +000184a0: 293b 2030 2e31 3537 3830 3873 2028 7468  ); 0.157808s (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
│ │ │ │ @@ -1066,17 +1066,17 @@
│ │ │ │  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
│ │ │ │ -00004320: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00004300: 6420 302e 3239 3837 3337 7320 2863 7075  d 0.298737s (cpu
│ │ │ │ +00004310: 293b 2030 2e31 3332 3436 7320 2874 6872  ); 0.13246s (thr
│ │ │ │ +00004320: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 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                  
│ │ │ │  00004390: 2020 2020 2020 7c0a 7c20 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 3631 3134 3737 7320 2863 7075  d 0.611477s (cpu
│ │ │ │ +00004860: 293b 2030 2e32 3934 3232 3773 2028 7468  ); 0.294227s (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 2031 2e34 3238 3636 7320  - used 1.42866s 
│ │ │ │ +00010180: 2863 7075 293b 2030 2e34 3339 3139 3573  (cpu); 0.439195s
│ │ │ │  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: 2032 2e35 3039 3136 7320 2863 7075 293b   2.50916s (cpu);
│ │ │ │ +00010530: 2032 2e31 3136 3438 7320 2874 6872 6561   2.11648s (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 3433 3432 3133 7320 2863 7075 293b  0.434213s (cpu);
│ │ │ │ +00010ab0: 2030 2e32 3236 3132 3573 2028 7468 7265   0.226125s (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 3237 3832  | -- used 0.2782
│ │ │ │ +00010e50: 3937 7320 2863 7075 293b 2030 2e31 3331  97s (cpu); 0.131
│ │ │ │ +00010e60: 3334 7320 2874 6872 6561 6429 3b20 3073  34s (thread); 0s
│ │ │ │ +00010e70: 2028 6763 2920 2020 2020 2020 2020 2020   (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 2031 2e37 3635 3936 7320  - used 1.76596s 
│ │ │ │ +00014b00: 2863 7075 293b 2030 2e34 3933 3038 3473  (cpu); 0.493084s
│ │ │ │  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    
│ │ │ │ @@ -5467,17 +5467,17 @@
│ │ │ │  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
│ │ │ │ -00015630: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +00015610: 3237 3539 3937 7320 2863 7075 293b 2030  275997s (cpu); 0
│ │ │ │ +00015620: 2e30 3935 3532 3232 7320 2874 6872 6561  .0955222s (threa
│ │ │ │ +00015630: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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    
│ │ │ │  000156a0: 3220 2020 2020 2020 2020 2020 2032 2020  2            2  
│ │ │ │ @@ -6424,16 +6424,16 @@
│ │ │ │  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
│ │ │ │ +000191e0: 2075 7365 6420 302e 3139 3236 3439 7320   used 0.192649s 
│ │ │ │ +000191f0: 2863 7075 293b 2030 2e30 3632 3334 3737  (cpu); 0.0623477
│ │ │ │  00019200: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  00019210: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  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                  
│ │ │ │ @@ -6449,16 +6449,16 @@
│ │ │ │  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
│ │ │ │ +00019370: 2075 7365 6420 302e 3336 3634 3534 7320   used 0.366454s 
│ │ │ │ +00019380: 2863 7075 293b 2030 2e31 3732 3131 3773  (cpu); 0.172117s
│ │ │ │  00019390: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  000193a0: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │  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                  
│ │ │ │ @@ -6653,16 +6653,16 @@
│ │ │ │  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
│ │ │ │ +0001a030: 3336 3237 3138 7320 2863 7075 293b 2030  362718s (cpu); 0
│ │ │ │ +0001a040: 2e31 3136 3936 3673 2028 7468 7265 6164  .116966s (thread
│ │ │ │  0001a050: 293b 2030 7320 2867 6329 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            
│ │ │ │ @@ -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 3235 3637 3035 7320 2863 7075  d 0.256705s (cpu
│ │ │ │ +0001a190: 293b 2030 2e31 3035 3736 3873 2028 7468  ); 0.105768s (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                  
│ │ │ │ @@ -7280,16 +7280,16 @@
│ │ │ │  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
│ │ │ │ +0001c760: 6564 2036 2e34 3534 3234 7320 2863 7075  ed 6.45424s (cpu
│ │ │ │ +0001c770: 293b 2031 2e35 3131 3737 7320 2874 6872  ); 1.51177s (thr
│ │ │ │  0001c780: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 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
│ │ │ │ @@ -7342,17 +7342,17 @@
│ │ │ │  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
│ │ │ │ -0001cb60: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0001cb40: 7573 6564 2036 2e35 3330 3573 2028 6370  used 6.5305s (cp
│ │ │ │ +0001cb50: 7529 3b20 312e 3533 3239 3673 2028 7468  u); 1.53296s (th
│ │ │ │ +0001cb60: 7265 6164 293b 2030 7320 2867 6329 2020  read); 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         |.|      
│ │ │ │  0001cbd0: 2032 2032 2020 2020 2032 2020 2020 2020   2 2     2      
│ │ │ │ @@ -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 2031 2e35 3239 3133 7320  - used 1.52913s 
│ │ │ │ +0001d3d0: 2863 7075 293b 2030 2e35 3738 3532 3573  (cpu); 0.578525s
│ │ │ │  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 2e31 3537 3836 3173 2028  used 0.157861s (
│ │ │ │ +0001d6f0: 6370 7529 3b20 302e 3035 3737 3437 3173  cpu); 0.0577471s
│ │ │ │  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                  
│ │ │ │ @@ -7588,16 +7588,16 @@
│ │ │ │  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     
│ │ │ │  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
│ │ │ │ +0001daa0: 3634 3336 3433 7320 2863 7075 293b 2030  643643s (cpu); 0
│ │ │ │ +0001dab0: 2e30 3338 3733 3137 7320 2874 6872 6561  .0387317s (threa
│ │ │ │  0001dac0: 6429 3b20 3073 2028 6763 297c 0a7c 2020  d); 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
│ │ │ │  0001db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7998,18 +7998,18 @@
│ │ │ │  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)    |.|       
│ │ │ │ +0001f440: 2d20 7573 6564 2034 2e32 3238 3732 7320  - used 4.22872s 
│ │ │ │ +0001f450: 2863 7075 293b 2031 2e33 3436 3531 7320  (cpu); 1.34651s 
│ │ │ │ +0001f460: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +0001f470: 2920 2020 2020 7c0a 7c20 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
│ │ │ │ @@ -8050,16 +8050,16 @@
│ │ │ │  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
│ │ │ │ +0001f780: 2030 2e39 3137 3335 3273 2028 6370 7529   0.917352s (cpu)
│ │ │ │ +0001f790: 3b20 302e 3235 3538 3734 7320 2874 6872  ; 0.255874s (thr
│ │ │ │  0001f7a0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  0001f7b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001f7f0: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │  0001f800: 2020 2033 2020 2020 2020 2020 2020 2020     3
│ │ ├── ./usr/share/info/CodingTheory.info.gz
│ │ │ ├── CodingTheory.info
│ │ │ │ @@ -3965,71 +3965,71 @@
│ │ │ │  0000f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f7d0: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │  0000f7e0: 6163 6865 203d 3e20 4361 6368 6554 6162  ache => CacheTab
│ │ │ │  0000f7f0: 6c65 7b7d 2020 2020 2020 2020 2020 207c  le{}           |
│ │ │ │  0000f800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f820: 2020 2020 2020 2020 2020 2020 2020 2043                 C
│ │ │ │ -0000f830: 6f64 6520 3d3e 2069 6d61 6765 207c 2031  ode => image | 1
│ │ │ │ -0000f840: 2020 2061 2b31 207c 2020 2020 2020 207c     a+1 |       |
│ │ │ │ +0000f830: 6f64 6520 3d3e 2069 6d61 6765 207c 2061  ode => image | a
│ │ │ │ +0000f840: 2b31 2030 2020 207c 2020 2020 2020 207c  +1 0   |       |
│ │ │ │  0000f850: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f880: 2020 2020 2020 2020 2020 2020 207c 2031               | 1
│ │ │ │ -0000f890: 2020 2030 2020 207c 2020 2020 2020 207c     0   |       |
│ │ │ │ +0000f880: 2020 2020 2020 2020 2020 2020 207c 2061               | a
│ │ │ │ +0000f890: 2b31 2030 2020 207c 2020 2020 2020 207c  +1 0   |       |
│ │ │ │  0000f8a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f8d0: 2020 2020 2020 2020 2020 2020 207c 2030               | 0
│ │ │ │ -0000f8e0: 2020 2030 2020 207c 2020 2020 2020 207c     0   |       |
│ │ │ │ +0000f8d0: 2020 2020 2020 2020 2020 2020 207c 2061               | a
│ │ │ │ +0000f8e0: 2020 2061 2020 207c 2020 2020 2020 207c     a   |       |
│ │ │ │  0000f8f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f920: 2020 2020 2020 2020 2020 2020 207c 2030               | 0
│ │ │ │ -0000f930: 2020 2030 2020 207c 2020 2020 2020 207c     0   |       |
│ │ │ │ +0000f920: 2020 2020 2020 2020 2020 2020 207c 2061               | a
│ │ │ │ +0000f930: 2020 2061 2020 207c 2020 2020 2020 207c     a   |       |
│ │ │ │  0000f940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f970: 2020 2020 2020 2020 2020 2020 207c 2031               | 1
│ │ │ │ -0000f980: 2020 2031 2020 207c 2020 2020 2020 207c     1   |       |
│ │ │ │ +0000f980: 2020 2061 2b31 207c 2020 2020 2020 207c     a+1 |       |
│ │ │ │  0000f990: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f9c0: 2020 2020 2020 2020 2020 2020 207c 2061               | a
│ │ │ │ -0000f9d0: 2b31 2030 2020 207c 2020 2020 2020 207c  +1 0   |       |
│ │ │ │ +0000f9c0: 2020 2020 2020 2020 2020 2020 207c 2031               | 1
│ │ │ │ +0000f9d0: 2020 2030 2020 207c 2020 2020 2020 207c     0   |       |
│ │ │ │  0000f9e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fa10: 2020 2020 2020 2020 2020 2020 207c 2061               | a
│ │ │ │ -0000fa20: 2b31 2030 2020 207c 2020 2020 2020 207c  +1 0   |       |
│ │ │ │ +0000fa10: 2020 2020 2020 2020 2020 2020 207c 2030               | 0
│ │ │ │ +0000fa20: 2020 2030 2020 207c 2020 2020 2020 207c     0   |       |
│ │ │ │  0000fa30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fa60: 2020 2020 2020 2020 2020 2020 207c 2061               | a
│ │ │ │ -0000fa70: 2020 2061 2020 207c 2020 2020 2020 207c     a   |       |
│ │ │ │ +0000fa60: 2020 2020 2020 2020 2020 2020 207c 2030               | 0
│ │ │ │ +0000fa70: 2020 2030 2020 207c 2020 2020 2020 207c     0   |       |
│ │ │ │  0000fa80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000fab0: 2020 2020 2020 2020 2020 2020 207c 2061               | a
│ │ │ │ -0000fac0: 2020 2061 2020 207c 2020 2020 2020 207c     a   |       |
│ │ │ │ +0000fab0: 2020 2020 2020 2020 2020 2020 207c 2031               | 1
│ │ │ │ +0000fac0: 2020 2031 2020 207c 2020 2020 2020 207c     1   |       |
│ │ │ │  0000fad0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000faf0: 2020 2020 2020 2020 2020 2020 2020 2047                 G
│ │ │ │  0000fb00: 656e 6572 6174 6f72 4d61 7472 6978 203d  eneratorMatrix =
│ │ │ │ -0000fb10: 3e20 7c20 3120 2020 3120 3020 3020 207c  > | 1   1 0 0  |
│ │ │ │ +0000fb10: 3e20 7c20 612b 3120 612b 3120 6120 207c  > | a+1 a+1 a  |
│ │ │ │  0000fb20: 0a7c 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 7c20 612b 3120 3020 3020 3020 207c    | a+1 0 0 0  |
│ │ │ │ +0000fb60: 2020 7c20 3020 2020 3020 2020 6120 207c    | 0   0   a  |
│ │ │ │  0000fb70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fb90: 2020 2020 2020 2020 2020 2020 2020 2047                 G
│ │ │ │ -0000fba0: 656e 6572 6174 6f72 7320 3d3e 207b 7b31  enerators => {{1
│ │ │ │ -0000fbb0: 2c20 312c 2030 2c20 302c 2031 2c20 207c  , 1, 0, 0, 1,  |
│ │ │ │ +0000fba0: 656e 6572 6174 6f72 7320 3d3e 207b 7b61  enerators => {{a
│ │ │ │ +0000fbb0: 202b 2031 2c20 6120 2b20 312c 2061 207c   + 1, a + 1, a |
│ │ │ │  0000fbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fbe0: 2020 2020 2020 2020 2020 2020 2020 2050                 P
│ │ │ │  0000fbf0: 6172 6974 7943 6865 636b 4d61 7472 6978  arityCheckMatrix
│ │ │ │  0000fc00: 203d 3e20 7c20 3120 3020 3020 3020 207c   => | 1 0 0 0  |
│ │ │ │  0000fc10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4064,17 +4064,17 @@
│ │ │ │  0000fdf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fe10: 2020 2020 2020 2020 2020 2020 2020 2050                 P
│ │ │ │  0000fe20: 6172 6974 7943 6865 636b 526f 7773 203d  arityCheckRows =
│ │ │ │  0000fe30: 3e20 7b7b 312c 2030 2c20 302c 2030 207c  > {{1, 0, 0, 0 |
│ │ │ │  0000fe40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fe50: 2020 2020 2020 506f 696e 7473 203d 3e20        Points => 
│ │ │ │ -0000fe60: 7b7b 612c 2061 7d2c 207b 302c 2030 7d2c  {{a, a}, {0, 0},
│ │ │ │ -0000fe70: 207b 312c 2030 7d2c 207b 302c 2031 7d2c   {1, 0}, {0, 1},
│ │ │ │ -0000fe80: 207b 312c 2031 7d2c 207b 612c 2030 207c   {1, 1}, {a, 0 |
│ │ │ │ +0000fe60: 7b7b 302c 2061 7d2c 207b 612c 2030 7d2c  {{0, a}, {a, 0},
│ │ │ │ +0000fe70: 207b 612c 2031 7d2c 207b 312c 2061 7d2c   {a, 1}, {1, a},
│ │ │ │ +0000fe80: 207b 612c 2061 7d2c 207b 302c 2030 207c   {a, a}, {0, 0 |
│ │ │ │  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 + 
│ │ │ │  0000fec0: 312c 2078 2a79 7d20 2020 2020 2020 2020  1, x*y}         
│ │ │ │  0000fed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000fee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fef0: 2020 2020 2020 5365 7473 203d 3e20 7b7b        Sets => {{
│ │ │ │ @@ -4172,71 +4172,71 @@
│ │ │ │  000104b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000104c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000104d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000104e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000104f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00010520: 0a7c 3120 612b 3120 612b 3120 6120 6120  .|1 a+1 a+1 a a 
│ │ │ │ +00010520: 0a7c 6120 3120 2020 3120 3020 3020 3120  .|a 1   1 0 0 1 
│ │ │ │  00010530: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00010540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00010570: 0a7c 3120 3020 2020 3020 2020 6120 6120  .|1 0   0   a a 
│ │ │ │ +00010570: 0a7c 6120 612b 3120 3020 3020 3020 3120  .|a a+1 0 0 0 1 
│ │ │ │  00010580: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00010590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000105a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000105b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000105c0: 0a7c 6120 2b20 312c 2061 202b 2031 2c20  .|a + 1, a + 1, 
│ │ │ │ -000105d0: 612c 2061 7d2c 207b 6120 2b20 312c 2030  a, a}, {a + 1, 0
│ │ │ │ -000105e0: 2c20 302c 2030 2c20 312c 2030 2c20 302c  , 0, 0, 1, 0, 0,
│ │ │ │ -000105f0: 2061 2c20 617d 7d20 2020 2020 2020 2020   a, a}}         
│ │ │ │ +000105c0: 0a7c 2c20 612c 2031 2c20 312c 2030 2c20  .|, a, 1, 1, 0, 
│ │ │ │ +000105d0: 302c 2031 7d2c 207b 302c 2030 2c20 612c  0, 1}, {0, 0, a,
│ │ │ │ +000105e0: 2061 2c20 6120 2b20 312c 2030 2c20 302c   a, a + 1, 0, 0,
│ │ │ │ +000105f0: 2030 2c20 317d 7d20 2020 2020 2020 2020   0, 1}}         
│ │ │ │  00010600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00010610: 0a7c 3020 612b 3120 3020 6120 2020 3020  .|0 a+1 0 a   0 
│ │ │ │ +00010610: 0a7c 3020 612b 3120 3020 3020 3020 2020  .|0 a+1 0 0 0   
│ │ │ │  00010620: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00010630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010650: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00010660: 0a7c 3020 6120 2020 3020 3020 2020 3020  .|0 a   0 0   0 
│ │ │ │ +00010660: 0a7c 3020 612b 3120 3020 3020 3020 2020  .|0 a+1 0 0 0   
│ │ │ │  00010670: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00010680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000106a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000106b0: 0a7c 3020 3020 2020 3020 3020 2020 3020  .|0 0   0 0   0 
│ │ │ │ +000106b0: 0a7c 3020 3020 2020 3020 3020 6120 2020  .|0 0   0 0 a   
│ │ │ │  000106c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000106d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000106e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000106f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00010700: 0a7c 3020 3020 2020 3020 3020 2020 3020  .|0 0   0 0   0 
│ │ │ │ +00010700: 0a7c 3020 3020 2020 3020 3020 6120 2020  .|0 0   0 0 a   
│ │ │ │  00010710: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00010720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00010750: 0a7c 3120 3020 2020 3020 612b 3120 3020  .|1 0   0 a+1 0 
│ │ │ │ +00010750: 0a7c 3120 6120 2020 3020 3020 612b 3120  .|1 a   0 0 a+1 
│ │ │ │  00010760: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00010770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000107a0: 0a7c 3020 3120 2020 3120 3020 2020 3020  .|0 1   1 0   0 
│ │ │ │ +000107a0: 0a7c 3020 3020 2020 3120 3020 3020 2020  .|0 0   1 0 0   
│ │ │ │  000107b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000107c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000107d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000107e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000107f0: 0a7c 3020 3020 2020 3020 3120 2020 3120  .|0 0   0 1   1 
│ │ │ │ +000107f0: 0a7c 3020 3020 2020 3020 3120 3020 2020  .|0 0   0 1 0   
│ │ │ │  00010800: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00010810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00010840: 0a7c 2c20 302c 2061 202b 2031 2c20 302c  .|, 0, a + 1, 0,
│ │ │ │ -00010850: 2061 2c20 307d 2c20 7b30 2c20 312c 2030   a, 0}, {0, 1, 0
│ │ │ │ -00010860: 2c20 302c 2030 2c20 612c 2030 2c20 302c  , 0, 0, a, 0, 0,
│ │ │ │ -00010870: 2030 7d2c 207b 302c 2030 2c20 312c 2030   0}, {0, 0, 1, 0
│ │ │ │ -00010880: 2c20 302c 2030 2c20 302c 2030 2c20 207c  , 0, 0, 0, 0,  |
│ │ │ │ -00010890: 0a7c 7d2c 207b 302c 2061 7d2c 207b 612c  .|}, {0, a}, {a,
│ │ │ │ -000108a0: 2031 7d2c 207b 312c 2061 7d7d 2020 2020   1}, {1, a}}    
│ │ │ │ +00010850: 2030 2c20 307d 2c20 7b30 2c20 312c 2030   0, 0}, {0, 1, 0
│ │ │ │ +00010860: 2c20 302c 2030 2c20 6120 2b20 312c 2030  , 0, 0, a + 1, 0
│ │ │ │ +00010870: 2c20 302c 2030 7d2c 207b 302c 2030 2c20  , 0, 0}, {0, 0, 
│ │ │ │ +00010880: 312c 2030 2c20 302c 2030 2c20 302c 207c  1, 0, 0, 0, 0, |
│ │ │ │ +00010890: 0a7c 7d2c 207b 312c 2030 7d2c 207b 302c  .|}, {1, 0}, {0,
│ │ │ │ +000108a0: 2031 7d2c 207b 312c 2031 7d7d 2020 2020   1}, {1, 1}}    
│ │ │ │  000108b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000108c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000108d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000108e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000108f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4377,38 +4377,38 @@
│ │ │ │  00011180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000111a0: 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000111e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000111f0: 0a7c 307d 2c20 7b30 2c20 302c 2030 2c20  .|0}, {0, 0, 0, 
│ │ │ │ -00011200: 312c 2030 2c20 302c 2030 2c20 302c 2030  1, 0, 0, 0, 0, 0
│ │ │ │ -00011210: 7d2c 207b 302c 2030 2c20 302c 2030 2c20  }, {0, 0, 0, 0, 
│ │ │ │ -00011220: 312c 2030 2c20 302c 2061 202b 2031 2c20  1, 0, 0, a + 1, 
│ │ │ │ -00011230: 307d 2c20 7b30 2c20 302c 2030 2c20 207c  0}, {0, 0, 0,  |
│ │ │ │ +000111f0: 0a7c 302c 2061 7d2c 207b 302c 2030 2c20  .|0, a}, {0, 0, 
│ │ │ │ +00011200: 302c 2031 2c20 302c 2030 2c20 302c 2030  0, 1, 0, 0, 0, 0
│ │ │ │ +00011210: 2c20 617d 2c20 7b30 2c20 302c 2030 2c20  , a}, {0, 0, 0, 
│ │ │ │ +00011220: 302c 2031 2c20 612c 2030 2c20 302c 2061  0, 1, a, 0, 0, a
│ │ │ │ +00011230: 202b 2031 7d2c 207b 302c 2030 2c20 207c   + 1}, {0, 0,  |
│ │ │ │  00011240: 0a7c 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │  00011290: 0a7c 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 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │ +000112c0: 2020 2020 207d 2020 2020 2020 2020 2020       }          
│ │ │ │  000112d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000112e0: 0a7c 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 207c                 |
│ │ │ │  00011330: 0a7c 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: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +00011360: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │  00011370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00011380: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000113d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ @@ -4507,18 +4507,18 @@
│ │ │ │  000119a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 302c 2030 2c20 302c 2030 2c20 312c  .|0, 0, 0, 0, 1,
│ │ │ │ +00011a20: 2030 2c20 307d 2c20 7b30 2c20 302c 2030   0, 0}, {0, 0, 0
│ │ │ │ +00011a30: 2c20 302c 2030 2c20 302c 2030 2c20 312c  , 0, 0, 0, 0, 1,
│ │ │ │ +00011a40: 2030 7d7d 2020 2020 2020 2020 2020 2020   0}}            
│ │ │ │  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 612b 3120 7c20 2020 2020  | 1   a+1 |     
│ │ │ │  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 3120 2020 3020 2020 7c20 2020 2020  | 1   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 3020 2020 3020 2020 7c20 2020 2020  | 0   0   |     
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020   a              
│ │ │ │  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
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│ │ │ │ +00011fd0: 2020 2020 207c 2030 2020 2030 2020 2061       | 0   0   a
│ │ │ │ +00011fe0: 2061 2020 2020 2020 2020 2020 2020 2020   a              
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020  a,              
│ │ │ │ +00012040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020   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 2020 2020 2020 2020 2020 2020 2020   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 2020 2020 2020 2020 2020 2020 2020  0,              
│ │ │ │ +000122c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  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               
│ │ │ │ @@ -4722,69 +4722,69 @@
│ │ │ │  00012710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00012730: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00012780: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00012780: 0a7c 3120 2020 3120 3020 3020 3120 7c20  .|1   1 0 0 1 | 
│ │ │ │  00012790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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  .|              
│ │ │ │ +000127d0: 0a7c 612b 3120 3020 3020 3020 3120 7c20  .|a+1 0 0 0 1 | 
│ │ │ │  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}}      
│ │ │ │ -00012840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012820: 0a7c 612c 2031 2c20 312c 2030 2c20 302c  .|a, 1, 1, 0, 0,
│ │ │ │ +00012830: 2031 7d2c 207b 302c 2030 2c20 612c 2061   1}, {0, 0, a, a
│ │ │ │ +00012840: 2c20 6120 2b20 312c 2030 2c20 302c 2030  , a + 1, 0, 0, 0
│ │ │ │ +00012850: 2c20 317d 7d20 2020 2020 2020 2020 2020  , 1}}           
│ │ │ │  00012860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00012870: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00012870: 0a7c 612b 3120 3020 3020 3020 2020 7c20  .|a+1 0 0 0   | 
│ │ │ │  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                  
│ │ │ │  000128b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000128c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000128c0: 0a7c 612b 3120 3020 3020 3020 2020 7c20  .|a+1 0 0 0   | 
│ │ │ │  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 2020 2020 2020 2020 207c                 |
│ │ │ │ -00012910: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00012910: 0a7c 3020 2020 3020 3020 6120 2020 7c20  .|0   0 0 a   | 
│ │ │ │  00012920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00012960: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00012960: 0a7c 3020 2020 3020 3020 6120 2020 7c20  .|0   0 0 a   | 
│ │ │ │  00012970: 2020 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 2020 2020 207c                 |
│ │ │ │ -000129b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000129b0: 0a7c 6120 2020 3020 3020 612b 3120 7c20  .|a   0 0 a+1 | 
│ │ │ │  000129c0: 2020 2020 2020 2020 2020 2020 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 207c                 |
│ │ │ │ -00012a00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00012a00: 0a7c 3020 2020 3120 3020 3020 2020 7c20  .|0   1 0 0   | 
│ │ │ │  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 207c                 |
│ │ │ │ -00012a50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00012a50: 0a7c 3020 2020 3020 3120 3020 2020 7c20  .|0   0 1 0   | 
│ │ │ │  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 2061 202b 2031 2c20 302c 2030  .|0, a + 1, 0, 0
│ │ │ │ +00012ab0: 2c20 307d 2c20 7b30 2c20 312c 2030 2c20  , 0}, {0, 1, 0, 
│ │ │ │ +00012ac0: 302c 2030 2c20 6120 2b20 312c 2030 2c20  0, 0, a + 1, 0, 
│ │ │ │ +00012ad0: 302c 2030 7d2c 207b 302c 2030 2c20 312c  0, 0}, {0, 0, 1,
│ │ │ │ +00012ae0: 2030 2c20 302c 2030 2c20 302c 2030 2c7c   0, 0, 0, 0, 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                  
│ │ │ │ @@ -4892,28 +4892,28 @@
│ │ │ │  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 + 
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│ │ │ │ -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 617d 2c20 7b30 2c20 302c 2030 2c20  .|a}, {0, 0, 0, 
│ │ │ │ +00013230: 312c 2030 2c20 302c 2030 2c20 302c 2061  1, 0, 0, 0, 0, a
│ │ │ │ +00013240: 7d2c 207b 302c 2030 2c20 302c 2030 2c20  }, {0, 0, 0, 0, 
│ │ │ │ +00013250: 312c 2061 2c20 302c 2030 2c20 6120 2b20  1, a, 0, 0, a + 
│ │ │ │ +00013260: 317d 2c20 7b30 2c20 302c 2030 2c20 207c  1}, {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      }           
│ │ │ │ +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: 207d 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                  
│ │ │ │  00013350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00013360: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ @@ -5012,18 +5012,18 @@
│ │ │ │  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}}            
│ │ │ │ -000139c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000139d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000139a0: 0a7c 302c 2030 2c20 302c 2031 2c20 302c  .|0, 0, 0, 1, 0,
│ │ │ │ +000139b0: 2030 7d2c 207b 302c 2030 2c20 302c 2030   0}, {0, 0, 0, 0
│ │ │ │ +000139c0: 2c20 302c 2030 2c20 302c 2031 2c20 307d  , 0, 0, 0, 1, 0}
│ │ │ │ +000139d0: 7d20 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  ----------------
│ │ │ │  00013a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00013a40: 0a0a 6120 7269 6e67 2c20 6120 6c69 7374  ..a ring, a list
│ │ │ │ @@ -5131,66 +5131,66 @@
│ │ │ │  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   |       |.| 
│ │ │ │  000140e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014110: 2020 2020 2020 2020 2020 7c20 3020 2020            | 0   
│ │ │ │ -00014120: 3020 2020 7c20 2020 2020 2020 7c0a 7c20  0   |       |.| 
│ │ │ │ +00014110: 2020 2020 2020 2020 2020 7c20 3120 2020            | 1   
│ │ │ │ +00014120: 612b 3120 7c20 2020 2020 2020 7c0a 7c20  a+1 |       |.| 
│ │ │ │  00014130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014160: 2020 2020 2020 2020 2020 7c20 612b 3120            | a+1 
│ │ │ │ -00014170: 3120 2020 7c20 2020 2020 2020 7c0a 7c20  1   |       |.| 
│ │ │ │ +00014160: 2020 2020 2020 2020 2020 7c20 3020 2020            | 0   
│ │ │ │ +00014170: 3020 2020 7c20 2020 2020 2020 7c0a 7c20  0   |       |.| 
│ │ │ │  00014180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000141a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000141b0: 2020 2020 2020 2020 2020 7c20 6120 2020            | a   
│ │ │ │ -000141c0: 612b 3120 7c20 2020 2020 2020 7c0a 7c20  a+1 |       |.| 
│ │ │ │ +000141b0: 2020 2020 2020 2020 2020 7c20 3020 2020            | 0   
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│ │ │ │  000141d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000141e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000141f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014200: 2020 2020 2020 2020 2020 7c20 3120 2020            | 1   
│ │ │ │ -00014210: 612b 3120 7c20 2020 2020 2020 7c0a 7c20  a+1 |       |.| 
│ │ │ │ +00014200: 2020 2020 2020 2020 2020 7c20 3020 2020            | 0   
│ │ │ │ +00014210: 3020 2020 7c20 2020 2020 2020 7c0a 7c20  0   |       |.| 
│ │ │ │  00014220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014250: 2020 2020 2020 2020 2020 7c20 3020 2020            | 0   
│ │ │ │ -00014260: 3020 2020 7c20 2020 2020 2020 7c0a 7c20  0   |       |.| 
│ │ │ │ +00014250: 2020 2020 2020 2020 2020 7c20 6120 2020            | a   
│ │ │ │ +00014260: 612b 3120 7c20 2020 2020 2020 7c0a 7c20  a+1 |       |.| 
│ │ │ │  00014270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000142a0: 2020 2020 2020 2020 2020 7c20 3020 2020            | 0   
│ │ │ │  000142b0: 3020 2020 7c20 2020 2020 2020 7c0a 7c20  0   |       |.| 
│ │ │ │  000142c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000142d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000142e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000142f0: 2020 2020 2020 2020 2020 7c20 3020 2020            | 0   
│ │ │ │ -00014300: 3020 2020 7c20 2020 2020 2020 7c0a 7c20  0   |       |.| 
│ │ │ │ +000142f0: 2020 2020 2020 2020 2020 7c20 612b 3120            | a+1 
│ │ │ │ +00014300: 3120 2020 7c20 2020 2020 2020 7c0a 7c20  1   |       |.| 
│ │ │ │  00014310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014340: 2020 2020 2020 2020 2020 7c20 3120 2020            | 1   
│ │ │ │  00014350: 3120 2020 7c20 2020 2020 2020 7c0a 7c20  1   |       |.| 
│ │ │ │  00014360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014380: 2020 2020 2020 2020 2020 2020 4765 6e65              Gene
│ │ │ │  00014390: 7261 746f 724d 6174 7269 7820 3d3e 207c  ratorMatrix => |
│ │ │ │ -000143a0: 2030 2030 2061 2b31 2061 2020 7c0a 7c20   0 0 a+1 a  |.| 
│ │ │ │ +000143a0: 2030 2031 2020 2030 2030 2020 7c0a 7c20   0 1   0 0  |.| 
│ │ │ │  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 207c                 |
│ │ │ │ -000143f0: 2030 2030 2031 2020 2061 2b20 7c0a 7c20   0 0 1   a+ |.| 
│ │ │ │ +000143f0: 2030 2061 2b31 2030 2030 2020 7c0a 7c20   0 a+1 0 0  |.| 
│ │ │ │  00014400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014420: 2020 2020 2020 2020 2020 2020 4765 6e65              Gene
│ │ │ │ -00014430: 7261 746f 7273 203d 3e20 7b7b 302c 2030  rators => {{0, 0
│ │ │ │ -00014440: 2c20 6120 2b20 312c 2061 2c20 7c0a 7c20  , a + 1, a, |.| 
│ │ │ │ +00014430: 7261 746f 7273 203d 3e20 7b7b 302c 2031  rators => {{0, 1
│ │ │ │ +00014440: 2c20 302c 2030 2c20 302c 2020 7c0a 7c20  , 0, 0, 0,  |.| 
│ │ │ │  00014450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014470: 2020 2020 2020 2020 2020 2020 5061 7269              Pari
│ │ │ │  00014480: 7479 4368 6563 6b4d 6174 7269 7820 3d3e  tyCheckMatrix =>
│ │ │ │  00014490: 207c 2031 2030 2030 2030 2020 7c0a 7c20   | 1 0 0 0  |.| 
│ │ │ │  000144a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5225,17 +5225,17 @@
│ │ │ │  00014680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000146a0: 2020 2020 2020 2020 2020 2020 5061 7269              Pari
│ │ │ │  000146b0: 7479 4368 6563 6b52 6f77 7320 3d3e 207b  tyCheckRows => {
│ │ │ │  000146c0: 7b31 2c20 302c 2030 2c20 3020 7c0a 7c20  {1, 0, 0, 0 |.| 
│ │ │ │  000146d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000146e0: 2020 2050 6f69 6e74 7320 3d3e 207b 7b30     Points => {{0
│ │ │ │ -000146f0: 2c20 617d 2c20 7b61 2c20 307d 2c20 7b31  , a}, {a, 0}, {1
│ │ │ │ -00014700: 2c20 617d 2c20 7b61 2c20 317d 2c20 7b61  , a}, {a, 1}, {a
│ │ │ │ -00014710: 2c20 617d 2c20 7b30 2c20 3020 7c0a 7c20  , a}, {0, 0 |.| 
│ │ │ │ +000146f0: 2c20 617d 2c20 7b61 2c20 617d 2c20 7b61  , a}, {a, a}, {a
│ │ │ │ +00014700: 2c20 307d 2c20 7b30 2c20 307d 2c20 7b30  , 0}, {0, 0}, {0
│ │ │ │ +00014710: 2c20 317d 2c20 7b61 2c20 3120 7c0a 7c20  , 1}, {a, 1 |.| 
│ │ │ │  00014720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014740: 2020 2020 2020 2020 3220 2020 3220 3320          2   2 3 
│ │ │ │  00014750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014760: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00014770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014780: 2020 2050 6f6c 796e 6f6d 6961 6c53 6574     PolynomialSet
│ │ │ │ @@ -5347,71 +5347,71 @@
│ │ │ │  00014e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00014e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00014e90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00014ea0: 2031 2020 2030 2030 2030 2031 207c 2020   1   0 0 0 1 |  
│ │ │ │ +00014e90: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +00014ea0: 2061 2020 2030 2061 2b31 2031 207c 2020   a   0 a+1 1 |  
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│ │ │ │  00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00014ef0: 2061 2b31 2030 2030 2030 2031 207c 2020   a+1 0 0 0 1 |  
│ │ │ │ +00014ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +00014ef0: 2061 2b31 2030 2031 2020 2031 207c 2020   a+1 0 1   1 |  
│ │ │ │  00014f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20              |.| 
│ │ │ │ -00014f40: 312c 2030 2c20 302c 2030 2c20 317d 2c20  1, 0, 0, 0, 1}, 
│ │ │ │ -00014f50: 7b30 2c20 302c 2031 2c20 6120 2b20 312c  {0, 0, 1, a + 1,
│ │ │ │ -00014f60: 2061 202b 2031 2c20 302c 2030 2c20 302c   a + 1, 0, 0, 0,
│ │ │ │ +00014f30: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +00014f40: 2c20 302c 2061 202b 2031 2c20 317d 2c20  , 0, a + 1, 1}, 
│ │ │ │ +00014f50: 7b30 2c20 6120 2b20 312c 2030 2c20 302c  {0, a + 1, 0, 0,
│ │ │ │ +00014f60: 2030 2c20 6120 2b20 312c 2030 2c20 312c   0, a + 1, 0, 1,
│ │ │ │  00014f70: 2031 7d7d 2020 2020 2020 2020 2020 2020   1}}            
│ │ │ │  00014f80: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ -00014f90: 2020 2030 2030 2030 2030 207c 2020 2020     0 0 0 0 |    
│ │ │ │ +00014f90: 2030 2030 2030 2020 2030 207c 2020 2020   0 0 0   0 |    
│ │ │ │  00014fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ -00014fe0: 2020 2030 2030 2030 2030 207c 2020 2020     0 0 0 0 |    
│ │ │ │ +00014fe0: 2030 2030 2031 2020 2061 207c 2020 2020   0 0 1   a |    
│ │ │ │  00014ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015020: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ -00015030: 2020 2030 2030 2030 2061 207c 2020 2020     0 0 0 a |    
│ │ │ │ +00015020: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +00015030: 2030 2030 2030 2020 2030 207c 2020 2020   0 0 0   0 |    
│ │ │ │  00015040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015070: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ -00015080: 2b31 2030 2030 2030 2031 207c 2020 2020  +1 0 0 0 1 |    
│ │ │ │ +00015070: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +00015080: 2030 2030 2030 2020 2030 207c 2020 2020   0 0 0   0 |    
│ │ │ │  00015090: 2020 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 7c0a 7c30              |.|0
│ │ │ │ -000150d0: 2020 2031 2030 2030 2030 207c 2020 2020     1 0 0 0 |    
│ │ │ │ +000150c0: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ +000150d0: 2030 2030 2030 2020 2030 207c 2020 2020   0 0 0   0 |    
│ │ │ │  000150e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015110: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ -00015120: 2020 2030 2031 2030 2030 207c 2020 2020     0 1 0 0 |    
│ │ │ │ +00015120: 2031 2030 2061 2b31 2030 207c 2020 2020   1 0 a+1 0 |    
│ │ │ │  00015130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015160: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ -00015170: 2020 2030 2030 2031 2030 207c 2020 2020     0 0 1 0 |    
│ │ │ │ +00015170: 2030 2031 2030 2020 2030 207c 2020 2020   0 1 0   0 |    
│ │ │ │  00015180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000151a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000151b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2c              |.|,
│ │ │ │  000151c0: 2030 2c20 302c 2030 2c20 302c 2030 7d2c   0, 0, 0, 0, 0},
│ │ │ │  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
│ │ │ │ -000151f0: 2c20 302c 2031 2c20 302c 2031 2c20 302c  , 0, 1, 0, 1, 0,
│ │ │ │ -00015200: 2030 2c20 302c 2061 7d2c 2020 7c0a 7c7d   0, 0, a},  |.|}
│ │ │ │ -00015210: 2c20 7b30 2c20 317d 2c20 7b31 2c20 307d  , {0, 1}, {1, 0}
│ │ │ │ +000151e0: 2030 2c20 302c 2031 2c20 617d 2c20 7b30   0, 0, 1, a}, {0
│ │ │ │ +000151f0: 2c20 302c 2031 2c20 302c 2030 2c20 302c  , 0, 1, 0, 0, 0,
│ │ │ │ +00015200: 2030 2c20 302c 2030 7d2c 2020 7c0a 7c7d   0, 0, 0},  |.|}
│ │ │ │ +00015210: 2c20 7b31 2c20 307d 2c20 7b31 2c20 617d  , {1, 0}, {1, a}
│ │ │ │  00015220: 2c20 7b31 2c20 317d 7d20 2020 2020 2020  , {1, 1}}       
│ │ │ │  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                  
│ │ │ │ @@ -5568,37 +5568,37 @@
│ │ │ │  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}, 
│ │ │ │ -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,|.|-
│ │ │ │ +00015c60: 302c 2030 2c20 302c 2031 2c20 302c 2030  0, 0, 0, 1, 0, 0
│ │ │ │ +00015c70: 2c20 302c 2030 2c20 307d 2c20 7b30 2c20  , 0, 0, 0}, {0, 
│ │ │ │ +00015c80: 302c 2030 2c20 302c 2031 2c20 302c 2030  0, 0, 0, 1, 0, 0
│ │ │ │ +00015c90: 2c20 302c 2030 7d2c 207b 302c 2030 2c20  , 0, 0}, {0, 0, 
│ │ │ │ +00015ca0: 302c 2030 2c20 302c 2031 2c20 7c0a 7c2d  0, 0, 0, 1, |.|-
│ │ │ │  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  ----------------
│ │ │ │  00015cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  00015d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d20: 2020 2020 2020 2020 207d 2020 2020 2020           }      
│ │ │ │ +00015d20: 2020 2020 2020 2020 2020 7d20 2020 2020            }     
│ │ │ │  00015d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00015d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00015da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015dc0: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │ +00015dc0: 2020 2020 2020 2020 207d 2020 2020 2020           }      
│ │ │ │  00015dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015de0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00015df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015e00: 2020 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 7c0a 7c20              |.| 
│ │ │ │ @@ -5698,17 +5698,17 @@
│ │ │ │  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, 
│ │ │ │ -00016490: 302c 2030 2c20 302c 2030 2c20 302c 2030  0, 0, 0, 0, 0, 0
│ │ │ │ -000164a0: 2c20 312c 2030 7d7d 2020 2020 2020 2020  , 1, 0}}        
│ │ │ │ +00016480: 2c20 6120 2b20 312c 2030 7d2c 207b 302c  , a + 1, 0}, {0,
│ │ │ │ +00016490: 2030 2c20 302c 2030 2c20 302c 2030 2c20   0, 0, 0, 0, 0, 
│ │ │ │ +000164a0: 312c 2030 2c20 307d 7d20 2020 2020 2020  1, 0, 0}}       
│ │ │ │  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 |     
│ │ │ │ +00016da0: 7c20 3120 2020 3020 2020 7c20 2020 2020  | 1   0   |     
│ │ │ │  00016db0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00016dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016df0: 7c20 612b 3120 3020 2020 7c20 2020 2020  | a+1 0   |     
│ │ │ │ +00016df0: 7c20 3020 2020 3020 2020 7c20 2020 2020  | 0   0   |     
│ │ │ │  00016e00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00016e10: 2020 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: 7c20 612b 3120 3020 2020 7c20 2020 2020  | a+1 0   |     
│ │ │ │ +00016e40: 7c20 3020 2020 3020 2020 7c20 2020 2020  | 0   0   |     
│ │ │ │  00016e50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  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: 7c20 6120 2020 6120 2020 7c20 2020 2020  | a   a   |     
│ │ │ │ +00016e90: 7c20 612b 3120 3020 2020 7c20 2020 2020  | a+1 0   |     
│ │ │ │  00016ea0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ee0: 7c20 3120 2020 3020 2020 7c20 2020 2020  | 1   0   |     
│ │ │ │ +00016ee0: 7c20 612b 3120 3020 2020 7c20 2020 2020  | a+1 0   |     
│ │ │ │  00016ef0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00016f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f30: 7c20 6120 2020 6120 2020 7c20 2020 2020  | a   a   |     
│ │ │ │  00016f40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  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: 7c20 3020 2020 3020 2020 7c20 2020 2020  | 0   0   |     
│ │ │ │ +00016f80: 7c20 3120 2020 612b 3120 7c20 2020 2020  | 1   a+1 |     
│ │ │ │  00016f90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016fd0: 7c20 3020 2020 3020 2020 7c20 2020 2020  | 0   0   |     
│ │ │ │ +00016fd0: 7c20 6120 2020 6120 2020 7c20 2020 2020  | a   a   |     
│ │ │ │  00016fe0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00016ff0: 2020 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: 7c20 3120 2020 3120 2020 7c20 2020 2020  | 1   1   |     
│ │ │ │  00017030: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017060: 2020 4765 6e65 7261 746f 724d 6174 7269    GeneratorMatri
│ │ │ │ -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                  
│ │ │ │ -000170c0: 2020 2020 207c 2061 2b31 2030 2020 2030       | a+1 0   0
│ │ │ │ +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                  
│ │ │ │  00017100: 2020 4765 6e65 7261 746f 7273 203d 3e20    Generators => 
│ │ │ │ -00017110: 7b7b 312c 2061 202b 2031 2c20 6120 2b20  {{1, a + 1, a + 
│ │ │ │ -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                  
│ │ │ │  00017150: 2020 5061 7269 7479 4368 6563 6b4d 6174    ParityCheckMat
│ │ │ │  00017160: 7269 7820 3d3e 207c 2031 2030 2030 2030  rix => | 1 0 0 0
│ │ │ │  00017170: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5942,18 +5942,18 @@
│ │ │ │  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                  
│ │ │ │  00017380: 2020 5061 7269 7479 4368 6563 6b52 6f77    ParityCheckRow
│ │ │ │  00017390: 7320 3d3e 207b 7b31 2c20 302c 2030 2c20  s => {{1, 0, 0, 
│ │ │ │  000173a0: 3020 7c0a 7c20 2020 2020 2020 2020 2020  0 |.|           
│ │ │ │  000173b0: 2020 2020 2020 2020 2050 6f69 6e74 7320           Points 
│ │ │ │ -000173c0: 3d3e 207b 7b61 2c20 617d 2c20 7b30 2c20  => {{a, a}, {0, 
│ │ │ │ -000173d0: 617d 2c20 7b61 2c20 307d 2c20 7b61 2c20  a}, {a, 0}, {a, 
│ │ │ │ -000173e0: 317d 2c20 7b30 2c20 307d 2c20 7b31 2c20  1}, {0, 0}, {1, 
│ │ │ │ -000173f0: 6120 7c0a 7c20 2020 2020 2020 2020 2020  a |.|           
│ │ │ │ +000173c0: 3d3e 207b 7b30 2c20 307d 2c20 7b30 2c20  => {{0, 0}, {0, 
│ │ │ │ +000173d0: 317d 2c20 7b31 2c20 307d 2c20 7b30 2c20  1}, {1, 0}, {0, 
│ │ │ │ +000173e0: 617d 2c20 7b61 2c20 307d 2c20 7b61 2c20  a}, {a, 0}, {a, 
│ │ │ │ +000173f0: 3120 7c0a 7c20 2020 2020 2020 2020 2020  1 |.|           
│ │ │ │  00017400: 2020 2020 2020 2020 2050 6f6c 796e 6f6d           Polynom
│ │ │ │  00017410: 6961 6c53 6574 203d 3e20 7b78 202b 2079  ialSet => {x + y
│ │ │ │  00017420: 202b 2031 2c20 782a 797d 2020 2020 2020   + 1, x*y}      
│ │ │ │  00017430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017440: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017450: 2020 2020 2020 2020 2053 6574 7320 3d3e           Sets =>
│ │ │ │  00017460: 207b 7b30 2c20 312c 2061 7d2c 207b 302c   {{0, 1, a}, {0,
│ │ │ │ @@ -6050,71 +6050,71 @@
│ │ │ │  00017a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00017a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a80: 2020 7c0a 7c31 2061 2031 2061 2030 2030    |.|1 a 1 a 0 0
│ │ │ │ +00017a80: 2020 7c0a 7c61 2b31 2061 2031 2020 2061    |.|a+1 a 1   a
│ │ │ │  00017a90: 2031 207c 2020 2020 2020 2020 2020 2020   1 |            
│ │ │ │  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                  
│ │ │ │ -00017ad0: 2020 7c0a 7c20 2061 2030 2061 2030 2030    |.|  a 0 a 0 0
│ │ │ │ +00017ad0: 2020 7c0a 7c30 2020 2061 2061 2b31 2061    |.|0   a a+1 a
│ │ │ │  00017ae0: 2031 207c 2020 2020 2020 2020 2020 2020   1 |            
│ │ │ │  00017af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b20: 2020 7c0a 7c2c 2061 2c20 312c 2061 2c20    |.|, a, 1, a, 
│ │ │ │ -00017b30: 302c 2030 2c20 317d 2c20 7b61 202b 2031  0, 0, 1}, {a + 1
│ │ │ │ -00017b40: 2c20 302c 2030 2c20 612c 2030 2c20 612c  , 0, 0, a, 0, a,
│ │ │ │ -00017b50: 2030 2c20 302c 2031 7d7d 2020 2020 2020   0, 0, 1}}      
│ │ │ │ +00017b20: 2020 7c0a 7c20 6120 2b20 312c 2061 2c20    |.| a + 1, a, 
│ │ │ │ +00017b30: 312c 2061 2c20 317d 2c20 7b30 2c20 302c  1, a, 1}, {0, 0,
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│ │ │ │ +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                  
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│ │ │ │ +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}, 
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│ │ │ │  00017e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -6255,38 +6255,38 @@
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│ │ │ │  000187f0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │  00018800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018810: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +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                  
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│ │ │ │  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                  
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│ │ │ │  000188e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │  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                  
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│ │ │ │  00018f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00018f90: 2c20 302c 2030 2c20 302c 2030 2c20 312c  , 0, 0, 0, 0, 1,
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│ │ │ │ +00018fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  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
│ │ │ │ @@ -13181,42 +13181,42 @@
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│ │ │ │ +00033830: 612c 2031 2c20 6120 2b20 317d 2c20 7b30  a, 1, a + 1}, {0
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│ │ │ │ -000338d0: 202b 2031 2c20 6120 2b20 312c 2061 202b   + 1, a + 1, a +
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│ │ │ │ +000338a0: 317d 2c20 7b30 2c20 6120 2b20 312c 2031  1}, {0, a + 1, 1
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│ │ │ │ +000338c0: 2c20 7b61 202b 2031 2c20 302c 2061 7d2c  , {a + 1, 0, a},
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│ │ │ │  000338f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +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}, {
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│ │ │ │ +0005df30: 2020 207c 0a7c 6f33 203d 207b 7b30 7d2c     |.|o3 = {{0},
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│ │ │ │ +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  ----------------
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│ │ │ │  000675d0: 2020 2020 2020 2043 6f64 6520 3d3e 2069         Code => i
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│ │ │ │  000692b0: 2050 6172 6974 7943 6865 636b 4d61 7472   ParityCheckMatr
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│ │ │ │  000692e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  000693a0: 2050 6172 6974 7943 6865 636b 526f 7773   ParityCheckRows
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│ │ │ │  000693e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ @@ -27021,16 +27021,16 @@
│ │ │ │  000698c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000698d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  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 2033 2e31 3431 3233 7320 656c  | -- 3.14123s 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 3533 3839 3735 7320 656c  | -- .538975s 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 302e 3638 3033 7320 656c  | -- 10.6803s 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 202e 3531 3834 3435 7320 656c  | -- .518445s 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
│ │ │ │ -000072f0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │ +000072e0: 7c20 2d2d 202e 3430 3330 3773 2065 6c61  | -- .40307s ela
│ │ │ │ +000072f0: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  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 3430 3331 3132 7320 656c  | -- .403112s 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 302e 3734 3439 7320 2863 7075 293b   10.7449s (cpu);
│ │ │ │ +000100d0: 2035 2e39 3934 3433 7320 2874 6872 6561   5.99443s (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 3833 3139 3373 2028 6370 7529   0.183193s (cpu)
│ │ │ │ +00012290: 3b20 302e 3131 3039 3737 7320 2874 6872  ; 0.110977s (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 
│ │ │ │ @@ -4683,16 +4683,16 @@
│ │ │ │  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 (
│ │ │ │ +00012510: 7573 6564 2031 2e36 3839 3432 7320 2863  used 1.68942s (c
│ │ │ │ +00012520: 7075 293b 2030 2e39 3432 3135 3973 2028  pu); 0.942159s (
│ │ │ │  00012530: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 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     
│ │ │ │ @@ -4740,16 +4740,16 @@
│ │ │ │  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
│ │ │ │ +000128a0: 7365 6420 312e 3638 3638 3473 2028 6370  sed 1.68684s (cp
│ │ │ │ +000128b0: 7529 3b20 302e 3838 3034 3538 7320 2874  u); 0.880458s (t
│ │ │ │  000128c0: 6872 6561 6429 3b20 3073 2028 6763 297c  hread); 0s (gc)|
│ │ │ │  000128d0: 0a7c 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    
│ │ │ │ @@ -27637,24 +27637,24 @@
│ │ │ │  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 (
│ │ │ │ +0006bfb0: 7365 6420 302e 3738 3238 3131 7320 2863  sed 0.782811s (c
│ │ │ │ +0006bfc0: 7075 293b 2030 2e34 3131 3032 3773 2028  pu); 0.411027s (
│ │ │ │  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
│ │ │ │ -0006c010: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0006bff0: 302e 3335 3234 3673 2028 6370 7529 3b20  0.35246s (cpu); 
│ │ │ │ +0006c000: 302e 3136 3530 3173 2028 7468 7265 6164  0.16501s (thread
│ │ │ │ +0006c010: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  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
│ │ │ │ +0006c030: 3031 3738 3732 3873 2028 6370 7529 3b20  0178728s (cpu); 
│ │ │ │ +0006c040: 322e 3832 3665 2d30 3673 2028 7468 7265  2.826e-06s (thre
│ │ │ │  0006c050: 6164 293b 2030 7320 2867 6329 7c0a 7c32  ad); 0s (gc)|.|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}         
│ │ │ │ @@ -28166,17 +28166,17 @@
│ │ │ │  0006e050: 2b2d 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) 
│ │ │ │  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
│ │ │ │ +0006e0c0: 7c20 2d2d 2075 7365 6420 312e 3631 3731  | -- used 1.6171
│ │ │ │ +0006e0d0: 3173 2028 6370 7529 3b20 302e 3736 3938  1s (cpu); 0.7698
│ │ │ │ +0006e0e0: 3838 7320 2874 6872 6561 6429 3b20 3073  88s (thread); 0s
│ │ │ │  0006e0f0: 2028 6763 2920 7c0a 7c32 2020 2020 2020   (gc) |.|2      
│ │ │ │  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                  
│ │ │ │ @@ -28184,35 +28184,35 @@
│ │ │ │  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  |               
│ │ │ │  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)|.
│ │ │ │ +0006e1e0: 6420 302e 3934 3431 3373 2028 6370 7529  d 0.94413s (cpu)
│ │ │ │ +0006e1f0: 3b20 302e 3436 3337 3233 7320 2874 6872  ; 0.463723s (thr
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│ │ │ │  0006e210: 7c33 2020 2020 2020 2020 2020 2020 2020  |3              
│ │ │ │  0006e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0006e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006e320: 2867 6329 2020 7c0a 7c34 2020 2020 2020  (gc)  |.|4      
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│ │ │ │ +0006e310: 3639 3973 2028 7468 7265 6164 293b 2030  699s (thread); 0
│ │ │ │ +0006e320: 7320 2867 6329 7c0a 7c34 2020 2020 2020  s (gc)|.|4      
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│ │ │ │  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        |.+-------
│ │ ├── ./usr/share/info/Cremona.info.gz
│ │ │ ├── Cremona.info
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│ │ │ │  000009f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ @@ -322,16 +322,16 @@
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│ │ │ │  00001430: 696d 6520 4a20 3d20 6b65 726e 656c 2870  ime J = kernel(p
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│ │ │ │  00001450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001470: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
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│ │ │ │  000014c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000014d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000014e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000014f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -387,18 +387,18 @@
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│ │ │ │  00001860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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  )               
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│ │ │ │ +000018a0: 7529 3b20 302e 3035 3332 3632 3473 2028  u); 0.0532624s (
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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                  
│ │ │ │ @@ -412,16 +412,16 @@
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│ │ │ │  000019d0: 696d 6520 7072 6f6a 6563 7469 7665 4465  ime projectiveDe
│ │ │ │  000019e0: 6772 6565 7320 7068 6920 2020 2020 2020  grees phi       
│ │ │ │  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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│ │ │ │  00001a40: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00001a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00001a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -447,16 +447,16 @@
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│ │ │ │  00001c00: 696d 6520 7072 6f6a 6563 7469 7665 4465  ime projectiveDe
│ │ │ │  00001c10: 6772 6565 7328 7068 692c 4e75 6d44 6567  grees(phi,NumDeg
│ │ │ │  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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│ │ │ │  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                  
│ │ │ │ @@ -482,15 +482,15 @@
│ │ │ │  00001e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00001e30: 696d 6520 7068 6920 3d20 746f 4d61 7028  ime phi = toMap(
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│ │ │ │  00001e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001e70: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001e80: 6564 2030 2e30 3030 3138 3930 3635 7320  ed 0.000189065s 
│ │ │ │ +00001e80: 6564 2030 2e30 3030 3138 3139 3533 7320  ed 0.000181953s 
│ │ │ │  00001e90: 2863 7075 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                  
│ │ │ │ @@ -567,15 +567,15 @@
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│ │ │ │  00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 446f 6d69 6e61  -------|.|Domina
│ │ │ │  00002380: 6e74 3d3e 4a29 2020 2020 2020 2020 2020  nt=>J)          
│ │ │ │  00002390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000023a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000023b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000023c0: 2020 2020 2020 207c 0a7c 293b 2030 2e30         |.|); 0.0
│ │ │ │ -000023d0: 3032 3034 3239 3973 2028 7468 7265 6164  0204299s (thread
│ │ │ │ +000023d0: 3032 3430 3334 3773 2028 7468 7265 6164  0240347s (thread
│ │ │ │  000023e0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  000023f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002410: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -832,17 +832,17 @@
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│ │ │ │  00003400: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074  -------+.|i8 : t
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│ │ │ │  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
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│ │ │ │  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 @@
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│ │ │ │  000045d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074  -------+.|i9 : t
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│ │ │ │  000045f0: 2870 6869 2c70 7369 2920 2020 2020 2020  (phi,psi)       
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│ │ │ │  00004610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004620: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
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│ │ │ │ +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 @@
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│ │ │ │  00004760: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
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│ │ │ │  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 3830 3431 3573 2028 6370  ed 0.280415s (cp
│ │ │ │ +000047d0: 7529 3b20 302e 3230 3132 3738 7320 2874  u); 0.201278s (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 @@
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│ │ │ │  000048f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
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│ │ │ │  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
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│ │ │ │ -00004960: 293b 2034 2e32 3737 3236 7320 2874 6872  ); 4.27726s (thr
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│ │ │ │ +00004960: 293b 2035 2e31 3935 3132 7320 2874 6872  ); 5.19512s (thr
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│ │ │ │  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
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│ │ │ │ -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 3030  .| -- used 0.000
│ │ │ │ +00004c40: 3931 3433 3437 7320 2863 7075 293b 2030  914347s (cpu); 0
│ │ │ │ +00004c50: 2e30 3032 3334 3936 3173 2028 7468 7265  .00234961s (thre
│ │ │ │ +00004c60: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 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
│ │ │ │ @@ -1493,18 +1493,18 @@
│ │ │ │  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);
│ │ │ │ -00005de0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e31 3733  .| -- used 0.173
│ │ │ │ +00005dc0: 3333 3973 2028 6370 7529 3b20 302e 3039  339s (cpu); 0.09
│ │ │ │ +00005dd0: 3639 3236 7320 2874 6872 6561 6429 3b20  6926s (thread); 
│ │ │ │ +00005de0: 3073 2028 6763 2920 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                 |
│ │ │ │  00005e50: 0a7c 6f31 3320 3d20 2d2d 2072 6174 696f  .|o13 = -- ratio
│ │ │ │ @@ -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 3630  .| -- used 0.460
│ │ │ │ +00008700: 3138 3973 2028 6370 7529 3b20 302e 3435  189s (cpu); 0.45
│ │ │ │ +00008710: 3931 3137 7320 2874 6872 6561 6429 3b20  9117s (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                 |
│ │ │ │ @@ -2708,17 +2708,17 @@
│ │ │ │  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); 
│ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3432  .| -- used 0.442
│ │ │ │ +0000a9b0: 3532 3873 2028 6370 7529 3b20 302e 3335  528s (cpu); 0.35
│ │ │ │ +0000a9c0: 3430 3932 7320 2874 6872 6561 6429 3b20  4092s (thread); 
│ │ │ │  0000a9d0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (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                 |
│ │ │ │ @@ -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 3532  .| -- used 0.052
│ │ │ │ +0000abe0: 3633 3433 7320 2863 7075 293b 2030 2e30  6343s (cpu); 0.0
│ │ │ │ +0000abf0: 3238 3638 3738 7320 2874 6872 6561 6429  286878s (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
│ │ │ │ @@ -2779,16 +2779,16 @@
│ │ │ │  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
│ │ │ │ +0000ae10: 3533 3735 3473 2028 6370 7529 3b20 302e  53754s (cpu); 0.
│ │ │ │ +0000ae20: 3030 3436 3934 3334 7320 2874 6872 6561  00469434s (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 3130  .| -- used 0.010
│ │ │ │ +0000b220: 3639 3634 7320 2863 7075 293b 2030 2e30  6964s (cpu); 0.0
│ │ │ │ +0000b230: 3131 3837 3433 7320 2874 6872 6561 6429  118743s (thread)
│ │ │ │ +0000b240: 3b20 3073 2028 6763 2920 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
│ │ │ │ @@ -2923,17 +2923,17 @@
│ │ │ │  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
│ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3036  .| -- used 0.006
│ │ │ │ +0000b720: 3633 3632 7320 2863 7075 293b 2030 2e30  6362s (cpu); 0.0
│ │ │ │ +0000b730: 3039 3938 3039 3873 2028 7468 7265 6164  0998098s (thread
│ │ │ │  0000b740: 293b 2030 7320 2867 6329 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                 |
│ │ │ │ @@ -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 2e33 3434  .| -- used 1.344
│ │ │ │ +0000b950: 3336 7320 2863 7075 293b 2031 2e30 3532  36s (cpu); 1.052
│ │ │ │ +0000b960: 3639 7320 2874 6872 6561 6429 3b20 3073  69s (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: 3139 3136 3232 7320 2863 7075 293b 2031  191622s (cpu); 1
│ │ │ │ +0000bb90: 2e37 3931 3765 2d30 3573 2028 7468 7265  .7917e-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                 |
│ │ │ │ @@ -3018,18 +3018,18 @@
│ │ │ │  0000bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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
│ │ │ │ -0000bd30: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +0000bd00: 0a7c 202d 2d20 7573 6564 2038 2e38 3537  .| -- used 8.857
│ │ │ │ +0000bd10: 3665 2d30 3573 2028 6370 7529 3b20 302e  6e-05s (cpu); 0.
│ │ │ │ +0000bd20: 3030 3138 3137 7320 2874 6872 6561 6429  001817s (thread)
│ │ │ │ +0000bd30: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 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                 |
│ │ │ │  0000bda0: 0a7c 6f32 3320 3d20 7261 7469 6f6e 616c  .|o23 = rational
│ │ │ │ @@ -4588,18 +4588,18 @@
│ │ │ │  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)
│ │ │ │ -00011f50: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00011f20: 2d2d 2075 7365 6420 302e 3030 3335 3739  -- used 0.003579
│ │ │ │ +00011f30: 3873 2028 6370 7529 3b20 302e 3030 3033  8s (cpu); 0.0003
│ │ │ │ +00011f40: 3936 3932 3773 2028 7468 7265 6164 293b  96927s (thread);
│ │ │ │ +00011f50: 2030 7320 2867 6329 2020 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
│ │ │ │  00011fc0: 3420 3d20 2d2d 2072 6174 696f 6e61 6c20  4 = -- rational 
│ │ │ │ @@ -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
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│ │ │ │  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  ---------------+
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│ │ │ │  000124a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000124b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -000124f0: 2867 6329 2020 2020 2020 207c 0a7c 2020  (gc)       |.|  
│ │ │ │ +000124c0: 202d 2d20 7573 6564 2030 2e34 3138 3131   -- used 0.41811
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│ │ │ │ +000124e0: 3739 7320 2874 6872 6561 6429 3b20 3073  79s (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 =
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│ │ │ │  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
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│ │ │ │  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.
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│ │ │ │ -00013f50: 2e30 3635 3236 3533 7320 2874 6872 6561  .0652653s (threa
│ │ │ │ -00013f60: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00013f40: 3034 3739 3232 3473 2028 6370 7529 3b20  0479224s (cpu); 
│ │ │ │ +00013f50: 302e 3034 3837 3931 3373 2028 7468 7265  0.0487913s (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 @@
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│ │ │ │  000143a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000143b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
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│ │ │ │ -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 3430 3434 3173 2028 6370 7529 3b20  3.40441s (cpu); 
│ │ │ │ +00014410: 312e 3934 3832 3973 2028 7468 7265 6164  1.94829s (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
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│ │ │ │ +000144c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57  ------------+..W
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│ │ │ │  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 3938 3433 3173 2028 6370  0.000198431s (cp
│ │ │ │ +000145b0: 7529 3b20 322e 3838 3538 652d 3035 7320  u); 2.8858e-05s 
│ │ │ │ +000145c0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +000145d0: 297c 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
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ +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 352e 3338 3739 3673 2028   used 5.38796s (
│ │ │ │ +00014900: 6370 7529 3b20 332e 3035 3733 3173 2028  cpu); 3.05731s (
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│ │ │ │ +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
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│ │ │ │ -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 2034 2e33 3535 3336 7320  - used 4.35536s 
│ │ │ │ +00014e60: 2863 7075 293b 2032 2e35 3832 3332 7320  (cpu); 2.58232s 
│ │ │ │ +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                  
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│ │ │ │ +000158e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -00015930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00015950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00015970: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54  Caveat.======..T
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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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│ │ │ │ +00015a10: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c46  PolynomialRing,F
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│ │ │ │ +00015a40: 7469 6f6e 616c 4d61 7028 506f 6c79 6e6f  tionalMap(Polyno
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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.========
│ │ │ │ +00015ac0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +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
│ │ │ │ +00015b00: 616c 4d61 702c 2069 7320 6120 2a6e 6f74  alMap, is a *not
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│ │ │ │ +00015b20: 6e3a 2028 4d61 6361 756c 6179 3244 6f63  n: (Macaulay2Doc
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│ │ │ │ +00015b40: 2e0a 1f0a 4669 6c65 3a20 4372 656d 6f6e  ....File: Cremon
│ │ │ │ +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             |.|  
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│ │ │ │ +00016740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016750: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00016760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016770: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00016800: 2020 2020 2020 2020 2020 2035 2036 2020             5 6  
│ │ │ │ +00016810: 2020 3120 3820 2020 2020 2020 2020 2020    1 8           
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│ │ │ │ +00016c00: 207c 0a7c 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             |.|  
│ │ │ │ -00016c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c60: 2020 2020 7820 7820 202d 2078 2078 202c      x x  - x x ,
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│ │ │ │ -00016c90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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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                  
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│ │ │ │  00017b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017be0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -00017c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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
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│ │ │ │ +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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│ │ │ │ -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        
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│ │ │ │ -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 
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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
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│ │ │ │ +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      
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│ │ │ │ +00018040: 3620 2020 2020 2030 2032 2034 2036 2020  6      0 2 4 6  
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│ │ │ │ -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  
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│ │ │ │ +000180b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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  + 
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│ │ │ │ -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    |.|--
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│ │ │ │ +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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│ │ │ │ -000181e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00018210: 2020 2020 2020 2020 2032 2020 2020 2020           2      
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│ │ │ │ -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    
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│ │ │ │ -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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│ │ │ │ +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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│ │ │ │ +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  
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│ │ │ │ +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       
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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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│ │ │ │ -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    
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│ │ │ │ +00018360: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +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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│ │ │ │  00019460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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: 3032 3631 3373 2028 6370 7529 3b20 302e  02613s (cpu); 0.
│ │ │ │ +00019880: 3233 3238 3632 7320 2874 6872 6561 6429  232862s (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:
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│ │ │ │ -00019e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019db0: 207c 0a7c 2020 2020 2074 6172 6765 743a   |.|     target:
│ │ │ │ +00019dc0: 2073 7562 7661 7269 6574 7920 6f66 2050   subvariety of P
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│ │ │ │ +00019e00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ +00019e60: 207b 2020 2020 2020 2020 2020 2020 2020   {              
│ │ │ │  00019e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +00019ea0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  00019ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ee0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00019ef0: 2020 2020 2020 2020 2020 2020 7820 7820              x x 
│ │ │ │ -00019f00: 202b 2031 3478 2078 2078 2020 2b20 3236   + 14x x x  + 26
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│ │ │ │ -00019f30: 3131 7820 7820 7820 7820 207c 0a7c 2020  11x x x x  |.|  
│ │ │ │ -00019f40: 2020 2020 2020 2020 2020 2020 2031 2032               1 2
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│ │ │ │ +00019ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019ef0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00019f00: 2020 7820 7820 202b 2031 3478 2078 2078    x x  + 14x x x
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│ │ │ │  00019fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019fd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00019fe0: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
│ │ │ │ -00019ff0: 733a 207b 2020 2020 2020 2020 2020 2020  s: {            
│ │ │ │ +00019fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019fe0: 207c 0a7c 2020 2020 2064 6566 696e 696e   |.|     definin
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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                  
│ │ │ │ +0001a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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             |.|  
│ │ │ │ -0001a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a090: 2020 2020 2035 2037 2020 2020 3420 3820       5 7    4 8 
│ │ │ │ -0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001a090: 2020 2020 2020 2020 2020 2035 2037 2020             5 7  
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│ │ │ │ -0001a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +0001a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a120: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a130: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
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│ │ │ │  0001a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a160: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -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                  
│ │ │ │ +0001a170: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a180: 2020 2020 2020 2020 2020 2032 2037 2020             2 7  
│ │ │ │ +0001a190: 2020 3120 3820 2020 2020 2020 2020 2020    1 8           
│ │ │ │  0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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  -
│ │ │ │ +0001a230: 2074 2074 202c 2020 2020 2020 2020 2020   t t ,          
│ │ │ │  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                  
│ │ │ │  0001a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +0001a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001a400: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
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│ │ │ │  0001a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a430: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a450: 2020 2020 2032 2036 2020 2020 3020 3820       2 6    0 8 
│ │ │ │ -0001a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001a450: 2020 2020 2020 2020 2020 2032 2036 2020             2 6  
│ │ │ │ +0001a460: 2020 3020 3820 2020 2020 2020 2020 2020    0 8           
│ │ │ │  0001a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ -0001a4d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4f0: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
│ │ │ │ -0001a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001a500: 2074 2074 202c 2020 2020 2020 2020 2020   t t ,          
│ │ │ │  0001a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a520: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a540: 2020 2020 2031 2036 2020 2020 3020 3720       1 6    0 7 
│ │ │ │ -0001a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a530: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a540: 2020 2020 2020 2020 2020 2031 2036 2020             1 6  
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│ │ │ │  0001a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │  0001a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a5c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a5e0: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
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│ │ │ │ +0001a5d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001a5e0: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
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│ │ │ │  0001a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a610: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a630: 2020 2020 2032 2034 2020 2020 3120 3520       2 4    1 5 
│ │ │ │ -0001a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001a630: 2020 2020 2020 2020 2020 2032 2034 2020             2 4  
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│ │ │ │ -0001a660: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001a670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a670: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  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 ,
│ │ │ │ -0001a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001a6d0: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
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│ │ │ │  0001a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 ,
│ │ │ │ -0001a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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  -
│ │ │ │ +0001a7d0: 2074 2074 202c 2020 2020 2020 2020 2020   t t ,          
│ │ │ │  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                  
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│ │ │ │ -0001aa40: 7175 6164 7261 7469 6320 7261 7469 6f6e  quadratic ration
│ │ │ │ -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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│ │ │ │  0001aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001ac30: 7820 7820 202d 2031 3478 2078 2078 2078  x x  - 14x x x x
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│ │ │ │ +0001abe0: 3220 2020 2020 2032 2032 2020 2020 2020  2      2 2      
│ │ │ │ +0001abf0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
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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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│ │ │ │  0001b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001b560: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -0001b600: 2020 2020 2020 2020 2020 207c 0a7c 202b             |.| +
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│ │ │ │ -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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│ │ │ │  0001b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001b750: 7820 7820 7820 7820 202b 2031 3478 2078  x x x x  + 14x x
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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   |.|            
│ │ │ │  0001b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001b970: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b980: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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             |.|  
│ │ │ │ -0001ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ba20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0001ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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   |.|            
│ │ │ │  0001ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001bab0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bac0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001bb00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001bb10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0001bb50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001bb60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0001bba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bbb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bbf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001bc00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0001bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001bc40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001bc50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0001be70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001be80: 207c 0a7c 2020 2020 2020 2020 2020 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                  
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│ │ │ │ -0001bfb0: 2020 2020 2020 2020 2020 207c 0a7c 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   
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│ │ │ │ -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  
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│ │ │ │ +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 
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│ │ │ │ +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  
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│ │ │ │ -0001c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001cc80: 2020 2020 2020 2020 2020 207c 0a7c 3435             |.|45
│ │ │ │ -0001cc90: 7420 202d 2034 3574 2074 2020 2b20 3230  t  - 45t t  + 20
│ │ │ │ -0001cca0: 7420 7420 202b 2034 3474 2074 2020 2d20  t t  + 44t t  - 
│ │ │ │ -0001ccb0: 3430 7420 7420 202d 2032 3274 2074 2020  40t t  - 22t t  
│ │ │ │ -0001ccc0: 2b20 3337 7420 7420 202b 2032 3274 2074  + 37t t  + 22t t
│ │ │ │ -0001ccd0: 2020 2b20 3238 7420 7420 207c 0a7c 2020    + 28t t  |.|  
│ │ │ │ -0001cce0: 2037 2020 2020 2020 3020 3820 2020 2020   7      0 8     
│ │ │ │ -0001ccf0: 2031 2038 2020 2020 2020 3220 3820 2020   1 8      2 8   
│ │ │ │ -0001cd00: 2020 2033 2038 2020 2020 2020 3420 3820     3 8      4 8 
│ │ │ │ -0001cd10: 2020 2020 2035 2038 2020 2020 2020 3620       5 8      6 
│ │ │ │ -0001cd20: 3820 2020 2020 2037 2038 207c 0a7c 2d2d  8      7 8 |.|--
│ │ │ │ -0001cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cc90: 207c 0a7c 3435 7420 202d 2034 3574 2074   |.|45t  - 45t t
│ │ │ │ +0001cca0: 2020 2b20 3230 7420 7420 202b 2034 3474    + 20t t  + 44t
│ │ │ │ +0001ccb0: 2074 2020 2d20 3430 7420 7420 202d 2032   t  - 40t t  - 2
│ │ │ │ +0001ccc0: 3274 2074 2020 2b20 3337 7420 7420 202b  2t t  + 37t t  +
│ │ │ │ +0001ccd0: 2032 3274 2074 2020 2b20 3238 7420 7420   22t t  + 28t t 
│ │ │ │ +0001cce0: 207c 0a7c 2020 2037 2020 2020 2020 3020   |.|   7      0 
│ │ │ │ +0001ccf0: 3820 2020 2020 2031 2038 2020 2020 2020  8      1 8      
│ │ │ │ +0001cd00: 3220 3820 2020 2020 2033 2038 2020 2020  2 8      3 8    
│ │ │ │ +0001cd10: 2020 3420 3820 2020 2020 2035 2038 2020    4 8      5 8  
│ │ │ │ +0001cd20: 2020 2020 3620 3820 2020 2020 2037 2038      6 8      7 8
│ │ │ │ +0001cd30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001cd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd90: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001cd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cd80: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0001cd90: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdc0: 2020 2020 2020 2020 2020 207c 0a7c 3131             |.|11
│ │ │ │ -0001cdd0: 7820 7820 7820 7820 202d 2032 3278 2078  x x x x  - 22x x
│ │ │ │ -0001cde0: 2078 2020 2b20 3130 7820 7820 7820 7820   x  + 10x x x x 
│ │ │ │ -0001cdf0: 202b 2031 3178 2078 2078 2078 2020 2d20   + 11x x x x  - 
│ │ │ │ -0001ce00: 3432 7820 7820 7820 7820 202d 2031 3078  42x x x x  - 10x
│ │ │ │ -0001ce10: 2078 2078 2078 2020 2b20 207c 0a7c 2020   x x x  +  |.|  
│ │ │ │ -0001ce20: 2030 2033 2034 2036 2020 2020 2020 3020   0 3 4 6      0 
│ │ │ │ -0001ce30: 3420 3620 2020 2020 2031 2032 2035 2036  4 6      1 2 5 6
│ │ │ │ -0001ce40: 2020 2020 2020 3020 3320 3520 3620 2020        0 3 5 6   
│ │ │ │ -0001ce50: 2020 2031 2033 2035 2036 2020 2020 2020     1 3 5 6      
│ │ │ │ -0001ce60: 3020 3420 3520 3620 2020 207c 0a7c 2020  0 4 5 6    |.|  
│ │ │ │ -0001ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cdd0: 207c 0a7c 3131 7820 7820 7820 7820 202d   |.|11x x x x  -
│ │ │ │ +0001cde0: 2032 3278 2078 2078 2020 2b20 3130 7820   22x x x  + 10x 
│ │ │ │ +0001cdf0: 7820 7820 7820 202b 2031 3178 2078 2078  x x x  + 11x x x
│ │ │ │ +0001ce00: 2078 2020 2d20 3432 7820 7820 7820 7820   x  - 42x x x x 
│ │ │ │ +0001ce10: 202d 2031 3078 2078 2078 2078 2020 2b20   - 10x x x x  + 
│ │ │ │ +0001ce20: 207c 0a7c 2020 2030 2033 2034 2036 2020   |.|   0 3 4 6  
│ │ │ │ +0001ce30: 2020 2020 3020 3420 3620 2020 2020 2031      0 4 6      1
│ │ │ │ +0001ce40: 2032 2035 2036 2020 2020 2020 3020 3320   2 5 6      0 3 
│ │ │ │ +0001ce50: 3520 3620 2020 2020 2031 2033 2035 2036  5 6      1 3 5 6
│ │ │ │ +0001ce60: 2020 2020 2020 3020 3420 3520 3620 2020        0 4 5 6   
│ │ │ │ +0001ce70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0001d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001d730: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0001d780: 2020 3220 2020 2020 2020 2020 2020 2020    2             
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│ │ │ │ -0001d7d0: 3274 2020 2020 2020 2020 2020 2020 2020  2t              
│ │ │ │ +0001d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d7d0: 207c 0a7c 2b20 3274 2020 2020 2020 2020   |.|+ 2t        
│ │ │ │  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 2020                  
│ │ │ │ -0001d810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001d820: 2020 3820 2020 2020 2020 2020 2020 2020    8             
│ │ │ │ +0001d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d820: 207c 0a7c 2020 2020 3820 2020 2020 2020   |.|    8       
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│ │ │ │  0001d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d860: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -0001d870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001d870: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001d880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001d8c0: 2020 2032 2020 2020 2020 2020 3220 3220     2        2 2 
│ │ │ │ -0001d8d0: 2020 2020 2020 2020 2032 2020 2020 2032           2     2
│ │ │ │ -0001d8e0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ -0001d8f0: 2020 2032 2032 2020 2020 2020 2020 2032     2 2         2
│ │ │ │ -0001d900: 2020 2020 2020 2020 2020 327c 0a7c 3432            2|.|42
│ │ │ │ -0001d910: 7820 7820 7820 202d 2033 3878 2078 2020  x x x  - 38x x  
│ │ │ │ -0001d920: 2d20 3337 7820 7820 7820 202b 2037 7820  - 37x x x  + 7x 
│ │ │ │ -0001d930: 7820 202b 2032 3878 2078 2078 2020 2d20  x  + 28x x x  - 
│ │ │ │ -0001d940: 3338 7820 7820 202d 2033 7820 7820 7820  38x x  - 3x x x 
│ │ │ │ -0001d950: 202d 2032 3978 2078 2078 207c 0a7c 2020   - 29x x x |.|  
│ │ │ │ -0001d960: 2030 2035 2036 2020 2020 2020 3220 3620   0 5 6      2 6 
│ │ │ │ -0001d970: 2020 2020 2032 2033 2036 2020 2020 2033       2 3 6     3
│ │ │ │ -0001d980: 2036 2020 2020 2020 3220 3420 3620 2020   6      2 4 6   
│ │ │ │ -0001d990: 2020 2034 2036 2020 2020 2032 2035 2036     4 6     2 5 6
│ │ │ │ -0001d9a0: 2020 2020 2020 3320 3520 367c 0a7c 2d2d        3 5 6|.|--
│ │ │ │ -0001d9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d8c0: 2d7c 0a7c 2020 2020 2032 2020 2020 2020  -|.|     2      
│ │ │ │ +0001d8d0: 2020 3220 3220 2020 2020 2020 2020 2032    2 2          2
│ │ │ │ +0001d8e0: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ +0001d8f0: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ +0001d900: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0001d910: 327c 0a7c 3432 7820 7820 7820 202d 2033  2|.|42x x x  - 3
│ │ │ │ +0001d920: 3878 2078 2020 2d20 3337 7820 7820 7820  8x x  - 37x x x 
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│ │ │ │ +0001d960: 207c 0a7c 2020 2030 2035 2036 2020 2020   |.|   0 5 6    
│ │ │ │ +0001d970: 2020 3220 3620 2020 2020 2032 2033 2036    2 6      2 3 6
│ │ │ │ +0001d980: 2020 2020 2033 2036 2020 2020 2020 3220       3 6      2 
│ │ │ │ +0001d990: 3420 3620 2020 2020 2034 2036 2020 2020  4 6      4 6    
│ │ │ │ +0001d9a0: 2032 2035 2036 2020 2020 2020 3320 3520   2 5 6      3 5 
│ │ │ │ +0001d9b0: 367c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  6|.|------------
│ │ │ │  0001d9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001da00: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ -0001da10: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0001da20: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0001da30: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0001da40: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
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│ │ │ │ -0001da90: 2020 2d20 2020 2020 2020 207c 0a7c 2020    -        |.|  
│ │ │ │ -0001daa0: 2020 2034 2035 2036 2020 2020 2020 3520     4 5 6      5 
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│ │ │ │ -0001dac0: 2020 2020 3120 3220 3720 2020 2020 2031      1 2 7      1
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│ │ │ │ -0001dae0: 3720 2020 2020 2020 2020 207c 0a7c 2d2d  7          |.|--
│ │ │ │ -0001daf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001da00: 2d7c 0a7c 2020 2020 2020 2020 2032 2020  -|.|         2  
│ │ │ │ +0001da10: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
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│ │ │ │ +0001da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001da40: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0001da50: 207c 0a7c 2d20 3437 7820 7820 7820 202b   |.|- 47x x x  +
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│ │ │ │ +0001dab0: 2020 2020 3520 3620 2020 2020 2030 2031      5 6      0 1
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│ │ │ │ +0001dad0: 2020 2020 2031 2032 2033 2037 2020 2020       1 2 3 7    
│ │ │ │ +0001dae0: 2020 3120 3320 3720 2020 2020 2020 2020    1 3 7         
│ │ │ │ +0001daf0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
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│ │ │ │  0001db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001db40: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0001db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001db40: 2d7c 0a7c 2020 2032 2020 2020 2020 2020  -|.|   2        
│ │ │ │  0001db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db60: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0001db60: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  0001db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db80: 2020 2020 2020 2020 2020 207c 0a7c 3330             |.|30
│ │ │ │ -0001db90: 7820 7820 7820 202b 2032 3278 2078 2078  x x x  + 22x x x
│ │ │ │ -0001dba0: 2078 2020 2b20 3230 7820 7820 7820 7820   x  + 20x x x x 
│ │ │ │ -0001dbb0: 202d 2034 7820 7820 7820 202b 2032 3678   - 4x x x  + 26x
│ │ │ │ -0001dbc0: 2078 2078 2078 2020 2b20 3431 7820 7820   x x x  + 41x x 
│ │ │ │ -0001dbd0: 7820 7820 202d 2020 2020 207c 0a7c 2020  x x  -     |.|  
│ │ │ │ -0001dbe0: 2030 2034 2037 2020 2020 2020 3020 3120   0 4 7      0 1 
│ │ │ │ -0001dbf0: 3420 3720 2020 2020 2030 2033 2034 2037  4 7      0 3 4 7
│ │ │ │ -0001dc00: 2020 2020 2033 2034 2037 2020 2020 2020       3 4 7      
│ │ │ │ -0001dc10: 3120 3220 3520 3720 2020 2020 2030 2033  1 2 5 7      0 3
│ │ │ │ -0001dc20: 2035 2037 2020 2020 2020 207c 0a7c 2d2d   5 7       |.|--
│ │ │ │ -0001dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001db90: 207c 0a7c 3330 7820 7820 7820 202b 2032   |.|30x x x  + 2
│ │ │ │ +0001dba0: 3278 2078 2078 2078 2020 2b20 3230 7820  2x x x x  + 20x 
│ │ │ │ +0001dbb0: 7820 7820 7820 202d 2034 7820 7820 7820  x x x  - 4x x x 
│ │ │ │ +0001dbc0: 202b 2032 3678 2078 2078 2078 2020 2b20   + 26x x x x  + 
│ │ │ │ +0001dbd0: 3431 7820 7820 7820 7820 202d 2020 2020  41x x x x  -    
│ │ │ │ +0001dbe0: 207c 0a7c 2020 2030 2034 2037 2020 2020   |.|   0 4 7    
│ │ │ │ +0001dbf0: 2020 3020 3120 3420 3720 2020 2020 2030    0 1 4 7      0
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│ │ │ │ +0001dc10: 2020 2020 2020 3120 3220 3520 3720 2020        1 2 5 7   
│ │ │ │ +0001dc20: 2020 2030 2033 2035 2037 2020 2020 2020     0 3 5 7      
│ │ │ │ +0001dc30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
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│ │ │ │  0001dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dc80: 2d7c 0a7c 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 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0001dcc0: 2032 2020 2020 2020 2020 207c 0a7c 3238   2         |.|28
│ │ │ │ -0001dcd0: 7820 7820 7820 7820 202b 2034 7820 7820  x x x x  + 4x x 
│ │ │ │ -0001dce0: 7820 7820 202d 2032 3678 2078 2078 2078  x x  - 26x x x x
│ │ │ │ -0001dcf0: 2020 2b20 3132 7820 7820 7820 7820 202b    + 12x x x x  +
│ │ │ │ -0001dd00: 2032 3878 2078 2078 2020 2d20 3132 7820   28x x x  - 12x 
│ │ │ │ -0001dd10: 7820 7820 202d 2020 2020 207c 0a7c 2020  x x  -     |.|  
│ │ │ │ -0001dd20: 2031 2033 2035 2037 2020 2020 2032 2033   1 3 5 7     2 3
│ │ │ │ -0001dd30: 2035 2037 2020 2020 2020 3020 3420 3520   5 7      0 4 5 
│ │ │ │ -0001dd40: 3720 2020 2020 2033 2034 2035 2037 2020  7      3 4 5 7  
│ │ │ │ -0001dd50: 2020 2020 3020 3520 3720 2020 2020 2032      0 5 7      2
│ │ │ │ -0001dd60: 2035 2037 2020 2020 2020 207c 0a7c 2d2d   5 7       |.|--
│ │ │ │ -0001dd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dcb0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0001dcc0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001dcd0: 207c 0a7c 3238 7820 7820 7820 7820 202b   |.|28x x x x  +
│ │ │ │ +0001dce0: 2034 7820 7820 7820 7820 202d 2032 3678   4x x x x  - 26x
│ │ │ │ +0001dcf0: 2078 2078 2078 2020 2b20 3132 7820 7820   x x x  + 12x x 
│ │ │ │ +0001dd00: 7820 7820 202b 2032 3878 2078 2078 2020  x x  + 28x x x  
│ │ │ │ +0001dd10: 2d20 3132 7820 7820 7820 202d 2020 2020  - 12x x x  -    
│ │ │ │ +0001dd20: 207c 0a7c 2020 2031 2033 2035 2037 2020   |.|   1 3 5 7  
│ │ │ │ +0001dd30: 2020 2032 2033 2035 2037 2020 2020 2020     2 3 5 7      
│ │ │ │ +0001dd40: 3020 3420 3520 3720 2020 2020 2033 2034  0 4 5 7      3 4
│ │ │ │ +0001dd50: 2035 2037 2020 2020 2020 3020 3520 3720   5 7      0 5 7 
│ │ │ │ +0001dd60: 2020 2020 2032 2035 2037 2020 2020 2020       2 5 7      
│ │ │ │ +0001dd70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001dd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ddc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0001ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ddf0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001de00: 2020 2020 2020 2020 2020 207c 0a7c 3231             |.|21
│ │ │ │ -0001de10: 7820 7820 7820 7820 202d 2032 3878 2078  x x x x  - 28x x
│ │ │ │ -0001de20: 2078 2078 2020 2b20 3337 7820 7820 7820   x x  + 37x x x 
│ │ │ │ -0001de30: 7820 202b 2032 3178 2078 2078 2078 2020  x  + 21x x x x  
│ │ │ │ -0001de40: 2d20 3131 7820 7820 7820 202d 2032 3878  - 11x x x  - 28x
│ │ │ │ -0001de50: 2078 2078 2078 2020 2d20 207c 0a7c 2020   x x x  -  |.|  
│ │ │ │ -0001de60: 2030 2032 2036 2037 2020 2020 2020 3120   0 2 6 7      1 
│ │ │ │ -0001de70: 3220 3620 3720 2020 2020 2030 2033 2036  2 6 7      0 3 6
│ │ │ │ -0001de80: 2037 2020 2020 2020 3220 3320 3620 3720   7      2 3 6 7 
│ │ │ │ -0001de90: 2020 2020 2033 2036 2037 2020 2020 2020       3 6 7      
│ │ │ │ -0001dea0: 3020 3420 3620 3720 2020 207c 0a7c 2d2d  0 4 6 7    |.|--
│ │ │ │ -0001deb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ddf0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0001de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001de10: 207c 0a7c 3231 7820 7820 7820 7820 202d   |.|21x x x x  -
│ │ │ │ +0001de20: 2032 3878 2078 2078 2078 2020 2b20 3337   28x x x x  + 37
│ │ │ │ +0001de30: 7820 7820 7820 7820 202b 2032 3178 2078  x x x x  + 21x x
│ │ │ │ +0001de40: 2078 2078 2020 2d20 3131 7820 7820 7820   x x  - 11x x x 
│ │ │ │ +0001de50: 202d 2032 3878 2078 2078 2078 2020 2d20   - 28x x x x  - 
│ │ │ │ +0001de60: 207c 0a7c 2020 2030 2032 2036 2037 2020   |.|   0 2 6 7  
│ │ │ │ +0001de70: 2020 2020 3120 3220 3620 3720 2020 2020      1 2 6 7     
│ │ │ │ +0001de80: 2030 2033 2036 2037 2020 2020 2020 3220   0 3 6 7      2 
│ │ │ │ +0001de90: 3320 3620 3720 2020 2020 2033 2036 2037  3 6 7      3 6 7
│ │ │ │ +0001dea0: 2020 2020 2020 3020 3420 3620 3720 2020        0 4 6 7   
│ │ │ │ +0001deb0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
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│ │ │ │  0001ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001df00: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0001df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df40: 2020 2020 2020 2020 2020 207c 0a7c 3231             |.|21
│ │ │ │ -0001df50: 7820 7820 7820 7820 202b 2033 7820 7820  x x x x  + 3x x 
│ │ │ │ -0001df60: 7820 7820 202b 2034 3778 2078 2078 2078  x x  + 47x x x x
│ │ │ │ -0001df70: 2020 2b20 3438 7820 7820 7820 7820 202b    + 48x x x x  +
│ │ │ │ -0001df80: 2032 7820 7820 7820 7820 202d 2034 3578   2x x x x  - 45x
│ │ │ │ -0001df90: 2078 2078 2078 2020 2d20 207c 0a7c 2020   x x x  -  |.|  
│ │ │ │ -0001dfa0: 2031 2034 2036 2037 2020 2020 2030 2035   1 4 6 7     0 5
│ │ │ │ -0001dfb0: 2036 2037 2020 2020 2020 3120 3520 3620   6 7      1 5 6 
│ │ │ │ -0001dfc0: 3720 2020 2020 2032 2035 2036 2037 2020  7      2 5 6 7  
│ │ │ │ -0001dfd0: 2020 2033 2035 2036 2037 2020 2020 2020     3 5 6 7      
│ │ │ │ -0001dfe0: 3420 3520 3620 3720 2020 207c 0a7c 2d2d  4 5 6 7    |.|--
│ │ │ │ -0001dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001df50: 207c 0a7c 3231 7820 7820 7820 7820 202b   |.|21x x x x  +
│ │ │ │ +0001df60: 2033 7820 7820 7820 7820 202b 2034 3778   3x x x x  + 47x
│ │ │ │ +0001df70: 2078 2078 2078 2020 2b20 3438 7820 7820   x x x  + 48x x 
│ │ │ │ +0001df80: 7820 7820 202b 2032 7820 7820 7820 7820  x x  + 2x x x x 
│ │ │ │ +0001df90: 202d 2034 3578 2078 2078 2078 2020 2d20   - 45x x x x  - 
│ │ │ │ +0001dfa0: 207c 0a7c 2020 2031 2034 2036 2037 2020   |.|   1 4 6 7  
│ │ │ │ +0001dfb0: 2020 2030 2035 2036 2037 2020 2020 2020     0 5 6 7      
│ │ │ │ +0001dfc0: 3120 3520 3620 3720 2020 2020 2032 2035  1 5 6 7      2 5
│ │ │ │ +0001dfd0: 2036 2037 2020 2020 2033 2035 2036 2037   6 7     3 5 6 7
│ │ │ │ +0001dfe0: 2020 2020 2020 3420 3520 3620 3720 2020        4 5 6 7   
│ │ │ │ +0001dff0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001e040: 2032 2020 2020 2020 2020 2020 3220 3220   2          2 2 
│ │ │ │ -0001e050: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -0001e060: 3220 3220 2020 2020 2020 2020 2032 2020  2 2          2  
│ │ │ │ -0001e070: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ -0001e080: 2032 2020 2020 2020 2020 207c 0a7c 3333   2         |.|33
│ │ │ │ -0001e090: 7820 7820 7820 202d 2033 3878 2078 2020  x x x  - 38x x  
│ │ │ │ -0001e0a0: 2b20 3238 7820 7820 7820 202d 2033 3878  + 28x x x  - 38x
│ │ │ │ -0001e0b0: 2078 2020 2d20 3231 7820 7820 7820 202b   x  - 21x x x  +
│ │ │ │ -0001e0c0: 2034 3878 2078 2020 2d20 3438 7820 7820   48x x  - 48x x 
│ │ │ │ -0001e0d0: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -0001e0e0: 2035 2036 2037 2020 2020 2020 3020 3720   5 6 7      0 7 
│ │ │ │ -0001e0f0: 2020 2020 2030 2031 2037 2020 2020 2020       0 1 7      
│ │ │ │ -0001e100: 3120 3720 2020 2020 2030 2033 2037 2020  1 7      0 3 7  
│ │ │ │ -0001e110: 2020 2020 3320 3720 2020 2020 2030 2035      3 7      0 5
│ │ │ │ -0001e120: 2037 2020 2020 2020 2020 207c 0a7c 2d2d   7         |.|--
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│ │ │ │ +0001e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e040: 2d7c 0a7c 2020 2032 2020 2020 2020 2020  -|.|   2        
│ │ │ │ +0001e050: 2020 3220 3220 2020 2020 2020 2020 2032    2 2          2
│ │ │ │ +0001e060: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ +0001e070: 2020 2032 2020 2020 2020 3220 3220 2020     2      2 2   
│ │ │ │ +0001e080: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001e090: 207c 0a7c 3333 7820 7820 7820 202d 2033   |.|33x x x  - 3
│ │ │ │ +0001e0a0: 3878 2078 2020 2b20 3238 7820 7820 7820  8x x  + 28x x x 
│ │ │ │ +0001e0b0: 202d 2033 3878 2078 2020 2d20 3231 7820   - 38x x  - 21x 
│ │ │ │ +0001e0c0: 7820 7820 202b 2034 3878 2078 2020 2d20  x x  + 48x x  - 
│ │ │ │ +0001e0d0: 3438 7820 7820 7820 202b 2020 2020 2020  48x x x  +      
│ │ │ │ +0001e0e0: 207c 0a7c 2020 2035 2036 2037 2020 2020   |.|   5 6 7    
│ │ │ │ +0001e0f0: 2020 3020 3720 2020 2020 2030 2031 2037    0 7      0 1 7
│ │ │ │ +0001e100: 2020 2020 2020 3120 3720 2020 2020 2030        1 7      0
│ │ │ │ +0001e110: 2033 2037 2020 2020 2020 3320 3720 2020   3 7      3 7   
│ │ │ │ +0001e120: 2020 2030 2035 2037 2020 2020 2020 2020     0 5 7        
│ │ │ │ +0001e130: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001e180: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001e190: 3220 2020 2020 2032 2032 2020 2020 2020  2      2 2      
│ │ │ │ -0001e1a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0001e1b0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0001e1c0: 2020 2020 2020 2020 2020 207c 0a7c 3435             |.|45
│ │ │ │ -0001e1d0: 7820 7820 7820 202d 2034 3778 2078 2078  x x x  - 47x x x
│ │ │ │ -0001e1e0: 2020 2d20 3438 7820 7820 202b 2031 3178    - 48x x  + 11x
│ │ │ │ -0001e1f0: 2078 2078 2078 2020 2d20 3130 7820 7820   x x x  - 10x x 
│ │ │ │ -0001e200: 7820 202b 2032 3078 2078 2078 2020 2b20  x  + 20x x x  + 
│ │ │ │ -0001e210: 3131 7820 7820 7820 7820 207c 0a7c 2020  11x x x x  |.|  
│ │ │ │ -0001e220: 2031 2035 2037 2020 2020 2020 3320 3520   1 5 7      3 5 
│ │ │ │ -0001e230: 3720 2020 2020 2035 2037 2020 2020 2020  7      5 7      
│ │ │ │ -0001e240: 3020 3120 3220 3820 2020 2020 2031 2032  0 1 2 8      1 2
│ │ │ │ -0001e250: 2038 2020 2020 2020 3120 3220 3820 2020   8      1 2 8   
│ │ │ │ -0001e260: 2020 2030 2031 2033 2038 207c 0a7c 2d2d     0 1 3 8 |.|--
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│ │ │ │ +0001e180: 2d7c 0a7c 2020 2020 2020 2032 2020 2020  -|.|       2    
│ │ │ │ +0001e190: 2020 2020 2020 3220 2020 2020 2032 2032        2      2 2
│ │ │ │ +0001e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1b0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0001e1c0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001e1d0: 207c 0a7c 3435 7820 7820 7820 202d 2034   |.|45x x x  - 4
│ │ │ │ +0001e1e0: 3778 2078 2078 2020 2d20 3438 7820 7820  7x x x  - 48x x 
│ │ │ │ +0001e1f0: 202b 2031 3178 2078 2078 2078 2020 2d20   + 11x x x x  - 
│ │ │ │ +0001e200: 3130 7820 7820 7820 202b 2032 3078 2078  10x x x  + 20x x
│ │ │ │ +0001e210: 2078 2020 2b20 3131 7820 7820 7820 7820   x  + 11x x x x 
│ │ │ │ +0001e220: 207c 0a7c 2020 2031 2035 2037 2020 2020   |.|   1 5 7    
│ │ │ │ +0001e230: 2020 3320 3520 3720 2020 2020 2035 2037    3 5 7      5 7
│ │ │ │ +0001e240: 2020 2020 2020 3020 3120 3220 3820 2020        0 1 2 8   
│ │ │ │ +0001e250: 2020 2031 2032 2038 2020 2020 2020 3120     1 2 8      1 
│ │ │ │ +0001e260: 3220 3820 2020 2020 2030 2031 2033 2038  2 8      0 1 3 8
│ │ │ │ +0001e270: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001e2c0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0001e2d0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0001e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e2c0: 2d7c 0a7c 2020 2020 2032 2020 2020 2020  -|.|     2      
│ │ │ │ +0001e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e2e0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e300: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -0001e310: 3432 7820 7820 7820 202b 2034 3178 2078  42x x x  + 41x x
│ │ │ │ -0001e320: 2078 2078 2020 2d20 3131 7820 7820 7820   x x  - 11x x x 
│ │ │ │ -0001e330: 202b 2031 3078 2078 2078 2078 2020 2d20   + 10x x x x  - 
│ │ │ │ -0001e340: 3230 7820 7820 7820 7820 202d 2032 3678  20x x x x  - 26x
│ │ │ │ -0001e350: 2078 2078 2078 2020 2b20 207c 0a7c 2020   x x x  +  |.|  
│ │ │ │ -0001e360: 2020 2031 2033 2038 2020 2020 2020 3120     1 3 8      1 
│ │ │ │ -0001e370: 3220 3320 3820 2020 2020 2030 2034 2038  2 3 8      0 4 8
│ │ │ │ -0001e380: 2020 2020 2020 3020 3120 3420 3820 2020        0 1 4 8   
│ │ │ │ -0001e390: 2020 2030 2032 2034 2038 2020 2020 2020     0 2 4 8      
│ │ │ │ -0001e3a0: 3120 3220 3420 3820 2020 207c 0a7c 2d2d  1 2 4 8    |.|--
│ │ │ │ -0001e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e310: 207c 0a7c 2b20 3432 7820 7820 7820 202b   |.|+ 42x x x  +
│ │ │ │ +0001e320: 2034 3178 2078 2078 2078 2020 2d20 3131   41x x x x  - 11
│ │ │ │ +0001e330: 7820 7820 7820 202b 2031 3078 2078 2078  x x x  + 10x x x
│ │ │ │ +0001e340: 2078 2020 2d20 3230 7820 7820 7820 7820   x  - 20x x x x 
│ │ │ │ +0001e350: 202d 2032 3678 2078 2078 2078 2020 2b20   - 26x x x x  + 
│ │ │ │ +0001e360: 207c 0a7c 2020 2020 2031 2033 2038 2020   |.|     1 3 8  
│ │ │ │ +0001e370: 2020 2020 3120 3220 3320 3820 2020 2020      1 2 3 8     
│ │ │ │ +0001e380: 2030 2034 2038 2020 2020 2020 3020 3120   0 4 8      0 1 
│ │ │ │ +0001e390: 3420 3820 2020 2020 2030 2032 2034 2038  4 8      0 2 4 8
│ │ │ │ +0001e3a0: 2020 2020 2020 3120 3220 3420 3820 2020        1 2 4 8   
│ │ │ │ +0001e3b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e410: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0001e420: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0001e430: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0001e440: 2020 2020 2020 2020 2020 207c 0a7c 3238             |.|28
│ │ │ │ -0001e450: 7820 7820 7820 7820 202b 2034 7820 7820  x x x x  + 4x x 
│ │ │ │ -0001e460: 7820 7820 202b 2032 3678 2078 2078 2020  x x  + 26x x x  
│ │ │ │ -0001e470: 2d20 3132 7820 7820 7820 202d 2031 3178  - 12x x x  - 11x
│ │ │ │ -0001e480: 2078 2078 2020 2d20 3432 7820 7820 7820   x x  - 42x x x 
│ │ │ │ -0001e490: 7820 202d 2020 2020 2020 207c 0a7c 2020  x  -       |.|  
│ │ │ │ -0001e4a0: 2031 2033 2034 2038 2020 2020 2032 2033   1 3 4 8     2 3
│ │ │ │ -0001e4b0: 2034 2038 2020 2020 2020 3020 3420 3820   4 8      0 4 8 
│ │ │ │ -0001e4c0: 2020 2020 2033 2034 2038 2020 2020 2020       3 4 8      
│ │ │ │ -0001e4d0: 3020 3520 3820 2020 2020 2030 2031 2035  0 5 8      0 1 5
│ │ │ │ -0001e4e0: 2038 2020 2020 2020 2020 207c 0a7c 2d2d   8         |.|--
│ │ │ │ -0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e400: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0001e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e420: 2020 3220 2020 2020 2020 2020 2032 2020    2          2  
│ │ │ │ +0001e430: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0001e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e450: 207c 0a7c 3238 7820 7820 7820 7820 202b   |.|28x x x x  +
│ │ │ │ +0001e460: 2034 7820 7820 7820 7820 202b 2032 3678   4x x x x  + 26x
│ │ │ │ +0001e470: 2078 2078 2020 2d20 3132 7820 7820 7820   x x  - 12x x x 
│ │ │ │ +0001e480: 202d 2031 3178 2078 2078 2020 2d20 3432   - 11x x x  - 42
│ │ │ │ +0001e490: 7820 7820 7820 7820 202d 2020 2020 2020  x x x x  -      
│ │ │ │ +0001e4a0: 207c 0a7c 2020 2031 2033 2034 2038 2020   |.|   1 3 4 8  
│ │ │ │ +0001e4b0: 2020 2032 2033 2034 2038 2020 2020 2020     2 3 4 8      
│ │ │ │ +0001e4c0: 3020 3420 3820 2020 2020 2033 2034 2038  0 4 8      3 4 8
│ │ │ │ +0001e4d0: 2020 2020 2020 3020 3520 3820 2020 2020        0 5 8     
│ │ │ │ +0001e4e0: 2030 2031 2035 2038 2020 2020 2020 2020   0 1 5 8        
│ │ │ │ +0001e4f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001e540: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0001e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e540: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0001e550: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0001e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e580: 2020 2020 2020 2020 2020 207c 0a7c 3431             |.|41
│ │ │ │ -0001e590: 7820 7820 7820 7820 202d 2034 7820 7820  x x x x  - 4x x 
│ │ │ │ -0001e5a0: 7820 202d 2032 3878 2078 2078 2078 2020  x  - 28x x x x  
│ │ │ │ -0001e5b0: 2b20 3132 7820 7820 7820 7820 202b 2033  + 12x x x x  + 3
│ │ │ │ -0001e5c0: 3778 2078 2078 2078 2020 2b20 3378 2078  7x x x x  + 3x x
│ │ │ │ -0001e5d0: 2078 2078 2020 2d20 2020 207c 0a7c 2020   x x  -    |.|  
│ │ │ │ -0001e5e0: 2030 2032 2035 2038 2020 2020 2032 2035   0 2 5 8     2 5
│ │ │ │ -0001e5f0: 2038 2020 2020 2020 3020 3420 3520 3820   8      0 4 5 8 
│ │ │ │ -0001e600: 2020 2020 2032 2034 2035 2038 2020 2020       2 4 5 8    
│ │ │ │ -0001e610: 2020 3020 3220 3620 3820 2020 2020 3120    0 2 6 8     1 
│ │ │ │ -0001e620: 3220 3620 3820 2020 2020 207c 0a7c 2d2d  2 6 8      |.|--
│ │ │ │ -0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e590: 207c 0a7c 3431 7820 7820 7820 7820 202d   |.|41x x x x  -
│ │ │ │ +0001e5a0: 2034 7820 7820 7820 202d 2032 3878 2078   4x x x  - 28x x
│ │ │ │ +0001e5b0: 2078 2078 2020 2b20 3132 7820 7820 7820   x x  + 12x x x 
│ │ │ │ +0001e5c0: 7820 202b 2033 3778 2078 2078 2078 2020  x  + 37x x x x  
│ │ │ │ +0001e5d0: 2b20 3378 2078 2078 2078 2020 2d20 2020  + 3x x x x  -   
│ │ │ │ +0001e5e0: 207c 0a7c 2020 2030 2032 2035 2038 2020   |.|   0 2 5 8  
│ │ │ │ +0001e5f0: 2020 2032 2035 2038 2020 2020 2020 3020     2 5 8      0 
│ │ │ │ +0001e600: 3420 3520 3820 2020 2020 2032 2034 2035  4 5 8      2 4 5
│ │ │ │ +0001e610: 2038 2020 2020 2020 3020 3220 3620 3820   8      0 2 6 8 
│ │ │ │ +0001e620: 2020 2020 3120 3220 3620 3820 2020 2020      1 2 6 8     
│ │ │ │ +0001e630: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001e680: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0001e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e680: 2d7c 0a7c 2020 2032 2020 2020 2020 2020  -|.|   2        
│ │ │ │  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 207c 0a7c 3231             |.|21
│ │ │ │ -0001e6d0: 7820 7820 7820 202d 2031 3478 2078 2078  x x x  - 14x x x
│ │ │ │ -0001e6e0: 2078 2020 2b20 3239 7820 7820 7820 7820   x  + 29x x x x 
│ │ │ │ -0001e6f0: 202b 2031 3178 2078 2078 2078 2020 2b20   + 11x x x x  + 
│ │ │ │ -0001e700: 3437 7820 7820 7820 7820 202d 2034 3878  47x x x x  - 48x
│ │ │ │ -0001e710: 2078 2078 2078 2020 2d20 207c 0a7c 2020   x x x  -  |.|  
│ │ │ │ -0001e720: 2032 2036 2038 2020 2020 2020 3020 3320   2 6 8      0 3 
│ │ │ │ -0001e730: 3620 3820 2020 2020 2031 2033 2036 2038  6 8      1 3 6 8
│ │ │ │ -0001e740: 2020 2020 2020 3220 3320 3620 3820 2020        2 3 6 8   
│ │ │ │ -0001e750: 2020 2031 2034 2036 2038 2020 2020 2020     1 4 6 8      
│ │ │ │ -0001e760: 3220 3420 3620 3820 2020 207c 0a7c 2d2d  2 4 6 8    |.|--
│ │ │ │ -0001e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e6d0: 207c 0a7c 3231 7820 7820 7820 202d 2031   |.|21x x x  - 1
│ │ │ │ +0001e6e0: 3478 2078 2078 2078 2020 2b20 3239 7820  4x x x x  + 29x 
│ │ │ │ +0001e6f0: 7820 7820 7820 202b 2031 3178 2078 2078  x x x  + 11x x x
│ │ │ │ +0001e700: 2078 2020 2b20 3437 7820 7820 7820 7820   x  + 47x x x x 
│ │ │ │ +0001e710: 202d 2034 3878 2078 2078 2078 2020 2d20   - 48x x x x  - 
│ │ │ │ +0001e720: 207c 0a7c 2020 2032 2036 2038 2020 2020   |.|   2 6 8    
│ │ │ │ +0001e730: 2020 3020 3320 3620 3820 2020 2020 2031    0 3 6 8      1
│ │ │ │ +0001e740: 2033 2036 2038 2020 2020 2020 3220 3320   3 6 8      2 3 
│ │ │ │ +0001e750: 3620 3820 2020 2020 2031 2034 2036 2038  6 8      1 4 6 8
│ │ │ │ +0001e760: 2020 2020 2020 3220 3420 3620 3820 2020        2 4 6 8   
│ │ │ │ +0001e770: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001e7c0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e7c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0001e7d0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0001e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7f0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0001e800: 2020 2020 2020 2020 2020 207c 0a7c 3278             |.|2x
│ │ │ │ -0001e810: 2078 2078 2078 2020 2b20 3435 7820 7820   x x x  + 45x x 
│ │ │ │ -0001e820: 7820 202b 2032 3978 2078 2078 2078 2020  x  + 29x x x x  
│ │ │ │ -0001e830: 2b20 3235 7820 7820 7820 7820 202b 2033  + 25x x x x  + 3
│ │ │ │ -0001e840: 3378 2078 2078 2078 2020 2d20 3337 7820  3x x x x  - 37x 
│ │ │ │ -0001e850: 7820 7820 202d 2020 2020 207c 0a7c 2020  x x  -     |.|  
│ │ │ │ -0001e860: 3320 3420 3620 3820 2020 2020 2034 2036  3 4 6 8      4 6
│ │ │ │ -0001e870: 2038 2020 2020 2020 3020 3520 3620 3820   8      0 5 6 8 
│ │ │ │ -0001e880: 2020 2020 2031 2035 2036 2038 2020 2020       1 5 6 8    
│ │ │ │ -0001e890: 2020 3420 3520 3620 3820 2020 2020 2030    4 5 6 8      0
│ │ │ │ -0001e8a0: 2037 2038 2020 2020 2020 207c 0a7c 2d2d   7 8       |.|--
│ │ │ │ -0001e8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e800: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0001e810: 207c 0a7c 3278 2078 2078 2078 2020 2b20   |.|2x x x x  + 
│ │ │ │ +0001e820: 3435 7820 7820 7820 202b 2032 3978 2078  45x x x  + 29x x
│ │ │ │ +0001e830: 2078 2078 2020 2b20 3235 7820 7820 7820   x x  + 25x x x 
│ │ │ │ +0001e840: 7820 202b 2033 3378 2078 2078 2078 2020  x  + 33x x x x  
│ │ │ │ +0001e850: 2d20 3337 7820 7820 7820 202d 2020 2020  - 37x x x  -    
│ │ │ │ +0001e860: 207c 0a7c 2020 3320 3420 3620 3820 2020   |.|  3 4 6 8   
│ │ │ │ +0001e870: 2020 2034 2036 2038 2020 2020 2020 3020     4 6 8      0 
│ │ │ │ +0001e880: 3520 3620 3820 2020 2020 2031 2035 2036  5 6 8      1 5 6
│ │ │ │ +0001e890: 2038 2020 2020 2020 3420 3520 3620 3820   8      4 5 6 8 
│ │ │ │ +0001e8a0: 2020 2020 2030 2037 2038 2020 2020 2020       0 7 8      
│ │ │ │ +0001e8b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001e8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001e900: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0001e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e900: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0001e910: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0001e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e940: 2020 2020 2020 2020 2020 207c 0a7c 3378             |.|3x
│ │ │ │ -0001e950: 2078 2078 2078 2020 2d20 3437 7820 7820   x x x  - 47x x 
│ │ │ │ -0001e960: 7820 202b 2032 3178 2078 2078 2078 2020  x  + 21x x x x  
│ │ │ │ -0001e970: 2b20 3131 7820 7820 7820 7820 202b 2078  + 11x x x x  + x
│ │ │ │ -0001e980: 2078 2078 2078 2020 2b20 3438 7820 7820   x x x  + 48x x 
│ │ │ │ -0001e990: 7820 7820 202d 2020 2020 207c 0a7c 2020  x x  -     |.|  
│ │ │ │ -0001e9a0: 3020 3120 3720 3820 2020 2020 2031 2037  0 1 7 8      1 7
│ │ │ │ -0001e9b0: 2038 2020 2020 2020 3020 3220 3720 3820   8      0 2 7 8 
│ │ │ │ -0001e9c0: 2020 2020 2030 2033 2037 2038 2020 2020       0 3 7 8    
│ │ │ │ -0001e9d0: 3220 3320 3720 3820 2020 2020 2030 2034  2 3 7 8      0 4
│ │ │ │ -0001e9e0: 2037 2038 2020 2020 2020 207c 0a7c 2d2d   7 8       |.|--
│ │ │ │ -0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e950: 207c 0a7c 3378 2078 2078 2078 2020 2d20   |.|3x x x x  - 
│ │ │ │ +0001e960: 3437 7820 7820 7820 202b 2032 3178 2078  47x x x  + 21x x
│ │ │ │ +0001e970: 2078 2078 2020 2b20 3131 7820 7820 7820   x x  + 11x x x 
│ │ │ │ +0001e980: 7820 202b 2078 2078 2078 2078 2020 2b20  x  + x x x x  + 
│ │ │ │ +0001e990: 3438 7820 7820 7820 7820 202d 2020 2020  48x x x x  -    
│ │ │ │ +0001e9a0: 207c 0a7c 2020 3020 3120 3720 3820 2020   |.|  0 1 7 8   
│ │ │ │ +0001e9b0: 2020 2031 2037 2038 2020 2020 2020 3020     1 7 8      0 
│ │ │ │ +0001e9c0: 3220 3720 3820 2020 2020 2030 2033 2037  2 7 8      0 3 7
│ │ │ │ +0001e9d0: 2038 2020 2020 3220 3320 3720 3820 2020   8    2 3 7 8   
│ │ │ │ +0001e9e0: 2020 2030 2034 2037 2038 2020 2020 2020     0 4 7 8      
│ │ │ │ +0001e9f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ea40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0001ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea80: 2020 2020 2020 2020 2020 207c 0a7c 3435             |.|45
│ │ │ │ -0001ea90: 7820 7820 7820 7820 202b 2034 3778 2078  x x x x  + 47x x
│ │ │ │ -0001eaa0: 2078 2078 2020 2d20 3278 2078 2078 2078   x x  - 2x x x x
│ │ │ │ -0001eab0: 2020 2b20 3333 7820 7820 7820 7820 202b    + 33x x x x  +
│ │ │ │ -0001eac0: 2034 3778 2078 2078 2078 2020 2d20 7820   47x x x x  - x 
│ │ │ │ -0001ead0: 7820 7820 7820 202b 2020 207c 0a7c 2020  x x x  +   |.|  
│ │ │ │ -0001eae0: 2031 2034 2037 2038 2020 2020 2020 3320   1 4 7 8      3 
│ │ │ │ -0001eaf0: 3420 3720 3820 2020 2020 3020 3520 3720  4 7 8     0 5 7 
│ │ │ │ -0001eb00: 3820 2020 2020 2031 2035 2037 2038 2020  8      1 5 7 8  
│ │ │ │ -0001eb10: 2020 2020 3220 3520 3720 3820 2020 2034      2 5 7 8    4
│ │ │ │ -0001eb20: 2035 2037 2038 2020 2020 207c 0a7c 2d2d   5 7 8     |.|--
│ │ │ │ -0001eb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ea90: 207c 0a7c 3435 7820 7820 7820 7820 202b   |.|45x x x x  +
│ │ │ │ +0001eaa0: 2034 3778 2078 2078 2078 2020 2d20 3278   47x x x x  - 2x
│ │ │ │ +0001eab0: 2078 2078 2078 2020 2b20 3333 7820 7820   x x x  + 33x x 
│ │ │ │ +0001eac0: 7820 7820 202b 2034 3778 2078 2078 2078  x x  + 47x x x x
│ │ │ │ +0001ead0: 2020 2d20 7820 7820 7820 7820 202b 2020    - x x x x  +  
│ │ │ │ +0001eae0: 207c 0a7c 2020 2031 2034 2037 2038 2020   |.|   1 4 7 8  
│ │ │ │ +0001eaf0: 2020 2020 3320 3420 3720 3820 2020 2020      3 4 7 8     
│ │ │ │ +0001eb00: 3020 3520 3720 3820 2020 2020 2031 2035  0 5 7 8      1 5
│ │ │ │ +0001eb10: 2037 2038 2020 2020 2020 3220 3520 3720   7 8      2 5 7 
│ │ │ │ +0001eb20: 3820 2020 2034 2035 2037 2038 2020 2020  8    4 5 7 8    
│ │ │ │ +0001eb30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001eb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001eb80: 3220 3220 2020 2020 2020 2020 2032 2020  2 2          2  
│ │ │ │ -0001eb90: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ -0001eba0: 2032 2020 2020 2020 3220 3220 2020 2020   2      2 2     
│ │ │ │ -0001ebb0: 2020 2020 3220 2020 2020 2020 2020 2032      2          2
│ │ │ │ -0001ebc0: 2020 2020 2020 2020 2020 327c 0a7c 3778            2|.|7x
│ │ │ │ -0001ebd0: 2078 2020 2d20 3239 7820 7820 7820 202b   x  - 29x x x  +
│ │ │ │ -0001ebe0: 2033 3678 2078 2020 2d20 3131 7820 7820   36x x  - 11x x 
│ │ │ │ -0001ebf0: 7820 202b 2034 3878 2078 2020 2b20 3278  x  + 48x x  + 2x
│ │ │ │ -0001ec00: 2078 2078 2020 2d20 3333 7820 7820 7820   x x  - 33x x x 
│ │ │ │ -0001ec10: 202d 2034 3778 2078 2078 207c 0a7c 2020   - 47x x x |.|  
│ │ │ │ -0001ec20: 3020 3820 2020 2020 2030 2031 2038 2020  0 8      0 1 8  
│ │ │ │ -0001ec30: 2020 2020 3120 3820 2020 2020 2030 2032      1 8      0 2
│ │ │ │ -0001ec40: 2038 2020 2020 2020 3220 3820 2020 2020   8      2 8     
│ │ │ │ -0001ec50: 3020 3420 3820 2020 2020 2031 2034 2038  0 4 8      1 4 8
│ │ │ │ -0001ec60: 2020 2020 2020 3220 3420 387c 0a7c 2d2d        2 4 8|.|--
│ │ │ │ -0001ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eb80: 2d7c 0a7c 2020 3220 3220 2020 2020 2020  -|.|  2 2       
│ │ │ │ +0001eb90: 2020 2032 2020 2020 2020 3220 3220 2020     2      2 2   
│ │ │ │ +0001eba0: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ +0001ebb0: 3220 2020 2020 2020 2020 3220 2020 2020  2         2     
│ │ │ │ +0001ebc0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0001ebd0: 327c 0a7c 3778 2078 2020 2d20 3239 7820  2|.|7x x  - 29x 
│ │ │ │ +0001ebe0: 7820 7820 202b 2033 3678 2078 2020 2d20  x x  + 36x x  - 
│ │ │ │ +0001ebf0: 3131 7820 7820 7820 202b 2034 3878 2078  11x x x  + 48x x
│ │ │ │ +0001ec00: 2020 2b20 3278 2078 2078 2020 2d20 3333    + 2x x x  - 33
│ │ │ │ +0001ec10: 7820 7820 7820 202d 2034 3778 2078 2078  x x x  - 47x x x
│ │ │ │ +0001ec20: 207c 0a7c 2020 3020 3820 2020 2020 2030   |.|  0 8      0
│ │ │ │ +0001ec30: 2031 2038 2020 2020 2020 3120 3820 2020   1 8      1 8   
│ │ │ │ +0001ec40: 2020 2030 2032 2038 2020 2020 2020 3220     0 2 8      2 
│ │ │ │ +0001ec50: 3820 2020 2020 3020 3420 3820 2020 2020  8     0 4 8     
│ │ │ │ +0001ec60: 2031 2034 2038 2020 2020 2020 3220 3420   1 4 8      2 4 
│ │ │ │ +0001ec70: 387c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  8|.|------------
│ │ │ │  0001ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0001ecc0: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ +0001ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ecc0: 2d7c 0a7c 2020 2020 2032 2032 2020 2020  -|.|     2 2    
│ │ │ │  0001ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed00: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -0001ed10: 3438 7820 7820 202b 2034 3878 2078 2078  48x x  + 48x x x
│ │ │ │ -0001ed20: 2078 2020 2d20 3438 7820 7820 7820 7820   x  - 48x x x x 
│ │ │ │ -0001ed30: 202d 2034 3878 2078 2078 2078 2020 2b20   - 48x x x x  + 
│ │ │ │ -0001ed40: 3438 7820 7820 7820 7820 202b 2034 3878  48x x x x  + 48x
│ │ │ │ -0001ed50: 2078 2078 2078 2020 2d20 207c 0a7c 2020   x x x  -  |.|  
│ │ │ │ -0001ed60: 2020 2034 2038 2020 2020 2020 3320 3420     4 8      3 4 
│ │ │ │ -0001ed70: 3620 3920 2020 2020 2032 2035 2036 2039  6 9      2 5 6 9
│ │ │ │ -0001ed80: 2020 2020 2020 3120 3320 3720 3920 2020        1 3 7 9   
│ │ │ │ -0001ed90: 2020 2030 2035 2037 2039 2020 2020 2020     0 5 7 9      
│ │ │ │ -0001eda0: 3120 3220 3820 3920 2020 207c 0a7c 2d2d  1 2 8 9    |.|--
│ │ │ │ -0001edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ed10: 207c 0a7c 2d20 3438 7820 7820 202b 2034   |.|- 48x x  + 4
│ │ │ │ +0001ed20: 3878 2078 2078 2078 2020 2d20 3438 7820  8x x x x  - 48x 
│ │ │ │ +0001ed30: 7820 7820 7820 202d 2034 3878 2078 2078  x x x  - 48x x x
│ │ │ │ +0001ed40: 2078 2020 2b20 3438 7820 7820 7820 7820   x  + 48x x x x 
│ │ │ │ +0001ed50: 202b 2034 3878 2078 2078 2078 2020 2d20   + 48x x x x  - 
│ │ │ │ +0001ed60: 207c 0a7c 2020 2020 2034 2038 2020 2020   |.|     4 8    
│ │ │ │ +0001ed70: 2020 3320 3420 3620 3920 2020 2020 2032    3 4 6 9      2
│ │ │ │ +0001ed80: 2035 2036 2039 2020 2020 2020 3120 3320   5 6 9      1 3 
│ │ │ │ +0001ed90: 3720 3920 2020 2020 2030 2035 2037 2039  7 9      0 5 7 9
│ │ │ │ +0001eda0: 2020 2020 2020 3120 3220 3820 3920 2020        1 2 8 9   
│ │ │ │ +0001edb0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0001edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3438  -----------|.|48
│ │ │ │ -0001ee00: 7820 7820 7820 7820 2020 2020 2020 2020  x x x x         
│ │ │ │ +0001edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ee00: 2d7c 0a7c 3438 7820 7820 7820 7820 2020  -|.|48x x x x   
│ │ │ │  0001ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001ee50: 2030 2034 2038 2039 2020 2020 2020 2020   0 4 8 9        
│ │ │ │ +0001ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ee50: 207c 0a7c 2020 2030 2034 2038 2039 2020   |.|   0 4 8 9  
│ │ │ │  0001ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0001eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eea0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -0001eef0: 203a 2061 7373 6572 7428 7068 6920 2a20   : assert(phi * 
│ │ │ │ -0001ef00: 7073 6920 3d3d 2031 2061 6e64 2070 7369  psi == 1 and psi
│ │ │ │ -0001ef10: 202a 2070 6869 203d 3d20 3129 2020 2020   * phi == 1)    
│ │ │ │ -0001ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0001ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eef0: 2d2b 0a7c 6934 203a 2061 7373 6572 7428  -+.|i4 : assert(
│ │ │ │ +0001ef00: 7068 6920 2a20 7073 6920 3d3d 2031 2061  phi * psi == 1 a
│ │ │ │ +0001ef10: 6e64 2070 7369 202a 2070 6869 203d 3d20  nd psi * phi == 
│ │ │ │ +0001ef20: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │ +0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001ef50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ef60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ef70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ef80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -0001ef90: 203a 2074 696d 6520 7073 6927 203d 2061   : time psi' = a
│ │ │ │ -0001efa0: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ -0001efb0: 654d 6170 2870 6869 2c43 6f64 696d 4273  eMap(phi,CodimBs
│ │ │ │ -0001efc0: 496e 763d 3e35 293b 2020 2020 2020 2020  Inv=>5);        
│ │ │ │ -0001efd0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0001efe0: 2d20 7573 6564 2030 2e32 3138 3939 3973  - used 0.218999s
│ │ │ │ -0001eff0: 2028 6370 7529 3b20 302e 3134 3637 3133   (cpu); 0.146713
│ │ │ │ -0001f000: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -0001f010: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ -0001f020: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -0001f030: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -0001f040: 7273 654d 6170 3a20 7374 6570 2031 206f  rseMap: step 1 o
│ │ │ │ -0001f050: 6620 3320 2020 2020 2020 2020 2020 2020  f 3             
│ │ │ │ +0001ef80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ef90: 2d2b 0a7c 6935 203a 2074 696d 6520 7073  -+.|i5 : time ps
│ │ │ │ +0001efa0: 6927 203d 2061 7070 726f 7869 6d61 7465  i' = approximate
│ │ │ │ +0001efb0: 496e 7665 7273 654d 6170 2870 6869 2c43  InverseMap(phi,C
│ │ │ │ +0001efc0: 6f64 696d 4273 496e 763d 3e35 293b 2020  odimBsInv=>5);  
│ │ │ │ +0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001efe0: 207c 0a7c 202d 2d20 7573 6564 2030 2e32   |.| -- used 0.2
│ │ │ │ +0001eff0: 3530 3035 3973 2028 6370 7529 3b20 302e  50059s (cpu); 0.
│ │ │ │ +0001f000: 3137 3435 3973 2028 7468 7265 6164 293b  17459s (thread);
│ │ │ │ +0001f010: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +0001f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f030: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +0001f040: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +0001f050: 6570 2031 206f 6620 3320 2020 2020 2020  ep 1 of 3       
│ │ │ │  0001f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f070: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -0001f080: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -0001f090: 7273 654d 6170 3a20 7374 6570 2032 206f  rseMap: step 2 o
│ │ │ │ -0001f0a0: 6620 3320 2020 2020 2020 2020 2020 2020  f 3             
│ │ │ │ +0001f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f080: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +0001f090: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +0001f0a0: 6570 2032 206f 6620 3320 2020 2020 2020  ep 2 of 3       
│ │ │ │  0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0c0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -0001f0d0: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -0001f0e0: 7273 654d 6170 3a20 7374 6570 2033 206f  rseMap: step 3 o
│ │ │ │ -0001f0f0: 6620 3320 2020 2020 2020 2020 2020 2020  f 3             
│ │ │ │ +0001f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f0d0: 207c 0a7c 2d2d 2061 7070 726f 7869 6d61   |.|-- approxima
│ │ │ │ +0001f0e0: 7465 496e 7665 7273 654d 6170 3a20 7374  teInverseMap: st
│ │ │ │ +0001f0f0: 6570 2033 206f 6620 3320 2020 2020 2020  ep 3 of 3       
│ │ │ │  0001f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f110: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f120: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f160: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -0001f170: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ -0001f180: 7175 6164 7261 7469 6320 7261 7469 6f6e  quadratic ration
│ │ │ │ -0001f190: 616c 206d 6170 2066 726f 6d20 5050 5e38  al map from PP^8
│ │ │ │ -0001f1a0: 2074 6f20 6879 7065 7273 7572 6661 6365   to hypersurface
│ │ │ │ -0001f1b0: 2069 6e20 5050 5e39 2920 207c 0a2b 2d2d   in PP^9)  |.+--
│ │ │ │ -0001f1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f170: 207c 0a7c 6f35 203a 2052 6174 696f 6e61   |.|o5 : Rationa
│ │ │ │ +0001f180: 6c4d 6170 2028 7175 6164 7261 7469 6320  lMap (quadratic 
│ │ │ │ +0001f190: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ +0001f1a0: 6d20 5050 5e38 2074 6f20 6879 7065 7273  m PP^8 to hypers
│ │ │ │ +0001f1b0: 7572 6661 6365 2069 6e20 5050 5e39 2920  urface in PP^9) 
│ │ │ │ +0001f1c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001f1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -0001f210: 203a 2061 7373 6572 7428 7073 6920 3d3d   : assert(psi ==
│ │ │ │ -0001f220: 2070 7369 2729 2020 2020 2020 2020 2020   psi')          
│ │ │ │ +0001f200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f210: 2d2b 0a7c 6936 203a 2061 7373 6572 7428  -+.|i6 : assert(
│ │ │ │ +0001f220: 7073 6920 3d3d 2070 7369 2729 2020 2020  psi == psi')    
│ │ │ │  0001f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f250: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0001f260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f260: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0001f270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4120  -----------+..A 
│ │ │ │ -0001f2b0: 6d6f 7265 2063 6f6d 706c 6963 6174 6564  more complicated
│ │ │ │ -0001f2c0: 2065 7861 6d70 6c65 2069 7320 7468 6520   example is the 
│ │ │ │ -0001f2d0: 666f 6c6c 6f77 696e 6720 2868 6572 6520  following (here 
│ │ │ │ -0001f2e0: 2a6e 6f74 6520 696e 7665 7273 654d 6170  *note inverseMap
│ │ │ │ -0001f2f0: 3a20 696e 7665 7273 654d 6170 2c0a 7461  : inverseMap,.ta
│ │ │ │ -0001f300: 6b65 7320 6120 6c6f 7420 6f66 2074 696d  kes a lot of tim
│ │ │ │ -0001f310: 6521 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  e!)...+---------
│ │ │ │ +0001f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f2b0: 2d2b 0a0a 4120 6d6f 7265 2063 6f6d 706c  -+..A more compl
│ │ │ │ +0001f2c0: 6963 6174 6564 2065 7861 6d70 6c65 2069  icated example i
│ │ │ │ +0001f2d0: 7320 7468 6520 666f 6c6c 6f77 696e 6720  s the following 
│ │ │ │ +0001f2e0: 2868 6572 6520 2a6e 6f74 6520 696e 7665  (here *note inve
│ │ │ │ +0001f2f0: 7273 654d 6170 3a20 696e 7665 7273 654d  rseMap: inverseM
│ │ │ │ +0001f300: 6170 2c0a 7461 6b65 7320 6120 6c6f 7420  ap,.takes a lot 
│ │ │ │ +0001f310: 6f66 2074 696d 6521 292e 0a0a 2b2d 2d2d  of time!)...+---
│ │ │ │  0001f320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f360: 2d2d 2d2d 2b0a 7c69 3720 3a20 7068 6920  ----+.|i7 : phi 
│ │ │ │ -0001f370: 3d20 7261 7469 6f6e 616c 4d61 7020 6d61  = rationalMap ma
│ │ │ │ -0001f380: 7028 5038 2c5a 5a2f 3937 5b78 5f30 2e2e  p(P8,ZZ/97[x_0..
│ │ │ │ -0001f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +0001f370: 3a20 7068 6920 3d20 7261 7469 6f6e 616c  : phi = rational
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│ │ │ │ -0001f3b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001f3b0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │ -0001f400: 2020 2020 7c0a 7c6f 3720 3d20 2d2d 2072      |.|o7 = -- r
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│ │ │ │ -0001f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f400: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +0001f410: 3d20 2d2d 2072 6174 696f 6e61 6c20 6d61  = -- rational ma
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│ │ │ │  0001f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001f460: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ -0001f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f450: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  0001f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001f4b0: 6365 3a20 5072 6f6a 282d 2d5b 7420 2c20  ce: Proj(--[t , 
│ │ │ │ -0001f4c0: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ -0001f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f4a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f4b0: 2020 736f 7572 6365 3a20 5072 6f6a 282d    source: Proj(-
│ │ │ │ +0001f4c0: 2d5b 7420 2c20 7420 2c20 7420 2c20 7420  -[t , t , t , t 
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│ │ │ │  0001f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001f510: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -0001f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f4f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ +0001f510: 3720 2030 2020 2031 2020 2032 2020 2033  7  0   1   2   3
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│ │ │ │  0001f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f540: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001f540: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f560: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +0001f560: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │  0001f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f590: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ -0001f5a0: 6574 3a20 7375 6276 6172 6965 7479 206f  et: subvariety o
│ │ │ │ -0001f5b0: 6620 5072 6f6a 282d 2d5b 7820 2c20 7820  f Proj(--[x , x 
│ │ │ │ -0001f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f590: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f5a0: 2020 7461 7267 6574 3a20 7375 6276 6172    target: subvar
│ │ │ │ +0001f5b0: 6965 7479 206f 6620 5072 6f6a 282d 2d5b  iety of Proj(--[
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│ │ │ │  0001f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  0001f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f600: 2020 2020 2020 2039 3720 2030 2020 2031         97  0   1
│ │ │ │ -0001f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f600: 2020 2020 2020 2020 2020 2020 2039 3720               97 
│ │ │ │ +0001f610: 2030 2020 2031 2020 2020 2020 2020 2020   0   1          
│ │ │ │  0001f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f630: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001f640: 2020 2020 7b20 2020 2020 2020 2020 2020      {           
│ │ │ │ +0001f630: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f640: 2020 2020 2020 2020 2020 7b20 2020 2020            {     
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001f680: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6a0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001f6a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  0001f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001f6e0: 2020 2020 2078 2078 2020 2d20 3878 2078       x x  - 8x x
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│ │ │ │ -0001f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f6d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  0001f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001f730: 2020 2020 2020 3120 3320 2020 2020 3220        1 3     2 
│ │ │ │ -0001f740: 3320 2020 2020 2033 2020 2020 2020 3220  3      3      2 
│ │ │ │ -0001f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f720: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ +0001f750: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0001f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f770: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001f770: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ +0001f7c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f810: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001f820: 2020 2020 2078 2020 2b20 3137 7820 7820       x  + 17x x 
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│ │ │ │ -0001f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f810: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f820: 2020 2020 2020 2020 2020 2078 2020 2b20             x  + 
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│ │ │ │ +0001f840: 3133 7820 7820 2020 2020 2020 2020 2020  13x x           
│ │ │ │  0001f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f860: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ -0001f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f860: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f870: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0001f880: 2020 2032 2033 2020 2020 2020 3320 2020     2 3      3   
│ │ │ │ +0001f890: 2020 2032 2034 2020 2020 2020 2020 2020     2 4          
│ │ │ │  0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001f8b0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020                  
│ │ │ │  0001f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f900: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001f900: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f920: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0001f920: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0001f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f950: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001f960: 2020 2020 2078 2078 2020 2d20 3430 7820       x x  - 40x 
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│ │ │ │ -0001f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f950: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ +0001f970: 2d20 3430 7820 7820 202b 2032 3878 2020  - 40x x  + 28x  
│ │ │ │ +0001f980: 2d20 7820 7820 2020 2020 2020 2020 2020  - x x           
│ │ │ │  0001f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f9a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001f9b0: 2020 2020 2020 3120 3220 2020 2020 2032        1 2      2
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│ │ │ │ -0001f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f9a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f9b0: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
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│ │ │ │ +0001f9d0: 2020 2030 2034 2020 2020 2020 2020 2020     0 4          
│ │ │ │  0001f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f9f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001fa00: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +0001f9f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fa00: 2020 2020 2020 2020 2020 7d20 2020 2020            }     
│ │ │ │  0001fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fa40: 2020 2020 7c0a 7c20 2020 2020 6465 6669      |.|     defi
│ │ │ │ -0001fa50: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │ -0001fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fa40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fa50: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ +0001fa60: 3a20 7b20 2020 2020 2020 2020 2020 2020  : {             
│ │ │ │  0001fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fa90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001faa0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fa90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fab0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001faf0: 2020 2020 2020 2020 2020 2020 2074 2020               t  
│ │ │ │ -0001fb00: 2b20 7420 7420 202b 2074 2074 2020 2b20  + t t  + t t  + 
│ │ │ │ -0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb00: 2020 2074 2020 2b20 7420 7420 202b 2074     t  + t t  + t
│ │ │ │ +0001fb10: 2074 2020 2b20 2020 2020 2020 2020 2020   t  +           
│ │ │ │  0001fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001fb40: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ -0001fb50: 2020 2030 2035 2020 2020 3120 3520 2020     0 5    1 5   
│ │ │ │ -0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb50: 2020 2020 3420 2020 2030 2035 2020 2020      4    0 5    
│ │ │ │ +0001fb60: 3120 3520 2020 2020 2020 2020 2020 2020  1 5             
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│ │ │ │ -0001fb80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001fb80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001fbd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001fc30: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -0001fc40: 2020 2b20 3474 2074 2020 2b20 3339 7420    + 4t t  + 39t 
│ │ │ │ -0001fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc40: 2020 2074 2074 2020 2b20 3474 2074 2020     t t  + 4t t  
│ │ │ │ +0001fc50: 2b20 3339 7420 2020 2020 2020 2020 2020  + 39t           
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│ │ │ │ -0001fc70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001fc80: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -0001fc90: 3420 2020 2020 3020 3520 2020 2020 2031  4     0 5      1
│ │ │ │ -0001fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc90: 2020 2020 3320 3420 2020 2020 3020 3520      3 4     0 5 
│ │ │ │ +0001fca0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │  0001fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fcc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001fcc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001fd10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001fd20: 2020 2020 2020 2020 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 2020                  
│ │ │ │ -0001fd60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ +0001fde0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
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│ │ │ │ -0001fe00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001fe00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001fe50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001feb0: 2020 2020 2020 2020 2020 2020 2074 2074               t t
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│ │ │ │ +0001fed0: 2038 7420 7420 2020 2020 2020 2020 2020   8t t           
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│ │ │ │ +0001fef0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ff10: 2020 2020 3020 3420 2020 2030 2035 2020      0 4    0 5  
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│ │ │ │ +0001ff40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001ff90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ -0001ffe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001fff0: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -00020000: 2020 2d20 7420 7420 202d 2074 2074 2020    - t t  - t t  
│ │ │ │ -00020010: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020000: 2020 2074 2074 2020 2d20 7420 7420 202d     t t  - t t  -
│ │ │ │ +00020010: 2074 2074 2020 2020 2020 2020 2020 2020   t t            
│ │ │ │  00020020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020030: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ +00020150: 2b20 7420 7420 2020 2020 2020 2020 2020  + t t           
│ │ │ │  00020160: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000201a0: 2020 2031 2036 2020 2020 2020 2020 2020     1 6          
│ │ │ │  000201b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ -00020260: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00020270: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -00020280: 2020 2d20 3436 7420 7420 202d 2033 3974    - 46t t  - 39t
│ │ │ │ -00020290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020260: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020280: 2020 2074 2074 2020 2d20 3436 7420 7420     t t  - 46t t 
│ │ │ │ +00020290: 202d 2033 3974 2020 2020 2020 2020 2020   - 39t          
│ │ │ │  000202a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000202b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ -000202d0: 3320 2020 2020 2030 2035 2020 2020 2020  3      0 5      
│ │ │ │ +000202b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ +000202d0: 2020 2020 3020 3320 2020 2020 2030 2035      0 3      0 5
│ │ │ │  000202e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020300: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020300: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ -00020370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020350: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020360: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000203b0: 2020 2020 2020 2020 2020 2020 2074 2020               t  
│ │ │ │ -000203c0: 2d20 3436 7420 7420 202d 2033 3374 2074  - 46t t  - 33t t
│ │ │ │ -000203d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000203a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000203b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000203c0: 2020 2074 2020 2d20 3436 7420 7420 202d     t  - 46t t  -
│ │ │ │ +000203d0: 2033 3374 2074 2020 2020 2020 2020 2020   33t t          
│ │ │ │  000203e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00020400: 2020 2020 2020 2020 2020 2020 2020 3220                2 
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│ │ │ │ +000203f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ +00020410: 2020 2020 3220 2020 2020 2031 2034 2020      2      1 4  
│ │ │ │ +00020420: 2020 2020 3020 2020 2020 2020 2020 2020      0           
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│ │ │ │ -00020490: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000204a0: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -000204b0: 2020 2b20 3674 2074 2020 2b20 3574 2074    + 6t t  + 5t t
│ │ │ │ -000204c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000204a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000204c0: 2b20 3574 2074 2020 2020 2020 2020 2020  + 5t t          
│ │ │ │  000204d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000204e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000204f0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00020500: 3220 2020 2020 3120 3520 2020 2020 3020  2     1 5     0 
│ │ │ │ -00020510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000204e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000204f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020500: 2020 2020 3120 3220 2020 2020 3120 3520      1 2     1 5 
│ │ │ │ +00020510: 2020 2020 3020 2020 2020 2020 2020 2020      0           
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│ │ │ │ -00020530: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  00020540: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020580: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ -000205d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000205e0: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -000205f0: 2020 2b20 7420 7420 202b 2035 7420 7420    + t t  + 5t t 
│ │ │ │ -00020600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000205d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000205e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000205f0: 2020 2074 2074 2020 2b20 7420 7420 202b     t t  + t t  +
│ │ │ │ +00020600: 2035 7420 7420 2020 2020 2020 2020 2020   5t t           
│ │ │ │  00020610: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020640: 3220 2020 2031 2034 2020 2020 2030 2035  2    1 4     0 5
│ │ │ │ -00020650: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020670: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ -000206c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000206d0: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -000206e0: 2020 2d20 3339 7420 7420 202b 2034 3074    - 39t t  + 40t
│ │ │ │ -000206f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000206d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206e0: 2020 2074 2074 2020 2d20 3339 7420 7420     t t  - 39t t 
│ │ │ │ +000206f0: 202b 2034 3074 2020 2020 2020 2020 2020   + 40t          
│ │ │ │  00020700: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020720: 2020 2020 2020 2020 2020 2020 2020 3020                0 
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│ │ │ │ +00020720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020730: 2020 2020 3020 3120 2020 2020 2031 2034      0 1      1 4
│ │ │ │  00020740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020760: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020760: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00020780: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000207c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000207d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000207f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020800: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00020810: 2020 2020 2020 2020 2020 2020 2074 2020               t  
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│ │ │ │ -00020830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020800: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020810: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00020830: 2032 3074 2074 2020 2020 2020 2020 2020   20t t          
│ │ │ │  00020840: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020940: 2020 2020 7c0a 7c6f 3720 3a20 5261 7469      |.|o7 : Rati
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│ │ │ │ -00020c10: 2020 2020 7c0a 7c2c 2078 202c 2078 202c      |.|, x , x ,
│ │ │ │ -00020c20: 2078 202c 2078 202c 2078 202c 2078 202c   x , x , x , x ,
│ │ │ │ -00020c30: 2078 202c 2078 202c 2078 2020 2c20 7820   x , x , x  , x 
│ │ │ │ -00020c40: 205d 2920 6465 6669 6e65 6420 6279 2020   ]) defined by  
│ │ │ │ -00020c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c60: 2020 2020 7c0a 7c20 2020 3220 2020 3320      |.|   2   3 
│ │ │ │ -00020c70: 2020 3420 2020 3520 2020 3620 2020 3720    4   5   6   7 
│ │ │ │ -00020c80: 2020 3820 2020 3920 2020 3130 2020 2031    8   9   10   1
│ │ │ │ -00020c90: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00020c10: 2020 2020 2020 2020 2020 7c0a 7c2c 2078            |.|, x
│ │ │ │ +00020c20: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ +00020c30: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ +00020c40: 2020 2c20 7820 205d 2920 6465 6669 6e65    , x  ]) define
│ │ │ │ +00020c50: 6420 6279 2020 2020 2020 2020 2020 2020  d by            
│ │ │ │ +00020c60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020c70: 3220 2020 3320 2020 3420 2020 3520 2020  2   3   4   5   
│ │ │ │ +00020c80: 3620 2020 3720 2020 3820 2020 3920 2020  6   7   8   9   
│ │ │ │ +00020c90: 3130 2020 2031 3120 2020 2020 2020 2020  10   11         
│ │ │ │  00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020cb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00020cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020d00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00020d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d50: 2020 2020 7c0a 7c20 202d 2032 3278 2078      |.|  - 22x x
│ │ │ │ -00020d60: 2020 2b20 7820 7820 202b 2031 3378 2078    + x x  + 13x x
│ │ │ │ -00020d70: 2020 2b20 3431 7820 7820 202d 2078 2078    + 41x x  - x x
│ │ │ │ -00020d80: 2020 2b20 3132 7820 7820 202b 2032 3578    + 12x x  + 25x
│ │ │ │ -00020d90: 2078 2020 2b20 3235 7820 7820 202b 2032   x  + 25x x  + 2
│ │ │ │ -00020da0: 3378 2078 7c0a 7c34 2020 2020 2020 3320  3x x|.|4      3 
│ │ │ │ -00020db0: 3420 2020 2030 2035 2020 2020 2020 3220  4    0 5      2 
│ │ │ │ -00020dc0: 3520 2020 2020 2033 2035 2020 2020 3020  5      3 5    0 
│ │ │ │ -00020dd0: 3620 2020 2020 2032 2036 2020 2020 2020  6      2 6      
│ │ │ │ -00020de0: 3120 3720 2020 2020 2033 2037 2020 2020  1 7      3 7    
│ │ │ │ -00020df0: 2020 3520 7c0a 7c20 2020 2020 2020 2020    5 |.|         
│ │ │ │ +00020d50: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +00020d60: 2032 3278 2078 2020 2b20 7820 7820 202b   22x x  + x x  +
│ │ │ │ +00020d70: 2031 3378 2078 2020 2b20 3431 7820 7820   13x x  + 41x x 
│ │ │ │ +00020d80: 202d 2078 2078 2020 2b20 3132 7820 7820   - x x  + 12x x 
│ │ │ │ +00020d90: 202b 2032 3578 2078 2020 2b20 3235 7820   + 25x x  + 25x 
│ │ │ │ +00020da0: 7820 202b 2032 3378 2078 7c0a 7c34 2020  x  + 23x x|.|4  
│ │ │ │ +00020db0: 2020 2020 3320 3420 2020 2030 2035 2020      3 4    0 5  
│ │ │ │ +00020dc0: 2020 2020 3220 3520 2020 2020 2033 2035      2 5      3 5
│ │ │ │ +00020dd0: 2020 2020 3020 3620 2020 2020 2032 2036      0 6      2 6
│ │ │ │ +00020de0: 2020 2020 2020 3120 3720 2020 2020 2033        1 7      3
│ │ │ │ +00020df0: 2037 2020 2020 2020 3520 7c0a 7c20 2020   7      5 |.|   
│ │ │ │  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 2020 2020                  
│ │ │ │  00020e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020e40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00020e50: 2020 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 7c0a 7c20 2b20 3334 7820 7820      |.| + 34x x 
│ │ │ │ -00020ea0: 202b 2034 3478 2078 2020 2d20 3330 7820   + 44x x  - 30x 
│ │ │ │ -00020eb0: 7820 202b 2032 3778 2078 2020 2b20 3331  x  + 27x x  + 31
│ │ │ │ -00020ec0: 7820 7820 202d 2033 3678 2078 2020 2d20  x x  - 36x x  - 
│ │ │ │ -00020ed0: 7820 7820 202b 2031 3378 2078 2020 2b20  x x  + 13x x  + 
│ │ │ │ -00020ee0: 3878 2078 7c0a 7c20 2020 2020 2033 2034  8x x|.|      3 4
│ │ │ │ -00020ef0: 2020 2020 2020 3020 3520 2020 2020 2032        0 5      2
│ │ │ │ -00020f00: 2035 2020 2020 2020 3320 3520 2020 2020   5      3 5     
│ │ │ │ -00020f10: 2032 2036 2020 2020 2020 3320 3620 2020   2 6      3 6   
│ │ │ │ -00020f20: 2030 2037 2020 2020 2020 3120 3720 2020   0 7      1 7   
│ │ │ │ -00020f30: 2020 3320 7c0a 7c20 2020 2020 2020 2020    3 |.|         
│ │ │ │ +00020e90: 2020 2020 2020 2020 2020 7c0a 7c20 2b20            |.| + 
│ │ │ │ +00020ea0: 3334 7820 7820 202b 2034 3478 2078 2020  34x x  + 44x x  
│ │ │ │ +00020eb0: 2d20 3330 7820 7820 202b 2032 3778 2078  - 30x x  + 27x x
│ │ │ │ +00020ec0: 2020 2b20 3331 7820 7820 202d 2033 3678    + 31x x  - 36x
│ │ │ │ +00020ed0: 2078 2020 2d20 7820 7820 202b 2031 3378   x  - x x  + 13x
│ │ │ │ +00020ee0: 2078 2020 2b20 3878 2078 7c0a 7c20 2020   x  + 8x x|.|   
│ │ │ │ +00020ef0: 2020 2033 2034 2020 2020 2020 3020 3520     3 4      0 5 
│ │ │ │ +00020f00: 2020 2020 2032 2035 2020 2020 2020 3320       2 5      3 
│ │ │ │ +00020f10: 3520 2020 2020 2032 2036 2020 2020 2020  5      2 6      
│ │ │ │ +00020f20: 3320 3620 2020 2030 2037 2020 2020 2020  3 6    0 7      
│ │ │ │ +00020f30: 3120 3720 2020 2020 3320 7c0a 7c20 2020  1 7     3 |.|   
│ │ │ │  00020f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020f80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00020f90: 2020 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 7c0a 7c20 2b20 3578 2078 2020      |.| + 5x x  
│ │ │ │ -00020fe0: 2d20 3136 7820 7820 202b 2035 7820 7820  - 16x x  + 5x x 
│ │ │ │ -00020ff0: 202d 2033 3678 2078 2020 2b20 3337 7820   - 36x x  + 37x 
│ │ │ │ -00021000: 7820 202b 2034 3878 2078 2020 2d20 3578  x  + 48x x  - 5x
│ │ │ │ -00021010: 2078 2020 2d20 3578 2078 2020 2b20 7820   x  - 5x x  + x 
│ │ │ │ -00021020: 7820 202b 7c0a 7c20 2020 2020 3220 3420  x  +|.|     2 4 
│ │ │ │ -00021030: 2020 2020 2033 2034 2020 2020 2030 2035       3 4     0 5
│ │ │ │ -00021040: 2020 2020 2020 3220 3520 2020 2020 2033        2 5      3
│ │ │ │ -00021050: 2035 2020 2020 2020 3220 3620 2020 2020   5      2 6     
│ │ │ │ -00021060: 3120 3720 2020 2020 3320 3720 2020 2035  1 7     3 7    5
│ │ │ │ -00021070: 2037 2020 7c0a 7c20 2020 2020 2020 2020   7  |.|         
│ │ │ │ +00020fd0: 2020 2020 2020 2020 2020 7c0a 7c20 2b20            |.| + 
│ │ │ │ +00020fe0: 3578 2078 2020 2d20 3136 7820 7820 202b  5x x  - 16x x  +
│ │ │ │ +00020ff0: 2035 7820 7820 202d 2033 3678 2078 2020   5x x  - 36x x  
│ │ │ │ +00021000: 2b20 3337 7820 7820 202b 2034 3878 2078  + 37x x  + 48x x
│ │ │ │ +00021010: 2020 2d20 3578 2078 2020 2d20 3578 2078    - 5x x  - 5x x
│ │ │ │ +00021020: 2020 2b20 7820 7820 202b 7c0a 7c20 2020    + x x  +|.|   
│ │ │ │ +00021030: 2020 3220 3420 2020 2020 2033 2034 2020    2 4      3 4  
│ │ │ │ +00021040: 2020 2030 2035 2020 2020 2020 3220 3520     0 5      2 5 
│ │ │ │ +00021050: 2020 2020 2033 2035 2020 2020 2020 3220       3 5      2 
│ │ │ │ +00021060: 3620 2020 2020 3120 3720 2020 2020 3320  6     1 7     3 
│ │ │ │ +00021070: 3720 2020 2035 2037 2020 7c0a 7c20 2020  7    5 7  |.|   
│ │ │ │  00021080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000210c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000210d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021110: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00021120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021130: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00021130: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  00021140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021160: 2020 2020 7c0a 7c33 3574 2074 2020 2b20      |.|35t t  + 
│ │ │ │ -00021170: 3130 7420 7420 202b 2032 3574 2074 2020  10t t  + 25t t  
│ │ │ │ -00021180: 2d20 3574 2020 2d20 3134 7420 7420 202d  - 5t  - 14t t  -
│ │ │ │ -00021190: 2031 3474 2074 2020 2d20 3574 2074 2020   14t t  - 5t t  
│ │ │ │ -000211a0: 2d20 3133 7420 7420 202b 2033 3774 2074  - 13t t  + 37t t
│ │ │ │ -000211b0: 2020 2b20 7c0a 7c20 2020 3220 3520 2020    + |.|   2 5   
│ │ │ │ -000211c0: 2020 2033 2035 2020 2020 2020 3420 3520     3 5      4 5 
│ │ │ │ -000211d0: 2020 2020 3520 2020 2020 2030 2036 2020      5      0 6  
│ │ │ │ -000211e0: 2020 2020 3120 3620 2020 2020 3220 3620      1 6     2 6 
│ │ │ │ -000211f0: 2020 2020 2034 2036 2020 2020 2020 3520       4 6      5 
│ │ │ │ -00021200: 3620 2020 7c0a 7c20 2020 2020 2020 2020  6   |.|         
│ │ │ │ +00021160: 2020 2020 2020 2020 2020 7c0a 7c33 3574            |.|35t
│ │ │ │ +00021170: 2074 2020 2b20 3130 7420 7420 202b 2032   t  + 10t t  + 2
│ │ │ │ +00021180: 3574 2074 2020 2d20 3574 2020 2d20 3134  5t t  - 5t  - 14
│ │ │ │ +00021190: 7420 7420 202d 2031 3474 2074 2020 2d20  t t  - 14t t  - 
│ │ │ │ +000211a0: 3574 2074 2020 2d20 3133 7420 7420 202b  5t t  - 13t t  +
│ │ │ │ +000211b0: 2033 3774 2074 2020 2b20 7c0a 7c20 2020   37t t  + |.|   
│ │ │ │ +000211c0: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ +000211d0: 2020 3420 3520 2020 2020 3520 2020 2020    4 5     5     
│ │ │ │ +000211e0: 2030 2036 2020 2020 2020 3120 3620 2020   0 6      1 6   
│ │ │ │ +000211f0: 2020 3220 3620 2020 2020 2034 2036 2020    2 6      4 6  
│ │ │ │ +00021200: 2020 2020 3520 3620 2020 7c0a 7c20 2020      5 6   |.|   
│ │ │ │  00021210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021250: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021250: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00021260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021270: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00021280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021280: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00021290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212a0: 2020 2020 7c0a 7c74 2020 2d20 3430 7420      |.|t  - 40t 
│ │ │ │ -000212b0: 7420 202b 2034 3074 2074 2020 2b20 3236  t  + 40t t  + 26
│ │ │ │ -000212c0: 7420 7420 202d 2032 3074 2020 2b20 3431  t t  - 20t  + 41
│ │ │ │ -000212d0: 7420 7420 202b 2033 3674 2074 2020 2d20  t t  + 36t t  - 
│ │ │ │ -000212e0: 3232 7420 7420 202b 2033 3674 2074 2020  22t t  + 36t t  
│ │ │ │ -000212f0: 2d20 3330 7c0a 7c20 3520 2020 2020 2032  - 30|.| 5      2
│ │ │ │ -00021300: 2035 2020 2020 2020 3320 3520 2020 2020   5      3 5     
│ │ │ │ -00021310: 2034 2035 2020 2020 2020 3520 2020 2020   4 5      5     
│ │ │ │ -00021320: 2030 2036 2020 2020 2020 3120 3620 2020   0 6      1 6   
│ │ │ │ -00021330: 2020 2032 2036 2020 2020 2020 3420 3620     2 6      4 6 
│ │ │ │ -00021340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000212a0: 2020 2020 2020 2020 2020 7c0a 7c74 2020            |.|t  
│ │ │ │ +000212b0: 2d20 3430 7420 7420 202b 2034 3074 2074  - 40t t  + 40t t
│ │ │ │ +000212c0: 2020 2b20 3236 7420 7420 202d 2032 3074    + 26t t  - 20t
│ │ │ │ +000212d0: 2020 2b20 3431 7420 7420 202b 2033 3674    + 41t t  + 36t
│ │ │ │ +000212e0: 2074 2020 2d20 3232 7420 7420 202b 2033   t  - 22t t  + 3
│ │ │ │ +000212f0: 3674 2074 2020 2d20 3330 7c0a 7c20 3520  6t t  - 30|.| 5 
│ │ │ │ +00021300: 2020 2020 2032 2035 2020 2020 2020 3320       2 5      3 
│ │ │ │ +00021310: 3520 2020 2020 2034 2035 2020 2020 2020  5      4 5      
│ │ │ │ +00021320: 3520 2020 2020 2030 2036 2020 2020 2020  5      0 6      
│ │ │ │ +00021330: 3120 3620 2020 2020 2032 2036 2020 2020  1 6      2 6    
│ │ │ │ +00021340: 2020 3420 3620 2020 2020 7c0a 7c20 2020    4 6     |.|   
│ │ │ │  00021350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021390: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021390: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000213a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213b0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000213c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000213b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000213c0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  000213d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213e0: 2020 2020 7c0a 7c74 2020 2b20 3132 7420      |.|t  + 12t 
│ │ │ │ -000213f0: 7420 202b 2034 3774 2074 2020 2b20 3337  t  + 47t t  + 37
│ │ │ │ -00021400: 7420 7420 202b 2032 3574 2020 2d20 3237  t t  + 25t  - 27
│ │ │ │ -00021410: 7420 7420 202d 2032 3274 2074 2020 2b20  t t  - 22t t  + 
│ │ │ │ -00021420: 3237 7420 7420 202d 2032 3374 2074 2020  27t t  - 23t t  
│ │ │ │ -00021430: 2b20 3574 7c0a 7c20 3520 2020 2020 2032  + 5t|.| 5      2
│ │ │ │ -00021440: 2035 2020 2020 2020 3320 3520 2020 2020   5      3 5     
│ │ │ │ -00021450: 2034 2035 2020 2020 2020 3520 2020 2020   4 5      5     
│ │ │ │ -00021460: 2030 2036 2020 2020 2020 3120 3620 2020   0 6      1 6   
│ │ │ │ -00021470: 2020 2032 2036 2020 2020 2020 3420 3620     2 6      4 6 
│ │ │ │ -00021480: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000213e0: 2020 2020 2020 2020 2020 7c0a 7c74 2020            |.|t  
│ │ │ │ +000213f0: 2b20 3132 7420 7420 202b 2034 3774 2074  + 12t t  + 47t t
│ │ │ │ +00021400: 2020 2b20 3337 7420 7420 202b 2032 3574    + 37t t  + 25t
│ │ │ │ +00021410: 2020 2d20 3237 7420 7420 202d 2032 3274    - 27t t  - 22t
│ │ │ │ +00021420: 2074 2020 2b20 3237 7420 7420 202d 2032   t  + 27t t  - 2
│ │ │ │ +00021430: 3374 2074 2020 2b20 3574 7c0a 7c20 3520  3t t  + 5t|.| 5 
│ │ │ │ +00021440: 2020 2020 2032 2035 2020 2020 2020 3320       2 5      3 
│ │ │ │ +00021450: 3520 2020 2020 2034 2035 2020 2020 2020  5      4 5      
│ │ │ │ +00021460: 3520 2020 2020 2030 2036 2020 2020 2020  5      0 6      
│ │ │ │ +00021470: 3120 3620 2020 2020 2032 2036 2020 2020  1 6      2 6    
│ │ │ │ +00021480: 2020 3420 3620 2020 2020 7c0a 7c20 2020    4 6     |.|   
│ │ │ │  00021490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000214d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000214e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214f0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000214f0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  00021500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021520: 2020 2020 7c0a 7c20 2d20 3335 7420 7420      |.| - 35t t 
│ │ │ │ -00021530: 202d 2031 3074 2074 2020 2d20 3333 7420   - 10t t  - 33t 
│ │ │ │ -00021540: 7420 202b 2035 7420 202b 2031 3574 2074  t  + 5t  + 15t t
│ │ │ │ -00021550: 2020 2b20 3135 7420 7420 202b 2035 7420    + 15t t  + 5t 
│ │ │ │ -00021560: 7420 202b 2031 3574 2074 2020 2d20 3338  t  + 15t t  - 38
│ │ │ │ -00021570: 7420 7420 7c0a 7c20 2020 2020 2032 2035  t t |.|      2 5
│ │ │ │ -00021580: 2020 2020 2020 3320 3520 2020 2020 2034        3 5      4
│ │ │ │ -00021590: 2035 2020 2020 2035 2020 2020 2020 3020   5     5      0 
│ │ │ │ -000215a0: 3620 2020 2020 2031 2036 2020 2020 2032  6      1 6     2
│ │ │ │ -000215b0: 2036 2020 2020 2020 3420 3620 2020 2020   6      4 6     
│ │ │ │ -000215c0: 2035 2036 7c0a 7c20 2020 2020 2020 2020   5 6|.|         
│ │ │ │ +00021520: 2020 2020 2020 2020 2020 7c0a 7c20 2d20            |.| - 
│ │ │ │ +00021530: 3335 7420 7420 202d 2031 3074 2074 2020  35t t  - 10t t  
│ │ │ │ +00021540: 2d20 3333 7420 7420 202b 2035 7420 202b  - 33t t  + 5t  +
│ │ │ │ +00021550: 2031 3574 2074 2020 2b20 3135 7420 7420   15t t  + 15t t 
│ │ │ │ +00021560: 202b 2035 7420 7420 202b 2031 3574 2074   + 5t t  + 15t t
│ │ │ │ +00021570: 2020 2d20 3338 7420 7420 7c0a 7c20 2020    - 38t t |.|   
│ │ │ │ +00021580: 2020 2032 2035 2020 2020 2020 3320 3520     2 5      3 5 
│ │ │ │ +00021590: 2020 2020 2034 2035 2020 2020 2035 2020       4 5     5  
│ │ │ │ +000215a0: 2020 2020 3020 3620 2020 2020 2031 2036      0 6      1 6
│ │ │ │ +000215b0: 2020 2020 2032 2036 2020 2020 2020 3420       2 6      4 
│ │ │ │ +000215c0: 3620 2020 2020 2035 2036 7c0a 7c20 2020  6      5 6|.|   
│ │ │ │  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 2020 2020 2020 2020                  
│ │ │ │ -00021610: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021610: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00021620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021630: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00021630: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00021640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021660: 2020 2020 7c0a 7c2d 2033 3574 2074 2020      |.|- 35t t  
│ │ │ │ -00021670: 2d20 3130 7420 7420 202d 2033 3374 2074  - 10t t  - 33t t
│ │ │ │ -00021680: 2020 2b20 3574 2020 2b20 3134 7420 7420    + 5t  + 14t t 
│ │ │ │ -00021690: 202b 2031 3474 2074 2020 2b20 3574 2074   + 14t t  + 5t t
│ │ │ │ -000216a0: 2020 2b20 3134 7420 7420 202d 2033 3174    + 14t t  - 31t
│ │ │ │ -000216b0: 2074 2020 7c0a 7c20 2020 2020 3220 3520   t  |.|     2 5 
│ │ │ │ -000216c0: 2020 2020 2033 2035 2020 2020 2020 3420       3 5      4 
│ │ │ │ -000216d0: 3520 2020 2020 3520 2020 2020 2030 2036  5     5      0 6
│ │ │ │ -000216e0: 2020 2020 2020 3120 3620 2020 2020 3220        1 6     2 
│ │ │ │ -000216f0: 3620 2020 2020 2034 2036 2020 2020 2020  6      4 6      
│ │ │ │ -00021700: 3520 3620 7c0a 7c20 2020 2020 2020 2020  5 6 |.|         
│ │ │ │ +00021660: 2020 2020 2020 2020 2020 7c0a 7c2d 2033            |.|- 3
│ │ │ │ +00021670: 3574 2074 2020 2d20 3130 7420 7420 202d  5t t  - 10t t  -
│ │ │ │ +00021680: 2033 3374 2074 2020 2b20 3574 2020 2b20   33t t  + 5t  + 
│ │ │ │ +00021690: 3134 7420 7420 202b 2031 3474 2074 2020  14t t  + 14t t  
│ │ │ │ +000216a0: 2b20 3574 2074 2020 2b20 3134 7420 7420  + 5t t  + 14t t 
│ │ │ │ +000216b0: 202d 2033 3174 2074 2020 7c0a 7c20 2020   - 31t t  |.|   
│ │ │ │ +000216c0: 2020 3220 3520 2020 2020 2033 2035 2020    2 5      3 5  
│ │ │ │ +000216d0: 2020 2020 3420 3520 2020 2020 3520 2020      4 5     5   
│ │ │ │ +000216e0: 2020 2030 2036 2020 2020 2020 3120 3620     0 6      1 6 
│ │ │ │ +000216f0: 2020 2020 3220 3620 2020 2020 2034 2036      2 6      4 6
│ │ │ │ +00021700: 2020 2020 2020 3520 3620 7c0a 7c20 2020        5 6 |.|   
│ │ │ │  00021710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00021760: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00021770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021750: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00021760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021770: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00021780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217a0: 2020 2020 7c0a 7c20 2b20 7420 7420 202d      |.| + t t  -
│ │ │ │ -000217b0: 2037 7420 7420 202b 2032 7420 202d 2074   7t t  + 2t  - t
│ │ │ │ -000217c0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │ +000217a0: 2020 2020 2020 2020 2020 7c0a 7c20 2b20            |.| + 
│ │ │ │ +000217b0: 7420 7420 202d 2037 7420 7420 202b 2032  t t  - 7t t  + 2
│ │ │ │ +000217c0: 7420 202d 2074 2074 202c 2020 2020 2020  t  - t t ,      
│ │ │ │  000217d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000217e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217f0: 2020 2020 7c0a 7c20 2020 2034 2036 2020      |.|    4 6  
│ │ │ │ -00021800: 2020 2035 2036 2020 2020 2036 2020 2020     5 6     6    
│ │ │ │ -00021810: 3320 3720 2020 2020 2020 2020 2020 2020  3 7             
│ │ │ │ +000217f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00021800: 2034 2036 2020 2020 2035 2036 2020 2020   4 6     5 6    
│ │ │ │ +00021810: 2036 2020 2020 3320 3720 2020 2020 2020   6    3 7       
│ │ │ │  00021820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021840: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021840: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00021850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021890: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021890: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000218a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -000218c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000218b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000218c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  000218d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218e0: 2020 2020 7c0a 7c20 7420 202d 2034 3374      |.| t  - 43t
│ │ │ │ -000218f0: 2074 2020 2d20 3431 7420 7420 202d 2032   t  - 41t t  - 2
│ │ │ │ -00021900: 3674 2074 2020 2d20 3238 7420 202d 2033  6t t  - 28t  - 3
│ │ │ │ -00021910: 3574 2074 2020 2d20 3336 7420 7420 202b  5t t  - 36t t  +
│ │ │ │ -00021920: 2032 3074 2074 2020 2d20 3336 7420 7420   20t t  - 36t t 
│ │ │ │ -00021930: 202b 2039 7c0a 7c31 2035 2020 2020 2020   + 9|.|1 5      
│ │ │ │ -00021940: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ -00021950: 2020 3420 3520 2020 2020 2035 2020 2020    4 5      5    
│ │ │ │ -00021960: 2020 3020 3620 2020 2020 2031 2036 2020    0 6      1 6  
│ │ │ │ -00021970: 2020 2020 3220 3620 2020 2020 2034 2036      2 6      4 6
│ │ │ │ -00021980: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000218e0: 2020 2020 2020 2020 2020 7c0a 7c20 7420            |.| t 
│ │ │ │ +000218f0: 202d 2034 3374 2074 2020 2d20 3431 7420   - 43t t  - 41t 
│ │ │ │ +00021900: 7420 202d 2032 3674 2074 2020 2d20 3238  t  - 26t t  - 28
│ │ │ │ +00021910: 7420 202d 2033 3574 2074 2020 2d20 3336  t  - 35t t  - 36
│ │ │ │ +00021920: 7420 7420 202b 2032 3074 2074 2020 2d20  t t  + 20t t  - 
│ │ │ │ +00021930: 3336 7420 7420 202b 2039 7c0a 7c31 2035  36t t  + 9|.|1 5
│ │ │ │ +00021940: 2020 2020 2020 3220 3520 2020 2020 2033        2 5      3
│ │ │ │ +00021950: 2035 2020 2020 2020 3420 3520 2020 2020   5      4 5     
│ │ │ │ +00021960: 2035 2020 2020 2020 3020 3620 2020 2020   5      0 6     
│ │ │ │ +00021970: 2031 2036 2020 2020 2020 3220 3620 2020   1 6      2 6   
│ │ │ │ +00021980: 2020 2034 2036 2020 2020 7c0a 7c20 2020     4 6    |.|   
│ │ │ │  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 2020 2020                  
│ │ │ │  000219c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000219d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000219e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a00: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00021a00: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00021a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a20: 2020 2020 7c0a 7c20 202d 2034 3574 2074      |.|  - 45t t
│ │ │ │ -00021a30: 2020 2d20 3339 7420 7420 202d 2033 3974    - 39t t  - 39t
│ │ │ │ -00021a40: 2074 2020 2d20 3436 7420 7420 202d 2032   t  - 46t t  - 2
│ │ │ │ -00021a50: 3974 2020 2d20 3438 7420 7420 202d 2033  9t  - 48t t  - 3
│ │ │ │ -00021a60: 3874 2074 2020 2d20 3330 7420 7420 202b  8t t  - 30t t  +
│ │ │ │ -00021a70: 2031 3974 7c0a 7c35 2020 2020 2020 3120   19t|.|5      1 
│ │ │ │ -00021a80: 3520 2020 2020 2032 2035 2020 2020 2020  5      2 5      
│ │ │ │ -00021a90: 3320 3520 2020 2020 2034 2035 2020 2020  3 5      4 5    
│ │ │ │ -00021aa0: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ -00021ab0: 2020 3120 3620 2020 2020 2032 2036 2020    1 6      2 6  
│ │ │ │ -00021ac0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021a20: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +00021a30: 2034 3574 2074 2020 2d20 3339 7420 7420   45t t  - 39t t 
│ │ │ │ +00021a40: 202d 2033 3974 2074 2020 2d20 3436 7420   - 39t t  - 46t 
│ │ │ │ +00021a50: 7420 202d 2032 3974 2020 2d20 3438 7420  t  - 29t  - 48t 
│ │ │ │ +00021a60: 7420 202d 2033 3874 2074 2020 2d20 3330  t  - 38t t  - 30
│ │ │ │ +00021a70: 7420 7420 202b 2031 3974 7c0a 7c35 2020  t t  + 19t|.|5  
│ │ │ │ +00021a80: 2020 2020 3120 3520 2020 2020 2032 2035      1 5      2 5
│ │ │ │ +00021a90: 2020 2020 2020 3320 3520 2020 2020 2034        3 5      4
│ │ │ │ +00021aa0: 2035 2020 2020 2020 3520 2020 2020 2030   5      5      0
│ │ │ │ +00021ab0: 2036 2020 2020 2020 3120 3620 2020 2020   6      1 6     
│ │ │ │ +00021ac0: 2032 2036 2020 2020 2020 7c0a 7c20 2020   2 6      |.|   
│ │ │ │  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 2020 2020                  
│ │ │ │  00021b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b10: 2020 2020 7c0a 7c20 202d 2032 7420 7420      |.|  - 2t t 
│ │ │ │ -00021b20: 202d 2074 2074 2020 2d20 7420 7420 202b   - t t  - t t  +
│ │ │ │ -00021b30: 2035 7420 7420 202b 2074 2074 2020 2d20   5t t  + t t  - 
│ │ │ │ -00021b40: 3274 2074 2020 2d20 3774 2074 2020 2b20  2t t  - 7t t  + 
│ │ │ │ -00021b50: 3274 2074 2020 2d20 3274 2074 2020 2b20  2t t  - 2t t  + 
│ │ │ │ -00021b60: 3374 2074 7c0a 7c36 2020 2020 2031 2036  3t t|.|6     1 6
│ │ │ │ -00021b70: 2020 2020 3420 3620 2020 2035 2036 2020      4 6    5 6  
│ │ │ │ -00021b80: 2020 2030 2037 2020 2020 3120 3720 2020     0 7    1 7   
│ │ │ │ -00021b90: 2020 3220 3720 2020 2020 3520 3720 2020    2 7     5 7   
│ │ │ │ -00021ba0: 2020 3620 3720 2020 2020 3120 3820 2020    6 7     1 8   
│ │ │ │ -00021bb0: 2020 3720 7c0a 7c20 2020 2020 2020 2020    7 |.|         
│ │ │ │ +00021b10: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +00021b20: 2032 7420 7420 202d 2074 2074 2020 2d20   2t t  - t t  - 
│ │ │ │ +00021b30: 7420 7420 202b 2035 7420 7420 202b 2074  t t  + 5t t  + t
│ │ │ │ +00021b40: 2074 2020 2d20 3274 2074 2020 2d20 3774   t  - 2t t  - 7t
│ │ │ │ +00021b50: 2074 2020 2b20 3274 2074 2020 2d20 3274   t  + 2t t  - 2t
│ │ │ │ +00021b60: 2074 2020 2b20 3374 2074 7c0a 7c36 2020   t  + 3t t|.|6  
│ │ │ │ +00021b70: 2020 2031 2036 2020 2020 3420 3620 2020     1 6    4 6   
│ │ │ │ +00021b80: 2035 2036 2020 2020 2030 2037 2020 2020   5 6     0 7    
│ │ │ │ +00021b90: 3120 3720 2020 2020 3220 3720 2020 2020  1 7     2 7     
│ │ │ │ +00021ba0: 3520 3720 2020 2020 3620 3720 2020 2020  5 7     6 7     
│ │ │ │ +00021bb0: 3120 3820 2020 2020 3720 7c0a 7c20 2020  1 8     7 |.|   
│ │ │ │  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 2020 2020                  
│ │ │ │  00021bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021c00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00021c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c30: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00021c30: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00021c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c50: 2020 2020 7c0a 7c20 2b20 3332 7420 7420      |.| + 32t t 
│ │ │ │ -00021c60: 202d 2032 3074 2074 2020 2d20 3437 7420   - 20t t  - 47t 
│ │ │ │ -00021c70: 7420 202d 2033 3774 2074 2020 2d20 3235  t  - 37t t  - 25
│ │ │ │ -00021c80: 7420 202b 2031 3974 2074 2020 2b20 3232  t  + 19t t  + 22
│ │ │ │ -00021c90: 7420 7420 202d 2032 3574 2074 2020 2b20  t t  - 25t t  + 
│ │ │ │ -00021ca0: 3235 7420 7c0a 7c20 2020 2020 2031 2035  25t |.|      1 5
│ │ │ │ -00021cb0: 2020 2020 2020 3220 3520 2020 2020 2033        2 5      3
│ │ │ │ -00021cc0: 2035 2020 2020 2020 3420 3520 2020 2020   5      4 5     
│ │ │ │ -00021cd0: 2035 2020 2020 2020 3020 3620 2020 2020   5      0 6     
│ │ │ │ -00021ce0: 2031 2036 2020 2020 2020 3220 3620 2020   1 6      2 6   
│ │ │ │ -00021cf0: 2020 2034 7c0a 7c20 2020 2020 2020 2020     4|.|         
│ │ │ │ +00021c50: 2020 2020 2020 2020 2020 7c0a 7c20 2b20            |.| + 
│ │ │ │ +00021c60: 3332 7420 7420 202d 2032 3074 2074 2020  32t t  - 20t t  
│ │ │ │ +00021c70: 2d20 3437 7420 7420 202d 2033 3774 2074  - 47t t  - 37t t
│ │ │ │ +00021c80: 2020 2d20 3235 7420 202b 2031 3974 2074    - 25t  + 19t t
│ │ │ │ +00021c90: 2020 2b20 3232 7420 7420 202d 2032 3574    + 22t t  - 25t
│ │ │ │ +00021ca0: 2074 2020 2b20 3235 7420 7c0a 7c20 2020   t  + 25t |.|   
│ │ │ │ +00021cb0: 2020 2031 2035 2020 2020 2020 3220 3520     1 5      2 5 
│ │ │ │ +00021cc0: 2020 2020 2033 2035 2020 2020 2020 3420       3 5      4 
│ │ │ │ +00021cd0: 3520 2020 2020 2035 2020 2020 2020 3020  5      5      0 
│ │ │ │ +00021ce0: 3620 2020 2020 2031 2036 2020 2020 2020  6      1 6      
│ │ │ │ +00021cf0: 3220 3620 2020 2020 2034 7c0a 7c20 2020  2 6      4|.|   
│ │ │ │  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 2020                  
│ │ │ │  00021d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d40: 2020 2020 7c0a 7c20 7420 202d 2033 3774      |.| t  - 37t
│ │ │ │ -00021d50: 2074 2020 2d20 3339 7420 7420 202b 2031   t  - 39t t  + 1
│ │ │ │ -00021d60: 3974 2074 2020 2d20 3339 7420 7420 202d  9t t  - 39t t  -
│ │ │ │ -00021d70: 2033 3874 2074 2020 2b20 3339 7420 7420   38t t  + 39t t 
│ │ │ │ -00021d80: 202b 2031 3974 2074 2020 2b20 3138 7420   + 19t t  + 18t 
│ │ │ │ -00021d90: 7420 202d 7c0a 7c31 2035 2020 2020 2020  t  -|.|1 5      
│ │ │ │ -00021da0: 3020 3620 2020 2020 2031 2036 2020 2020  0 6      1 6    
│ │ │ │ -00021db0: 2020 3420 3620 2020 2020 2035 2036 2020    4 6      5 6  
│ │ │ │ -00021dc0: 2020 2020 3020 3720 2020 2020 2031 2037      0 7      1 7
│ │ │ │ -00021dd0: 2020 2020 2020 3220 3720 2020 2020 2035        2 7      5
│ │ │ │ -00021de0: 2037 2020 7c0a 7c20 2020 2020 2020 2020   7  |.|         
│ │ │ │ +00021d40: 2020 2020 2020 2020 2020 7c0a 7c20 7420            |.| t 
│ │ │ │ +00021d50: 202d 2033 3774 2074 2020 2d20 3339 7420   - 37t t  - 39t 
│ │ │ │ +00021d60: 7420 202b 2031 3974 2074 2020 2d20 3339  t  + 19t t  - 39
│ │ │ │ +00021d70: 7420 7420 202d 2033 3874 2074 2020 2b20  t t  - 38t t  + 
│ │ │ │ +00021d80: 3339 7420 7420 202b 2031 3974 2074 2020  39t t  + 19t t  
│ │ │ │ +00021d90: 2b20 3138 7420 7420 202d 7c0a 7c31 2035  + 18t t  -|.|1 5
│ │ │ │ +00021da0: 2020 2020 2020 3020 3620 2020 2020 2031        0 6      1
│ │ │ │ +00021db0: 2036 2020 2020 2020 3420 3620 2020 2020   6      4 6     
│ │ │ │ +00021dc0: 2035 2036 2020 2020 2020 3020 3720 2020   5 6      0 7   
│ │ │ │ +00021dd0: 2020 2031 2037 2020 2020 2020 3220 3720     1 7      2 7 
│ │ │ │ +00021de0: 2020 2020 2035 2037 2020 7c0a 7c20 2020       5 7  |.|   
│ │ │ │  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 2020 2020                  
│ │ │ │  00021e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021e30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00021e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e60: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00021e60: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00021e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e80: 2020 2020 7c0a 7c20 202b 2032 3774 2074      |.|  + 27t t
│ │ │ │ -00021e90: 2020 2d20 3874 2074 2020 2b20 3337 7420    - 8t t  + 37t 
│ │ │ │ -00021ea0: 7420 202b 2032 3874 2074 2020 2b20 3330  t  + 28t t  + 30
│ │ │ │ -00021eb0: 7420 202d 2034 3674 2074 2020 2b20 3234  t  - 46t t  + 24
│ │ │ │ -00021ec0: 7420 7420 202d 2034 3074 2074 2020 2b20  t t  - 40t t  + 
│ │ │ │ -00021ed0: 3235 7420 7c0a 7c35 2020 2020 2020 3120  25t |.|5      1 
│ │ │ │ -00021ee0: 3520 2020 2020 3220 3520 2020 2020 2033  5     2 5      3
│ │ │ │ -00021ef0: 2035 2020 2020 2020 3420 3520 2020 2020   5      4 5     
│ │ │ │ -00021f00: 2035 2020 2020 2020 3020 3620 2020 2020   5      0 6     
│ │ │ │ -00021f10: 2031 2036 2020 2020 2020 3220 3620 2020   1 6      2 6   
│ │ │ │ -00021f20: 2020 2034 7c0a 7c20 2020 2020 2020 2020     4|.|         
│ │ │ │ +00021e80: 2020 2020 2020 2020 2020 7c0a 7c20 202b            |.|  +
│ │ │ │ +00021e90: 2032 3774 2074 2020 2d20 3874 2074 2020   27t t  - 8t t  
│ │ │ │ +00021ea0: 2b20 3337 7420 7420 202b 2032 3874 2074  + 37t t  + 28t t
│ │ │ │ +00021eb0: 2020 2b20 3330 7420 202d 2034 3674 2074    + 30t  - 46t t
│ │ │ │ +00021ec0: 2020 2b20 3234 7420 7420 202d 2034 3074    + 24t t  - 40t
│ │ │ │ +00021ed0: 2074 2020 2b20 3235 7420 7c0a 7c35 2020   t  + 25t |.|5  
│ │ │ │ +00021ee0: 2020 2020 3120 3520 2020 2020 3220 3520      1 5     2 5 
│ │ │ │ +00021ef0: 2020 2020 2033 2035 2020 2020 2020 3420       3 5      4 
│ │ │ │ +00021f00: 3520 2020 2020 2035 2020 2020 2020 3020  5      5      0 
│ │ │ │ +00021f10: 3620 2020 2020 2031 2036 2020 2020 2020  6      1 6      
│ │ │ │ +00021f20: 3220 3620 2020 2020 2034 7c0a 7c20 2020  2 6      4|.|   
│ │ │ │  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 2020 2020                  
│ │ │ │  00021f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021f70: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  00021fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021fc0: 2020 2020 7c0a 7c66 726f 6d20 5050 5e38      |.|from PP^8
│ │ │ │ -00021fd0: 2074 6f20 382d 6469 6d65 6e73 696f 6e61   to 8-dimensiona
│ │ │ │ -00021fe0: 6c20 7375 6276 6172 6965 7479 206f 6620  l subvariety of 
│ │ │ │ -00021ff0: 5050 5e31 3129 2020 2020 2020 2020 2020  PP^11)          
│ │ │ │ +00021fc0: 2020 2020 2020 2020 2020 7c0a 7c66 726f            |.|fro
│ │ │ │ +00021fd0: 6d20 5050 5e38 2074 6f20 382d 6469 6d65  m PP^8 to 8-dime
│ │ │ │ +00021fe0: 6e73 696f 6e61 6c20 7375 6276 6172 6965  nsional subvarie
│ │ │ │ +00021ff0: 7479 206f 6620 5050 5e31 3129 2020 2020  ty of PP^11)    
│ │ │ │  00022000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022010: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00022010: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  00022020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000228f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00022910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022920: 2020 2020 7c0a 7c74 2074 2020 2d20 3133      |.|t t  - 13
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│ │ │ │ +00022c90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022ca0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ce0: 2020 2020 7c0a 7c2d 2032 3474 2020 2b20      |.|- 24t  + 
│ │ │ │ -00022cf0: 3332 7420 7420 202d 2032 3574 2074 2020  32t t  - 25t t  
│ │ │ │ -00022d00: 2d20 3139 7420 7420 202b 2034 3774 2074  - 19t t  + 47t t
│ │ │ │ -00022d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ce0: 2020 2020 2020 2020 2020 7c0a 7c2d 2032            |.|- 2
│ │ │ │ +00022cf0: 3474 2020 2b20 3332 7420 7420 202d 2032  4t  + 32t t  - 2
│ │ │ │ +00022d00: 3574 2074 2020 2d20 3139 7420 7420 202b  5t t  - 19t t  +
│ │ │ │ +00022d10: 2034 3774 2074 2020 2020 2020 2020 2020   47t t          
│ │ │ │  00022d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d30: 2020 2020 7c0a 7c20 2020 2020 3620 2020      |.|     6   
│ │ │ │ -00022d40: 2020 2033 2037 2020 2020 2020 3420 3720     3 7      4 7 
│ │ │ │ -00022d50: 2020 2020 2035 2037 2020 2020 2020 3620       5 7      6 
│ │ │ │ -00022d60: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +00022d30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022d40: 2020 3620 2020 2020 2033 2037 2020 2020    6      3 7    
│ │ │ │ +00022d50: 2020 3420 3720 2020 2020 2035 2037 2020    4 7      5 7  
│ │ │ │ +00022d60: 2020 2020 3620 3720 2020 2020 2020 2020      6 7         
│ │ │ │  00022d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022d80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022dd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022dd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022e20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022e70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ec0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022ec0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00022f20: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00022f10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022f20: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00022f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f60: 2020 2020 7c0a 7c74 2074 2020 2b20 3135      |.|t t  + 15
│ │ │ │ -00022f70: 7420 202b 2032 3674 2074 2020 2d20 3574  t  + 26t t  - 5t
│ │ │ │ -00022f80: 2074 2020 2b20 3335 7420 7420 202d 2031   t  + 35t t  - 1
│ │ │ │ -00022f90: 3074 2020 2020 2020 2020 2020 2020 2020  0t              
│ │ │ │ +00022f60: 2020 2020 2020 2020 2020 7c0a 7c74 2074            |.|t t
│ │ │ │ +00022f70: 2020 2b20 3135 7420 202b 2032 3674 2074    + 15t  + 26t t
│ │ │ │ +00022f80: 2020 2d20 3574 2074 2020 2b20 3335 7420    - 5t t  + 35t 
│ │ │ │ +00022f90: 7420 202d 2031 3074 2020 2020 2020 2020  t  - 10t        
│ │ │ │  00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fb0: 2020 2020 7c0a 7c20 3520 3620 2020 2020      |.| 5 6     
│ │ │ │ -00022fc0: 2036 2020 2020 2020 3320 3720 2020 2020   6      3 7     
│ │ │ │ -00022fd0: 3420 3720 2020 2020 2035 2037 2020 2020  4 7      5 7    
│ │ │ │ -00022fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022fb0: 2020 2020 2020 2020 2020 7c0a 7c20 3520            |.| 5 
│ │ │ │ +00022fc0: 3620 2020 2020 2036 2020 2020 2020 3320  6      6      3 
│ │ │ │ +00022fd0: 3720 2020 2020 3420 3720 2020 2020 2035  7     4 7      5
│ │ │ │ +00022fe0: 2037 2020 2020 2020 2020 2020 2020 2020   7              
│ │ │ │  00022ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023000: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023000: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023050: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00023060: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00023050: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023060: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00023070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230a0: 2020 2020 7c0a 7c20 7420 202d 2034 3474      |.| t  - 44t
│ │ │ │ -000230b0: 2074 2020 2d20 3437 7420 202d 2033 3674   t  - 47t  - 36t
│ │ │ │ -000230c0: 2074 2020 2d20 3436 7420 7420 202b 2074   t  - 46t t  + t
│ │ │ │ -000230d0: 2074 2020 2020 2020 2020 2020 2020 2020   t              
│ │ │ │ +000230a0: 2020 2020 2020 2020 2020 7c0a 7c20 7420            |.| t 
│ │ │ │ +000230b0: 202d 2034 3474 2074 2020 2d20 3437 7420   - 44t t  - 47t 
│ │ │ │ +000230c0: 202d 2033 3674 2074 2020 2d20 3436 7420   - 36t t  - 46t 
│ │ │ │ +000230d0: 7420 202b 2074 2074 2020 2020 2020 2020  t  + t t        
│ │ │ │  000230e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230f0: 2020 2020 7c0a 7c34 2036 2020 2020 2020      |.|4 6      
│ │ │ │ -00023100: 3520 3620 2020 2020 2036 2020 2020 2020  5 6      6      
│ │ │ │ -00023110: 3020 3720 2020 2020 2031 2037 2020 2020  0 7      1 7    
│ │ │ │ -00023120: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000230f0: 2020 2020 2020 2020 2020 7c0a 7c34 2036            |.|4 6
│ │ │ │ +00023100: 2020 2020 2020 3520 3620 2020 2020 2036        5 6      6
│ │ │ │ +00023110: 2020 2020 2020 3020 3720 2020 2020 2031        0 7      1
│ │ │ │ +00023120: 2037 2020 2020 3220 2020 2020 2020 2020   7    2         
│ │ │ │  00023130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023140: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023140: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  00023180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023190: 2020 2020 7c0a 7c20 2c20 2020 2020 2020      |.| ,       
│ │ │ │ +00023190: 2020 2020 2020 2020 2020 7c0a 7c20 2c20            |.| , 
│ │ │ │  000231a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000231b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000231c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000231d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000231e0: 2020 2020 7c0a 7c38 2020 2020 2020 2020      |.|8        
│ │ │ │ +000231e0: 2020 2020 2020 2020 2020 7c0a 7c38 2020            |.|8  
│ │ │ │  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 2020 2020                  
│ │ │ │  00023220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023230: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023230: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  00023270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023280: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00023290: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00023280: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023290: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  000232a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232d0: 2020 2020 7c0a 7c74 2020 2d20 3574 2074      |.|t  - 5t t
│ │ │ │ -000232e0: 2020 2b20 3133 7420 202b 2035 7420 7420    + 13t  + 5t t 
│ │ │ │ -000232f0: 202b 2074 2074 2020 2b20 3339 7420 7420   + t t  + 39t t 
│ │ │ │ -00023300: 202b 2020 2020 2020 2020 2020 2020 2020   +              
│ │ │ │ +000232d0: 2020 2020 2020 2020 2020 7c0a 7c74 2020            |.|t  
│ │ │ │ +000232e0: 2d20 3574 2074 2020 2b20 3133 7420 202b  - 5t t  + 13t  +
│ │ │ │ +000232f0: 2035 7420 7420 202b 2074 2074 2020 2b20   5t t  + t t  + 
│ │ │ │ +00023300: 3339 7420 7420 202b 2020 2020 2020 2020  39t t  +        
│ │ │ │  00023310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023320: 2020 2020 7c0a 7c20 3620 2020 2020 3520      |.| 6     5 
│ │ │ │ -00023330: 3620 2020 2020 2036 2020 2020 2030 2037  6      6     0 7
│ │ │ │ -00023340: 2020 2020 3120 3720 2020 2020 2033 2037      1 7      3 7
│ │ │ │ -00023350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023320: 2020 2020 2020 2020 2020 7c0a 7c20 3620            |.| 6 
│ │ │ │ +00023330: 2020 2020 3520 3620 2020 2020 2036 2020      5 6      6  
│ │ │ │ +00023340: 2020 2030 2037 2020 2020 3120 3720 2020     0 7    1 7   
│ │ │ │ +00023350: 2020 2033 2037 2020 2020 2020 2020 2020     3 7          
│ │ │ │  00023360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023370: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023370: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000233a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000233b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000233c0: 2020 2020 7c0a 7c20 3139 7420 7420 202b      |.| 19t t  +
│ │ │ │ -000233d0: 2031 3974 2074 2020 2b20 3230 7420 7420   19t t  + 20t t 
│ │ │ │ -000233e0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +000233c0: 2020 2020 2020 2020 2020 7c0a 7c20 3139            |.| 19
│ │ │ │ +000233d0: 7420 7420 202b 2031 3974 2074 2020 2b20  t t  + 19t t  + 
│ │ │ │ +000233e0: 3230 7420 7420 2c20 2020 2020 2020 2020  20t t ,         
│ │ │ │  000233f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023410: 2020 2020 7c0a 7c20 2020 2036 2037 2020      |.|    6 7  
│ │ │ │ -00023420: 2020 2020 3120 3820 2020 2020 2037 2038      1 8      7 8
│ │ │ │ -00023430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023410: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023420: 2036 2037 2020 2020 2020 3120 3820 2020   6 7      1 8   
│ │ │ │ +00023430: 2020 2037 2038 2020 2020 2020 2020 2020     7 8          
│ │ │ │  00023440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023460: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023460: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023480: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000234a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000234c0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000234b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000234c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  000234d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000234e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000234f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023500: 2020 2020 7c0a 7c74 2020 2b20 3136 7420      |.|t  + 16t 
│ │ │ │ -00023510: 7420 202d 2033 3574 2020 2b20 3239 7420  t  - 35t  + 29t 
│ │ │ │ -00023520: 7420 202b 2031 3274 2074 2020 2d20 3335  t  + 12t t  - 35
│ │ │ │ -00023530: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00023500: 2020 2020 2020 2020 2020 7c0a 7c74 2020            |.|t  
│ │ │ │ +00023510: 2b20 3136 7420 7420 202d 2033 3574 2020  + 16t t  - 35t  
│ │ │ │ +00023520: 2b20 3239 7420 7420 202b 2031 3274 2074  + 29t t  + 12t t
│ │ │ │ +00023530: 2020 2d20 3335 7420 2020 2020 2020 2020    - 35t         
│ │ │ │  00023540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023550: 2020 2020 7c0a 7c20 3620 2020 2020 2035      |.| 6      5
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│ │ │ │ +00023550: 2020 2020 2020 2020 2020 7c0a 7c20 3620            |.| 6 
│ │ │ │ +00023560: 2020 2020 2035 2036 2020 2020 2020 3620       5 6      6 
│ │ │ │ +00023570: 2020 2020 2030 2037 2020 2020 2020 3120       0 7      1 
│ │ │ │ +00023580: 3720 2020 2020 2032 2020 2020 2020 2020  7      2        
│ │ │ │  00023590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235a0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +000235a0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  000235b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000235c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000235d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000235e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000235f0: 2d2d 2d2d 7c0a 7c2b 3235 2a78 5f33 2a78  ----|.|+25*x_3*x
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│ │ │ │ -00023640: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000235f0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2b 3235  ----------|.|+25
│ │ │ │ +00023600: 2a78 5f33 2a78 5f37 2b32 332a 785f 352a  *x_3*x_7+23*x_5*
│ │ │ │ +00023610: 785f 372d 332a 785f 362a 785f 372b 322a  x_7-3*x_6*x_7+2*
│ │ │ │ +00023620: 785f 302a 785f 382b 3131 2a78 5f31 2a78  x_0*x_8+11*x_1*x
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│ │ │ │ +00023640: 2a78 5f34 2a78 2020 2020 7c0a 7c20 2020  *x_4*x    |.|   
│ │ │ │  00023650: 2020 2020 2020 2020 2020 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 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023690: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  000236d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000236e0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  00023720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023730: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023730: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
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│ │ │ │ -00023780: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023780: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000237d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000237e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00023800: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00023820: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  00023860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023870: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023870: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  000238b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000238c0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  00023900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023910: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  00023920: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00023940: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00023980: 7820 202d 2032 3578 2078 2020 2d20 3978  x  - 25x x  - 9x
│ │ │ │ -00023990: 2078 2020 2b20 3378 2078 2020 2b20 3234   x  + 3x x  + 24
│ │ │ │ -000239a0: 7820 7820 202d 2032 3778 2078 2020 2d20  x x  - 27x x  - 
│ │ │ │ -000239b0: 2020 2020 7c0a 7c38 2020 2020 2020 3620      |.|8      6 
│ │ │ │ -000239c0: 3820 2020 2020 3020 3920 2020 2020 2031  8     0 9      1
│ │ │ │ -000239d0: 2039 2020 2020 2020 3220 3920 2020 2020   9      2 9     
│ │ │ │ -000239e0: 3320 3920 2020 2020 3420 3920 2020 2020  3 9     4 9     
│ │ │ │ -000239f0: 2035 2039 2020 2020 2020 3620 3920 2020   5 9      6 9   
│ │ │ │ -00023a00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023960: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +00023970: 2033 3378 2078 2020 2b20 3878 2078 2020   33x x  + 8x x  
│ │ │ │ +00023980: 2b20 3130 7820 7820 202d 2032 3578 2078  + 10x x  - 25x x
│ │ │ │ +00023990: 2020 2d20 3978 2078 2020 2b20 3378 2078    - 9x x  + 3x x
│ │ │ │ +000239a0: 2020 2b20 3234 7820 7820 202d 2032 3778    + 24x x  - 27x
│ │ │ │ +000239b0: 2078 2020 2d20 2020 2020 7c0a 7c38 2020   x  -     |.|8  
│ │ │ │ +000239c0: 2020 2020 3620 3820 2020 2020 3020 3920      6 8     0 9 
│ │ │ │ +000239d0: 2020 2020 2031 2039 2020 2020 2020 3220       1 9      2 
│ │ │ │ +000239e0: 3920 2020 2020 3320 3920 2020 2020 3420  9     3 9     4 
│ │ │ │ +000239f0: 3920 2020 2020 2035 2039 2020 2020 2020  9      5 9      
│ │ │ │ +00023a00: 3620 3920 2020 2020 2020 7c0a 7c20 2020  6 9       |.|   
│ │ │ │  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 2020 2020                  
│ │ │ │  00023a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023a50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023aa0: 2020 2020 7c0a 7c20 202d 2039 7820 7820      |.|  - 9x x 
│ │ │ │ -00023ab0: 202d 2034 3678 2078 2020 2d20 3137 7820   - 46x x  - 17x 
│ │ │ │ -00023ac0: 7820 202b 2033 3278 2078 2020 2d20 3878  x  + 32x x  - 8x
│ │ │ │ -00023ad0: 2078 2020 2d20 3335 7820 7820 202d 2034   x  - 35x x  - 4
│ │ │ │ -00023ae0: 3678 2078 2020 2b20 3236 7820 7820 202b  6x x  + 26x x  +
│ │ │ │ -00023af0: 2020 2020 7c0a 7c38 2020 2020 2034 2038      |.|8     4 8
│ │ │ │ -00023b00: 2020 2020 2020 3620 3820 2020 2020 2030        6 8      0
│ │ │ │ -00023b10: 2039 2020 2020 2020 3120 3920 2020 2020   9      1 9     
│ │ │ │ -00023b20: 3220 3920 2020 2020 2033 2039 2020 2020  2 9      3 9    
│ │ │ │ -00023b30: 2020 3420 3920 2020 2020 2035 2039 2020    4 9      5 9  
│ │ │ │ -00023b40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023aa0: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +00023ab0: 2039 7820 7820 202d 2034 3678 2078 2020   9x x  - 46x x  
│ │ │ │ +00023ac0: 2d20 3137 7820 7820 202b 2033 3278 2078  - 17x x  + 32x x
│ │ │ │ +00023ad0: 2020 2d20 3878 2078 2020 2d20 3335 7820    - 8x x  - 35x 
│ │ │ │ +00023ae0: 7820 202d 2034 3678 2078 2020 2b20 3236  x  - 46x x  + 26
│ │ │ │ +00023af0: 7820 7820 202b 2020 2020 7c0a 7c38 2020  x x  +    |.|8  
│ │ │ │ +00023b00: 2020 2034 2038 2020 2020 2020 3620 3820     4 8      6 8 
│ │ │ │ +00023b10: 2020 2020 2030 2039 2020 2020 2020 3120       0 9      1 
│ │ │ │ +00023b20: 3920 2020 2020 3220 3920 2020 2020 2033  9     2 9      3
│ │ │ │ +00023b30: 2039 2020 2020 2020 3420 3920 2020 2020   9      4 9     
│ │ │ │ +00023b40: 2035 2039 2020 2020 2020 7c0a 7c20 2020   5 9      |.|   
│ │ │ │  00023b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023b90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023be0: 2020 2020 7c0a 7c20 7820 7820 202b 2034      |.| x x  + 4
│ │ │ │ -00023bf0: 3078 2078 2020 2d20 3332 7820 7820 202b  0x x  - 32x x  +
│ │ │ │ -00023c00: 2035 7820 7820 202d 2031 3178 2078 2020   5x x  - 11x x  
│ │ │ │ -00023c10: 2d20 3230 7820 7820 202b 2034 3578 2078  - 20x x  + 45x x
│ │ │ │ -00023c20: 2020 2d20 3134 7820 7820 202d 2032 3578    - 14x x  - 25x
│ │ │ │ -00023c30: 2020 2020 7c0a 7c20 2036 2038 2020 2020      |.|  6 8    
│ │ │ │ -00023c40: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ -00023c50: 2020 2032 2039 2020 2020 2020 3320 3920     2 9      3 9 
│ │ │ │ -00023c60: 2020 2020 2034 2039 2020 2020 2020 3520       4 9      5 
│ │ │ │ -00023c70: 3920 2020 2020 2036 2039 2020 2020 2020  9      6 9      
│ │ │ │ -00023c80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023be0: 2020 2020 2020 2020 2020 7c0a 7c20 7820            |.| x 
│ │ │ │ +00023bf0: 7820 202b 2034 3078 2078 2020 2d20 3332  x  + 40x x  - 32
│ │ │ │ +00023c00: 7820 7820 202b 2035 7820 7820 202d 2031  x x  + 5x x  - 1
│ │ │ │ +00023c10: 3178 2078 2020 2d20 3230 7820 7820 202b  1x x  - 20x x  +
│ │ │ │ +00023c20: 2034 3578 2078 2020 2d20 3134 7820 7820   45x x  - 14x x 
│ │ │ │ +00023c30: 202d 2032 3578 2020 2020 7c0a 7c20 2036   - 25x    |.|  6
│ │ │ │ +00023c40: 2038 2020 2020 2020 3020 3920 2020 2020   8      0 9     
│ │ │ │ +00023c50: 2031 2039 2020 2020 2032 2039 2020 2020   1 9     2 9    
│ │ │ │ +00023c60: 2020 3320 3920 2020 2020 2034 2039 2020    3 9      4 9  
│ │ │ │ +00023c70: 2020 2020 3520 3920 2020 2020 2036 2039      5 9      6 9
│ │ │ │ +00023c80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023cd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023cd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023d20: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020                  
│ │ │ │ -00023d70: 2020 2020 7c0a 7c34 3674 2074 2020 2b20      |.|46t t  + 
│ │ │ │ -00023d80: 3337 7420 7420 202b 2032 3874 2074 2020  37t t  + 28t t  
│ │ │ │ -00023d90: 2b20 3333 7420 7420 2c20 2020 2020 2020  + 33t t ,       
│ │ │ │ +00023d70: 2020 2020 2020 2020 2020 7c0a 7c34 3674            |.|46t
│ │ │ │ +00023d80: 2074 2020 2b20 3337 7420 7420 202b 2032   t  + 37t t  + 2
│ │ │ │ +00023d90: 3874 2074 2020 2b20 3333 7420 7420 2c20  8t t  + 33t t , 
│ │ │ │  00023da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023dc0: 2020 2020 7c0a 7c20 2020 3320 3820 2020      |.|   3 8   
│ │ │ │ -00023dd0: 2020 2034 2038 2020 2020 2020 3520 3820     4 8      5 8 
│ │ │ │ -00023de0: 2020 2020 2036 2038 2020 2020 2020 2020       6 8        
│ │ │ │ +00023dc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023dd0: 3320 3820 2020 2020 2034 2038 2020 2020  3 8      4 8    
│ │ │ │ +00023de0: 2020 3520 3820 2020 2020 2036 2038 2020    5 8      6 8  
│ │ │ │  00023df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023e10: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  00023e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023e60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023eb0: 2020 2020 7c0a 7c20 7420 202b 2031 3174      |.| t  + 11t
│ │ │ │ -00023ec0: 2074 2020 2b20 3436 7420 7420 202b 2032   t  + 46t t  + 2
│ │ │ │ -00023ed0: 3974 2074 2020 2b20 3238 7420 7420 2c20  9t t  + 28t t , 
│ │ │ │ -00023ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023eb0: 2020 2020 2020 2020 2020 7c0a 7c20 7420            |.| t 
│ │ │ │ +00023ec0: 202b 2031 3174 2074 2020 2b20 3436 7420   + 11t t  + 46t 
│ │ │ │ +00023ed0: 7420 202b 2032 3974 2074 2020 2b20 3238  t  + 29t t  + 28
│ │ │ │ +00023ee0: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │  00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f00: 2020 2020 7c0a 7c36 2037 2020 2020 2020      |.|6 7      
│ │ │ │ -00023f10: 3320 3820 2020 2020 2034 2038 2020 2020  3 8      4 8    
│ │ │ │ -00023f20: 2020 3520 3820 2020 2020 2036 2038 2020    5 8      6 8  
│ │ │ │ -00023f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023f00: 2020 2020 2020 2020 2020 7c0a 7c36 2037            |.|6 7
│ │ │ │ +00023f10: 2020 2020 2020 3320 3820 2020 2020 2034        3 8      4
│ │ │ │ +00023f20: 2038 2020 2020 2020 3520 3820 2020 2020   8      5 8     
│ │ │ │ +00023f30: 2036 2038 2020 2020 2020 2020 2020 2020   6 8            
│ │ │ │  00023f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023f50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023fa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023fa0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ff0: 2020 2020 7c0a 7c74 2020 2b20 3336 7420      |.|t  + 36t 
│ │ │ │ -00024000: 7420 202b 2031 3374 2074 2020 2b20 3236  t  + 13t t  + 26
│ │ │ │ -00024010: 7420 7420 202b 2033 3774 2074 202c 2020  t t  + 37t t ,  
│ │ │ │ -00024020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ff0: 2020 2020 2020 2020 2020 7c0a 7c74 2020            |.|t  
│ │ │ │ +00024000: 2b20 3336 7420 7420 202b 2031 3374 2074  + 36t t  + 13t t
│ │ │ │ +00024010: 2020 2b20 3236 7420 7420 202b 2033 3774    + 26t t  + 37t
│ │ │ │ +00024020: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │  00024030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024040: 2020 2020 7c0a 7c20 3720 2020 2020 2033      |.| 7      3
│ │ │ │ -00024050: 2038 2020 2020 2020 3420 3820 2020 2020   8      4 8     
│ │ │ │ -00024060: 2035 2038 2020 2020 2020 3620 3820 2020   5 8      6 8   
│ │ │ │ -00024070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024040: 2020 2020 2020 2020 2020 7c0a 7c20 3720            |.| 7 
│ │ │ │ +00024050: 2020 2020 2033 2038 2020 2020 2020 3420       3 8      4 
│ │ │ │ +00024060: 3820 2020 2020 2035 2038 2020 2020 2020  8      5 8      
│ │ │ │ +00024070: 3620 3820 2020 2020 2020 2020 2020 2020  6 8             
│ │ │ │  00024080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024090: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024090: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000240a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000240b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000240c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000240d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +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: 2020 2020 7c0a 7c20 2b20 3436 7420 7420      |.| + 46t t 
│ │ │ │ -00024140: 202d 2033 3674 2074 2020 2d20 3335 7420   - 36t t  - 35t 
│ │ │ │ -00024150: 7420 202d 2033 3174 2074 202c 2020 2020  t  - 31t t ,    
│ │ │ │ -00024160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024130: 2020 2020 2020 2020 2020 7c0a 7c20 2b20            |.| + 
│ │ │ │ +00024140: 3436 7420 7420 202d 2033 3674 2074 2020  46t t  - 36t t  
│ │ │ │ +00024150: 2d20 3335 7420 7420 202d 2033 3174 2074  - 35t t  - 31t t
│ │ │ │ +00024160: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  00024170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024180: 2020 2020 7c0a 7c20 2020 2020 2033 2038      |.|      3 8
│ │ │ │ -00024190: 2020 2020 2020 3420 3820 2020 2020 2035        4 8      5
│ │ │ │ -000241a0: 2038 2020 2020 2020 3620 3820 2020 2020   8      6 8     
│ │ │ │ -000241b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024180: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00024190: 2020 2033 2038 2020 2020 2020 3420 3820     3 8      4 8 
│ │ │ │ +000241a0: 2020 2020 2035 2038 2020 2020 2020 3620       5 8      6 
│ │ │ │ +000241b0: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │  000241c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000241d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000241d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000241e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000241f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024220: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024220: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024270: 2020 2020 7c0a 7c2b 2034 3674 2074 2020      |.|+ 46t t  
│ │ │ │ -00024280: 2d20 3336 7420 7420 202d 2033 3574 2074  - 36t t  - 35t t
│ │ │ │ -00024290: 2020 2d20 3331 7420 7420 2c20 2020 2020    - 31t t ,     
│ │ │ │ -000242a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024270: 2020 2020 2020 2020 2020 7c0a 7c2b 2034            |.|+ 4
│ │ │ │ +00024280: 3674 2074 2020 2d20 3336 7420 7420 202d  6t t  - 36t t  -
│ │ │ │ +00024290: 2033 3574 2074 2020 2d20 3331 7420 7420   35t t  - 31t t 
│ │ │ │ +000242a0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  000242b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000242c0: 2020 2020 7c0a 7c20 2020 2020 3320 3820      |.|     3 8 
│ │ │ │ -000242d0: 2020 2020 2034 2038 2020 2020 2020 3520       4 8      5 
│ │ │ │ -000242e0: 3820 2020 2020 2036 2038 2020 2020 2020  8      6 8      
│ │ │ │ +000242c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000242d0: 2020 3320 3820 2020 2020 2034 2038 2020    3 8      4 8  
│ │ │ │ +000242e0: 2020 2020 3520 3820 2020 2020 2036 2038      5 8      6 8
│ │ │ │  000242f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024310: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024310: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024360: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000243a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000243b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000243b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000243c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000243d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000243e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000243f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024400: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024400: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024450: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000244a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000244a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000244b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000244f0: 2020 2020 7c0a 7c20 7420 202d 2031 3074      |.| t  - 10t
│ │ │ │ -00024500: 2074 2020 2d20 3436 7420 7420 202b 2034   t  - 46t t  + 4
│ │ │ │ -00024510: 3774 2074 2020 2d20 3235 7420 7420 2c20  7t t  - 25t t , 
│ │ │ │ -00024520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244f0: 2020 2020 2020 2020 2020 7c0a 7c20 7420            |.| t 
│ │ │ │ +00024500: 202d 2031 3074 2074 2020 2d20 3436 7420   - 10t t  - 46t 
│ │ │ │ +00024510: 7420 202b 2034 3774 2074 2020 2d20 3235  t  + 47t t  - 25
│ │ │ │ +00024520: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │  00024530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024540: 2020 2020 7c0a 7c36 2037 2020 2020 2020      |.|6 7      
│ │ │ │ -00024550: 3320 3820 2020 2020 2034 2038 2020 2020  3 8      4 8    
│ │ │ │ -00024560: 2020 3520 3820 2020 2020 2036 2038 2020    5 8      6 8  
│ │ │ │ -00024570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024540: 2020 2020 2020 2020 2020 7c0a 7c36 2037            |.|6 7
│ │ │ │ +00024550: 2020 2020 2020 3320 3820 2020 2020 2034        3 8      4
│ │ │ │ +00024560: 2038 2020 2020 2020 3520 3820 2020 2020   8      5 8     
│ │ │ │ +00024570: 2036 2038 2020 2020 2020 2020 2020 2020   6 8            
│ │ │ │  00024580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024590: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024590: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020                  
│ │ │ │  000245d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000245e0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020                  
│ │ │ │  00024620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024630: 2020 2020 7c0a 7c20 202d 2034 3474 2074      |.|  - 44t t
│ │ │ │ -00024640: 2020 2b20 3438 7420 7420 202d 2031 3474    + 48t t  - 14t
│ │ │ │ -00024650: 2074 2020 2b20 3474 2074 2020 2d20 3336   t  + 4t t  - 36
│ │ │ │ -00024660: 7420 7420 202d 2034 3674 2074 2020 2b20  t t  - 46t t  + 
│ │ │ │ -00024670: 3437 7420 7420 202d 2033 3474 2074 2020  47t t  - 34t t  
│ │ │ │ -00024680: 2020 2020 7c0a 7c37 2020 2020 2020 3320      |.|7      3 
│ │ │ │ -00024690: 3720 2020 2020 2034 2037 2020 2020 2020  7      4 7      
│ │ │ │ -000246a0: 3520 3720 2020 2020 3620 3720 2020 2020  5 7     6 7     
│ │ │ │ -000246b0: 2030 2038 2020 2020 2020 3120 3820 2020   0 8      1 8   
│ │ │ │ -000246c0: 2020 2032 2038 2020 2020 2020 3320 3820     2 8      3 8 
│ │ │ │ -000246d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024630: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +00024640: 2034 3474 2074 2020 2b20 3438 7420 7420   44t t  + 48t t 
│ │ │ │ +00024650: 202d 2031 3474 2074 2020 2b20 3474 2074   - 14t t  + 4t t
│ │ │ │ +00024660: 2020 2d20 3336 7420 7420 202d 2034 3674    - 36t t  - 46t
│ │ │ │ +00024670: 2074 2020 2b20 3437 7420 7420 202d 2033   t  + 47t t  - 3
│ │ │ │ +00024680: 3474 2074 2020 2020 2020 7c0a 7c37 2020  4t t      |.|7  
│ │ │ │ +00024690: 2020 2020 3320 3720 2020 2020 2034 2037      3 7      4 7
│ │ │ │ +000246a0: 2020 2020 2020 3520 3720 2020 2020 3620        5 7     6 
│ │ │ │ +000246b0: 3720 2020 2020 2030 2038 2020 2020 2020  7      0 8      
│ │ │ │ +000246c0: 3120 3820 2020 2020 2032 2038 2020 2020  1 8      2 8    
│ │ │ │ +000246d0: 2020 3320 3820 2020 2020 7c0a 7c20 2020    3 8     |.|   
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00024720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024720: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024770: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024770: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000247a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000247b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000247c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000247d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000247e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000247f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024810: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024810: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024860: 2020 2020 7c0a 7c20 3238 7420 7420 202d      |.| 28t t  -
│ │ │ │ -00024870: 2039 7420 7420 202d 2033 3974 2074 2020   9t t  - 39t t  
│ │ │ │ -00024880: 2b20 3474 2074 2020 2b20 7420 7420 202d  + 4t t  + t t  -
│ │ │ │ -00024890: 2033 3674 2074 2020 2d20 3134 7420 7420   36t t  - 14t t 
│ │ │ │ -000248a0: 202d 2032 3674 2074 2020 2d20 3337 7420   - 26t t  - 37t 
│ │ │ │ -000248b0: 2020 2020 7c0a 7c20 2020 2034 2037 2020      |.|    4 7  
│ │ │ │ -000248c0: 2020 2035 2037 2020 2020 2020 3620 3720     5 7      6 7 
│ │ │ │ -000248d0: 2020 2020 3020 3820 2020 2031 2038 2020      0 8    1 8  
│ │ │ │ -000248e0: 2020 2020 3320 3820 2020 2020 2034 2038      3 8      4 8
│ │ │ │ -000248f0: 2020 2020 2020 3520 3820 2020 2020 2036        5 8      6
│ │ │ │ -00024900: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024860: 2020 2020 2020 2020 2020 7c0a 7c20 3238            |.| 28
│ │ │ │ +00024870: 7420 7420 202d 2039 7420 7420 202d 2033  t t  - 9t t  - 3
│ │ │ │ +00024880: 3974 2074 2020 2b20 3474 2074 2020 2b20  9t t  + 4t t  + 
│ │ │ │ +00024890: 7420 7420 202d 2033 3674 2074 2020 2d20  t t  - 36t t  - 
│ │ │ │ +000248a0: 3134 7420 7420 202d 2032 3674 2074 2020  14t t  - 26t t  
│ │ │ │ +000248b0: 2d20 3337 7420 2020 2020 7c0a 7c20 2020  - 37t     |.|   
│ │ │ │ +000248c0: 2034 2037 2020 2020 2035 2037 2020 2020   4 7     5 7    
│ │ │ │ +000248d0: 2020 3620 3720 2020 2020 3020 3820 2020    6 7     0 8   
│ │ │ │ +000248e0: 2031 2038 2020 2020 2020 3320 3820 2020   1 8      3 8   
│ │ │ │ +000248f0: 2020 2034 2038 2020 2020 2020 3520 3820     4 8      5 8 
│ │ │ │ +00024900: 2020 2020 2036 2020 2020 7c0a 7c20 2020       6    |.|   
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024950: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -000249a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000249a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000249b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000249f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024a40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a90: 2020 2020 7c0a 7c74 2020 2d20 3874 2074      |.|t  - 8t t
│ │ │ │ -00024aa0: 2020 2d20 3138 7420 7420 202b 2034 3274    - 18t t  + 42t
│ │ │ │ -00024ab0: 2074 2020 2d20 3132 7420 7420 202d 2036   t  - 12t t  - 6
│ │ │ │ -00024ac0: 7420 7420 202b 2031 3274 2074 2020 2d20  t t  + 12t t  - 
│ │ │ │ -00024ad0: 3135 7420 7420 202b 2039 7420 7420 202b  15t t  + 9t t  +
│ │ │ │ -00024ae0: 2020 2020 7c0a 7c20 3720 2020 2020 3320      |.| 7     3 
│ │ │ │ -00024af0: 3720 2020 2020 2034 2037 2020 2020 2020  7      4 7      
│ │ │ │ -00024b00: 3520 3720 2020 2020 2036 2037 2020 2020  5 7      6 7    
│ │ │ │ -00024b10: 2030 2038 2020 2020 2020 3120 3820 2020   0 8      1 8   
│ │ │ │ -00024b20: 2020 2033 2038 2020 2020 2034 2038 2020     3 8     4 8  
│ │ │ │ -00024b30: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00024a90: 2020 2020 2020 2020 2020 7c0a 7c74 2020            |.|t  
│ │ │ │ +00024aa0: 2d20 3874 2074 2020 2d20 3138 7420 7420  - 8t t  - 18t t 
│ │ │ │ +00024ab0: 202b 2034 3274 2074 2020 2d20 3132 7420   + 42t t  - 12t 
│ │ │ │ +00024ac0: 7420 202d 2036 7420 7420 202b 2031 3274  t  - 6t t  + 12t
│ │ │ │ +00024ad0: 2074 2020 2d20 3135 7420 7420 202b 2039   t  - 15t t  + 9
│ │ │ │ +00024ae0: 7420 7420 202b 2020 2020 7c0a 7c20 3720  t t  +    |.| 7 
│ │ │ │ +00024af0: 2020 2020 3320 3720 2020 2020 2034 2037      3 7      4 7
│ │ │ │ +00024b00: 2020 2020 2020 3520 3720 2020 2020 2036        5 7      6
│ │ │ │ +00024b10: 2037 2020 2020 2030 2038 2020 2020 2020   7     0 8      
│ │ │ │ +00024b20: 3120 3820 2020 2020 2033 2038 2020 2020  1 8      3 8    
│ │ │ │ +00024b30: 2034 2038 2020 2020 2020 7c0a 7c2d 2d2d   4 8      |.|---
│ │ │ │  00024b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024b80: 2d2d 2d2d 7c0a 7c5f 382d 3333 2a78 5f36  ----|.|_8-33*x_6
│ │ │ │ -00024b90: 2a78 5f38 2b38 2a78 5f30 2a78 5f39 2b31  *x_8+8*x_0*x_9+1
│ │ │ │ -00024ba0: 302a 785f 312a 785f 392d 3235 2a78 5f32  0*x_1*x_9-25*x_2
│ │ │ │ -00024bb0: 2a78 5f39 2d39 2a78 5f33 2a78 5f39 2b33  *x_9-9*x_3*x_9+3
│ │ │ │ -00024bc0: 2a78 5f34 2a78 5f39 2b32 342a 785f 352a  *x_4*x_9+24*x_5*
│ │ │ │ -00024bd0: 785f 2020 7c0a 7c20 2020 2020 2020 2020  x_  |.|         
│ │ │ │ +00024b80: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f 382d  ----------|.|_8-
│ │ │ │ +00024b90: 3333 2a78 5f36 2a78 5f38 2b38 2a78 5f30  33*x_6*x_8+8*x_0
│ │ │ │ +00024ba0: 2a78 5f39 2b31 302a 785f 312a 785f 392d  *x_9+10*x_1*x_9-
│ │ │ │ +00024bb0: 3235 2a78 5f32 2a78 5f39 2d39 2a78 5f33  25*x_2*x_9-9*x_3
│ │ │ │ +00024bc0: 2a78 5f39 2b33 2a78 5f34 2a78 5f39 2b32  *x_9+3*x_4*x_9+2
│ │ │ │ +00024bd0: 342a 785f 352a 785f 2020 7c0a 7c20 2020  4*x_5*x_  |.|   
│ │ │ │  00024be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024c20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024c70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024cc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024cc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024d10: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024d60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024db0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024db0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024e00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024e50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024ea0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ef0: 2020 2020 7c0a 7c35 7820 7820 2020 2b20      |.|5x x   + 
│ │ │ │ -00024f00: 3238 7820 7820 2020 2b20 3337 7820 7820  28x x   + 37x x 
│ │ │ │ -00024f10: 2020 2b20 3978 2078 2020 202b 2032 3778    + 9x x   + 27x
│ │ │ │ -00024f20: 2078 2020 202d 2032 3578 2078 2020 202b   x   - 25x x   +
│ │ │ │ -00024f30: 2039 7820 7820 2020 2b20 3237 7820 7820   9x x   + 27x x 
│ │ │ │ -00024f40: 2020 2020 7c0a 7c20 2030 2031 3020 2020      |.|  0 10   
│ │ │ │ -00024f50: 2020 2031 2031 3020 2020 2020 2032 2031     1 10      2 1
│ │ │ │ -00024f60: 3020 2020 2020 3420 3130 2020 2020 2020  0     4 10      
│ │ │ │ -00024f70: 3620 3130 2020 2020 2020 3020 3131 2020  6 10      0 11  
│ │ │ │ -00024f80: 2020 2032 2031 3120 2020 2020 2034 2031     2 11      4 1
│ │ │ │ -00024f90: 3120 2020 7c0a 7c20 2020 2020 2020 2020  1   |.|         
│ │ │ │ +00024ef0: 2020 2020 2020 2020 2020 7c0a 7c35 7820            |.|5x 
│ │ │ │ +00024f00: 7820 2020 2b20 3238 7820 7820 2020 2b20  x   + 28x x   + 
│ │ │ │ +00024f10: 3337 7820 7820 2020 2b20 3978 2078 2020  37x x   + 9x x  
│ │ │ │ +00024f20: 202b 2032 3778 2078 2020 202d 2032 3578   + 27x x   - 25x
│ │ │ │ +00024f30: 2078 2020 202b 2039 7820 7820 2020 2b20   x   + 9x x   + 
│ │ │ │ +00024f40: 3237 7820 7820 2020 2020 7c0a 7c20 2030  27x x     |.|  0
│ │ │ │ +00024f50: 2031 3020 2020 2020 2031 2031 3020 2020   10      1 10   
│ │ │ │ +00024f60: 2020 2032 2031 3020 2020 2020 3420 3130     2 10     4 10
│ │ │ │ +00024f70: 2020 2020 2020 3620 3130 2020 2020 2020        6 10      
│ │ │ │ +00024f80: 3020 3131 2020 2020 2032 2031 3120 2020  0 11     2 11   
│ │ │ │ +00024f90: 2020 2034 2031 3120 2020 7c0a 7c20 2020     4 11   |.|   
│ │ │ │  00024fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024fe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025030: 2020 2020 7c0a 7c20 3137 7820 7820 202b      |.| 17x x  +
│ │ │ │ -00025040: 2031 3578 2078 2020 202b 2033 3578 2078   15x x   + 35x x
│ │ │ │ -00025050: 2020 202b 2033 3478 2078 2020 202b 2032     + 34x x   + 2
│ │ │ │ -00025060: 3078 2078 2020 202b 2031 3478 2078 2020  0x x   + 14x x  
│ │ │ │ -00025070: 202b 2033 3678 2078 2020 202b 2033 3578   + 36x x   + 35x
│ │ │ │ -00025080: 2078 2020 7c0a 7c20 2020 2036 2039 2020   x  |.|    6 9  
│ │ │ │ -00025090: 2020 2020 3020 3130 2020 2020 2020 3120      0 10      1 
│ │ │ │ -000250a0: 3130 2020 2020 2020 3220 3130 2020 2020  10      2 10    
│ │ │ │ -000250b0: 2020 3420 3130 2020 2020 2020 3020 3131    4 10      0 11
│ │ │ │ -000250c0: 2020 2020 2020 3120 3131 2020 2020 2020        1 11      
│ │ │ │ -000250d0: 3220 2020 7c0a 7c20 2020 2020 2020 2020  2   |.|         
│ │ │ │ +00025030: 2020 2020 2020 2020 2020 7c0a 7c20 3137            |.| 17
│ │ │ │ +00025040: 7820 7820 202b 2031 3578 2078 2020 202b  x x  + 15x x   +
│ │ │ │ +00025050: 2033 3578 2078 2020 202b 2033 3478 2078   35x x   + 34x x
│ │ │ │ +00025060: 2020 202b 2032 3078 2078 2020 202b 2031     + 20x x   + 1
│ │ │ │ +00025070: 3478 2078 2020 202b 2033 3678 2078 2020  4x x   + 36x x  
│ │ │ │ +00025080: 202b 2033 3578 2078 2020 7c0a 7c20 2020   + 35x x  |.|   
│ │ │ │ +00025090: 2036 2039 2020 2020 2020 3020 3130 2020   6 9      0 10  
│ │ │ │ +000250a0: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ +000250b0: 3130 2020 2020 2020 3420 3130 2020 2020  10      4 10    
│ │ │ │ +000250c0: 2020 3020 3131 2020 2020 2020 3120 3131    0 11      1 11
│ │ │ │ +000250d0: 2020 2020 2020 3220 2020 7c0a 7c20 2020        2   |.|   
│ │ │ │  000250e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025120: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00025120: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00025130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00026490: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00026480: 2020 2020 2020 2020 2020 7c0a 7c2d 2032            |.|- 2
│ │ │ │ +00026490: 3778 2078 2020 2c20 2020 2020 2020 2020  7x x  ,         
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│ │ │ │  000264c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000265d0: 7820 202c 2020 2020 2020 2020 2020 2020  x  ,            
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│ │ │ │ +00026710: 2078 2020 2020 2020 2020 2020 2020 2020   x              
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│ │ │ │ -00026b50: 5f33 5e32 2d78 5f30 2a78 5f34 2b35 2a78  _3^2-x_0*x_4+5*x
│ │ │ │ -00026b60: 5f32 2a78 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  _2*x|.|---------
│ │ │ │ +00026b10: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 785f  ----------|.|*x_
│ │ │ │ +00026b20: 312a 785f 3131 2b33 352a 785f 322a 785f  1*x_11+35*x_2*x_
│ │ │ │ +00026b30: 3131 2d31 372a 785f 342a 785f 3131 2c78  11-17*x_4*x_11,x
│ │ │ │ +00026b40: 5f31 2a78 5f32 2d34 302a 785f 322a 785f  _1*x_2-40*x_2*x_
│ │ │ │ +00026b50: 332b 3238 2a78 5f33 5e32 2d78 5f30 2a78  3+28*x_3^2-x_0*x
│ │ │ │ +00026b60: 5f34 2b35 2a78 5f32 2a78 7c0a 7c2d 2d2d  _4+5*x_2*x|.|---
│ │ │ │  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 2d2d  ----------------
│ │ │ │  00026ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026bb0: 2d2d 2d2d 7c0a 7c5f 342d 3136 2a78 5f33  ----|.|_4-16*x_3
│ │ │ │ -00026bc0: 2a78 5f34 2b35 2a78 5f30 2a78 5f35 2d33  *x_4+5*x_0*x_5-3
│ │ │ │ -00026bd0: 362a 785f 322a 785f 352b 3337 2a78 5f33  6*x_2*x_5+37*x_3
│ │ │ │ -00026be0: 2a78 5f35 2b34 382a 785f 322a 785f 362d  *x_5+48*x_2*x_6-
│ │ │ │ -00026bf0: 352a 785f 312a 785f 372d 352a 785f 332a  5*x_1*x_7-5*x_3*
│ │ │ │ -00026c00: 785f 372b 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  x_7+|.|---------
│ │ │ │ +00026bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f 342d  ----------|.|_4-
│ │ │ │ +00026bc0: 3136 2a78 5f33 2a78 5f34 2b35 2a78 5f30  16*x_3*x_4+5*x_0
│ │ │ │ +00026bd0: 2a78 5f35 2d33 362a 785f 322a 785f 352b  *x_5-36*x_2*x_5+
│ │ │ │ +00026be0: 3337 2a78 5f33 2a78 5f35 2b34 382a 785f  37*x_3*x_5+48*x_
│ │ │ │ +00026bf0: 322a 785f 362d 352a 785f 312a 785f 372d  2*x_6-5*x_1*x_7-
│ │ │ │ +00026c00: 352a 785f 332a 785f 372b 7c0a 7c2d 2d2d  5*x_3*x_7+|.|---
│ │ │ │  00026c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026c50: 2d2d 2d2d 7c0a 7c78 5f35 2a78 5f37 2b32  ----|.|x_5*x_7+2
│ │ │ │ -00026c60: 302a 785f 362a 785f 372b 3130 2a78 5f30  0*x_6*x_7+10*x_0
│ │ │ │ -00026c70: 2a78 5f38 2b33 342a 785f 312a 785f 382b  *x_8+34*x_1*x_8+
│ │ │ │ -00026c80: 3431 2a78 5f33 2a78 5f38 2d78 5f34 2a78  41*x_3*x_8-x_4*x
│ │ │ │ -00026c90: 5f38 2b78 5f36 2a78 5f38 2b34 302a 785f  _8+x_6*x_8+40*x_
│ │ │ │ -00026ca0: 302a 785f 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  0*x_|.|---------
│ │ │ │ +00026c50: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c78 5f35  ----------|.|x_5
│ │ │ │ +00026c60: 2a78 5f37 2b32 302a 785f 362a 785f 372b  *x_7+20*x_6*x_7+
│ │ │ │ +00026c70: 3130 2a78 5f30 2a78 5f38 2b33 342a 785f  10*x_0*x_8+34*x_
│ │ │ │ +00026c80: 312a 785f 382b 3431 2a78 5f33 2a78 5f38  1*x_8+41*x_3*x_8
│ │ │ │ +00026c90: 2d78 5f34 2a78 5f38 2b78 5f36 2a78 5f38  -x_4*x_8+x_6*x_8
│ │ │ │ +00026ca0: 2b34 302a 785f 302a 785f 7c0a 7c2d 2d2d  +40*x_0*x_|.|---
│ │ │ │  00026cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026cf0: 2d2d 2d2d 7c0a 7c39 2d33 322a 785f 312a  ----|.|9-32*x_1*
│ │ │ │ -00026d00: 785f 392b 352a 785f 322a 785f 392d 3131  x_9+5*x_2*x_9-11
│ │ │ │ -00026d10: 2a78 5f33 2a78 5f39 2d32 302a 785f 342a  *x_3*x_9-20*x_4*
│ │ │ │ -00026d20: 785f 392b 3435 2a78 5f35 2a78 5f39 2d31  x_9+45*x_5*x_9-1
│ │ │ │ -00026d30: 342a 785f 362a 785f 392d 3235 2a78 5f30  4*x_6*x_9-25*x_0
│ │ │ │ -00026d40: 2a78 5f20 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *x_ |.|---------
│ │ │ │ +00026cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c39 2d33  ----------|.|9-3
│ │ │ │ +00026d00: 322a 785f 312a 785f 392b 352a 785f 322a  2*x_1*x_9+5*x_2*
│ │ │ │ +00026d10: 785f 392d 3131 2a78 5f33 2a78 5f39 2d32  x_9-11*x_3*x_9-2
│ │ │ │ +00026d20: 302a 785f 342a 785f 392b 3435 2a78 5f35  0*x_4*x_9+45*x_5
│ │ │ │ +00026d30: 2a78 5f39 2d31 342a 785f 362a 785f 392d  *x_9-14*x_6*x_9-
│ │ │ │ +00026d40: 3235 2a78 5f30 2a78 5f20 7c0a 7c2d 2d2d  25*x_0*x_ |.|---
│ │ │ │  00026d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026d90: 2d2d 2d2d 7c0a 7c31 302b 3435 2a78 5f31  ----|.|10+45*x_1
│ │ │ │ -00026da0: 2a78 5f31 302d 3431 2a78 5f32 2a78 5f31  *x_10-41*x_2*x_1
│ │ │ │ -00026db0: 302d 3436 2a78 5f34 2a78 5f31 302b 382a  0-46*x_4*x_10+8*
│ │ │ │ -00026dc0: 785f 362a 785f 3130 2d32 382a 785f 302a  x_6*x_10-28*x_0*
│ │ │ │ -00026dd0: 785f 3131 2b31 312a 785f 322a 785f 3131  x_11+11*x_2*x_11
│ │ │ │ -00026de0: 2b31 342a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  +14*|.|---------
│ │ │ │ +00026d90: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c31 302b  ----------|.|10+
│ │ │ │ +00026da0: 3435 2a78 5f31 2a78 5f31 302d 3431 2a78  45*x_1*x_10-41*x
│ │ │ │ +00026db0: 5f32 2a78 5f31 302d 3436 2a78 5f34 2a78  _2*x_10-46*x_4*x
│ │ │ │ +00026dc0: 5f31 302b 382a 785f 362a 785f 3130 2d32  _10+8*x_6*x_10-2
│ │ │ │ +00026dd0: 382a 785f 302a 785f 3131 2b31 312a 785f  8*x_0*x_11+11*x_
│ │ │ │ +00026de0: 322a 785f 3131 2b31 342a 7c0a 7c2d 2d2d  2*x_11+14*|.|---
│ │ │ │  00026df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026e30: 2d2d 2d2d 7c0a 7c78 5f34 2a78 5f31 312d  ----|.|x_4*x_11-
│ │ │ │ -00026e40: 382a 785f 352a 785f 3131 292c 7b74 5f34  8*x_5*x_11),{t_4
│ │ │ │ -00026e50: 5e32 2b74 5f30 2a74 5f35 2b74 5f31 2a74  ^2+t_0*t_5+t_1*t
│ │ │ │ -00026e60: 5f35 2b33 352a 745f 322a 745f 352b 3130  _5+35*t_2*t_5+10
│ │ │ │ -00026e70: 2a74 5f33 2a74 5f35 2b32 352a 745f 342a  *t_3*t_5+25*t_4*
│ │ │ │ -00026e80: 745f 352d 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  t_5-|.|---------
│ │ │ │ +00026e30: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c78 5f34  ----------|.|x_4
│ │ │ │ +00026e40: 2a78 5f31 312d 382a 785f 352a 785f 3131  *x_11-8*x_5*x_11
│ │ │ │ +00026e50: 292c 7b74 5f34 5e32 2b74 5f30 2a74 5f35  ),{t_4^2+t_0*t_5
│ │ │ │ +00026e60: 2b74 5f31 2a74 5f35 2b33 352a 745f 322a  +t_1*t_5+35*t_2*
│ │ │ │ +00026e70: 745f 352b 3130 2a74 5f33 2a74 5f35 2b32  t_5+10*t_3*t_5+2
│ │ │ │ +00026e80: 352a 745f 342a 745f 352d 7c0a 7c2d 2d2d  5*t_4*t_5-|.|---
│ │ │ │  00026e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026ed0: 2d2d 2d2d 7c0a 7c35 2a74 5f35 5e32 2d31  ----|.|5*t_5^2-1
│ │ │ │ -00026ee0: 342a 745f 302a 745f 362d 3134 2a74 5f31  4*t_0*t_6-14*t_1
│ │ │ │ -00026ef0: 2a74 5f36 2d35 2a74 5f32 2a74 5f36 2d31  *t_6-5*t_2*t_6-1
│ │ │ │ -00026f00: 332a 745f 342a 745f 362b 3337 2a74 5f35  3*t_4*t_6+37*t_5
│ │ │ │ -00026f10: 2a74 5f36 2b32 322a 745f 365e 322d 3331  *t_6+22*t_6^2-31
│ │ │ │ -00026f20: 2a74 5f33 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *t_3|.|---------
│ │ │ │ +00026ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c35 2a74  ----------|.|5*t
│ │ │ │ +00026ee0: 5f35 5e32 2d31 342a 745f 302a 745f 362d  _5^2-14*t_0*t_6-
│ │ │ │ +00026ef0: 3134 2a74 5f31 2a74 5f36 2d35 2a74 5f32  14*t_1*t_6-5*t_2
│ │ │ │ +00026f00: 2a74 5f36 2d31 332a 745f 342a 745f 362b  *t_6-13*t_4*t_6+
│ │ │ │ +00026f10: 3337 2a74 5f35 2a74 5f36 2b32 322a 745f  37*t_5*t_6+22*t_
│ │ │ │ +00026f20: 365e 322d 3331 2a74 5f33 7c0a 7c2d 2d2d  6^2-31*t_3|.|---
│ │ │ │  00026f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f70: 2d2d 2d2d 7c0a 7c2a 745f 372b 3236 2a74  ----|.|*t_7+26*t
│ │ │ │ -00026f80: 5f34 2a74 5f37 2b31 322a 745f 352a 745f  _4*t_7+12*t_5*t_
│ │ │ │ -00026f90: 372d 3435 2a74 5f36 2a74 5f37 2d34 362a  7-45*t_6*t_7-46*
│ │ │ │ -00026fa0: 745f 332a 745f 382b 3337 2a74 5f34 2a74  t_3*t_8+37*t_4*t
│ │ │ │ -00026fb0: 5f38 2b32 382a 745f 352a 745f 382b 3333  _8+28*t_5*t_8+33
│ │ │ │ -00026fc0: 2a74 5f36 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *t_6|.|---------
│ │ │ │ +00026f70: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00026f80: 372b 3236 2a74 5f34 2a74 5f37 2b31 322a  7+26*t_4*t_7+12*
│ │ │ │ +00026f90: 745f 352a 745f 372d 3435 2a74 5f36 2a74  t_5*t_7-45*t_6*t
│ │ │ │ +00026fa0: 5f37 2d34 362a 745f 332a 745f 382b 3337  _7-46*t_3*t_8+37
│ │ │ │ +00026fb0: 2a74 5f34 2a74 5f38 2b32 382a 745f 352a  *t_4*t_8+28*t_5*
│ │ │ │ +00026fc0: 745f 382b 3333 2a74 5f36 7c0a 7c2d 2d2d  t_8+33*t_6|.|---
│ │ │ │  00026fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027010: 2d2d 2d2d 7c0a 7c2a 745f 382c 745f 332a  ----|.|*t_8,t_3*
│ │ │ │ -00027020: 745f 342b 342a 745f 302a 745f 352b 3339  t_4+4*t_0*t_5+39
│ │ │ │ -00027030: 2a74 5f31 2a74 5f35 2d34 302a 745f 322a  *t_1*t_5-40*t_2*
│ │ │ │ -00027040: 745f 352b 3430 2a74 5f33 2a74 5f35 2b32  t_5+40*t_3*t_5+2
│ │ │ │ -00027050: 362a 745f 342a 745f 352d 3230 2a74 5f35  6*t_4*t_5-20*t_5
│ │ │ │ -00027060: 5e32 2b20 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  ^2+ |.|---------
│ │ │ │ +00027010: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00027020: 382c 745f 332a 745f 342b 342a 745f 302a  8,t_3*t_4+4*t_0*
│ │ │ │ +00027030: 745f 352b 3339 2a74 5f31 2a74 5f35 2d34  t_5+39*t_1*t_5-4
│ │ │ │ +00027040: 302a 745f 322a 745f 352b 3430 2a74 5f33  0*t_2*t_5+40*t_3
│ │ │ │ +00027050: 2a74 5f35 2b32 362a 745f 342a 745f 352d  *t_5+26*t_4*t_5-
│ │ │ │ +00027060: 3230 2a74 5f35 5e32 2b20 7c0a 7c2d 2d2d  20*t_5^2+ |.|---
│ │ │ │  00027070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000270a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000270b0: 2d2d 2d2d 7c0a 7c34 312a 745f 302a 745f  ----|.|41*t_0*t_
│ │ │ │ -000270c0: 362b 3336 2a74 5f31 2a74 5f36 2d32 322a  6+36*t_1*t_6-22*
│ │ │ │ -000270d0: 745f 322a 745f 362b 3336 2a74 5f34 2a74  t_2*t_6+36*t_4*t
│ │ │ │ -000270e0: 5f36 2d33 302a 745f 352a 745f 362d 3133  _6-30*t_5*t_6-13
│ │ │ │ -000270f0: 2a74 5f36 5e32 2d32 352a 745f 332a 745f  *t_6^2-25*t_3*t_
│ │ │ │ -00027100: 372b 352a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  7+5*|.|---------
│ │ │ │ +000270b0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34 312a  ----------|.|41*
│ │ │ │ +000270c0: 745f 302a 745f 362b 3336 2a74 5f31 2a74  t_0*t_6+36*t_1*t
│ │ │ │ +000270d0: 5f36 2d32 322a 745f 322a 745f 362b 3336  _6-22*t_2*t_6+36
│ │ │ │ +000270e0: 2a74 5f34 2a74 5f36 2d33 302a 745f 352a  *t_4*t_6-30*t_5*
│ │ │ │ +000270f0: 745f 362d 3133 2a74 5f36 5e32 2d32 352a  t_6-13*t_6^2-25*
│ │ │ │ +00027100: 745f 332a 745f 372b 352a 7c0a 7c2d 2d2d  t_3*t_7+5*|.|---
│ │ │ │  00027110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027150: 2d2d 2d2d 7c0a 7c74 5f34 2a74 5f37 2d33  ----|.|t_4*t_7-3
│ │ │ │ -00027160: 352a 745f 352a 745f 372b 3130 2a74 5f36  5*t_5*t_7+10*t_6
│ │ │ │ -00027170: 2a74 5f37 2b31 312a 745f 332a 745f 382b  *t_7+11*t_3*t_8+
│ │ │ │ -00027180: 3436 2a74 5f34 2a74 5f38 2b32 392a 745f  46*t_4*t_8+29*t_
│ │ │ │ -00027190: 352a 745f 382b 3238 2a74 5f36 2a74 5f38  5*t_8+28*t_6*t_8
│ │ │ │ -000271a0: 2c74 5f32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  ,t_2|.|---------
│ │ │ │ +00027150: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c74 5f34  ----------|.|t_4
│ │ │ │ +00027160: 2a74 5f37 2d33 352a 745f 352a 745f 372b  *t_7-35*t_5*t_7+
│ │ │ │ +00027170: 3130 2a74 5f36 2a74 5f37 2b31 312a 745f  10*t_6*t_7+11*t_
│ │ │ │ +00027180: 332a 745f 382b 3436 2a74 5f34 2a74 5f38  3*t_8+46*t_4*t_8
│ │ │ │ +00027190: 2b32 392a 745f 352a 745f 382b 3238 2a74  +29*t_5*t_8+28*t
│ │ │ │ +000271a0: 5f36 2a74 5f38 2c74 5f32 7c0a 7c2d 2d2d  _6*t_8,t_2|.|---
│ │ │ │  000271b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000271c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000271d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000271e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000271f0: 2d2d 2d2d 7c0a 7c2a 745f 342d 352a 745f  ----|.|*t_4-5*t_
│ │ │ │ -00027200: 302a 745f 352d 3430 2a74 5f31 2a74 5f35  0*t_5-40*t_1*t_5
│ │ │ │ -00027210: 2b31 322a 745f 322a 745f 352b 3437 2a74  +12*t_2*t_5+47*t
│ │ │ │ -00027220: 5f33 2a74 5f35 2b33 372a 745f 342a 745f  _3*t_5+37*t_4*t_
│ │ │ │ -00027230: 352b 3235 2a74 5f35 5e32 2d32 372a 745f  5+25*t_5^2-27*t_
│ │ │ │ -00027240: 302a 745f 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  0*t_|.|---------
│ │ │ │ +000271f0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00027200: 342d 352a 745f 302a 745f 352d 3430 2a74  4-5*t_0*t_5-40*t
│ │ │ │ +00027210: 5f31 2a74 5f35 2b31 322a 745f 322a 745f  _1*t_5+12*t_2*t_
│ │ │ │ +00027220: 352b 3437 2a74 5f33 2a74 5f35 2b33 372a  5+47*t_3*t_5+37*
│ │ │ │ +00027230: 745f 342a 745f 352b 3235 2a74 5f35 5e32  t_4*t_5+25*t_5^2
│ │ │ │ +00027240: 2d32 372a 745f 302a 745f 7c0a 7c2d 2d2d  -27*t_0*t_|.|---
│ │ │ │  00027250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027290: 2d2d 2d2d 7c0a 7c36 2d32 322a 745f 312a  ----|.|6-22*t_1*
│ │ │ │ -000272a0: 745f 362b 3237 2a74 5f32 2a74 5f36 2d32  t_6+27*t_2*t_6-2
│ │ │ │ -000272b0: 332a 745f 342a 745f 362b 352a 745f 352a  3*t_4*t_6+5*t_5*
│ │ │ │ -000272c0: 745f 362d 3133 2a74 5f36 5e32 2d33 392a  t_6-13*t_6^2-39*
│ │ │ │ -000272d0: 745f 332a 745f 372d 3239 2a74 5f34 2a74  t_3*t_7-29*t_4*t
│ │ │ │ -000272e0: 5f37 2b39 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  _7+9|.|---------
│ │ │ │ +00027290: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c36 2d32  ----------|.|6-2
│ │ │ │ +000272a0: 322a 745f 312a 745f 362b 3237 2a74 5f32  2*t_1*t_6+27*t_2
│ │ │ │ +000272b0: 2a74 5f36 2d32 332a 745f 342a 745f 362b  *t_6-23*t_4*t_6+
│ │ │ │ +000272c0: 352a 745f 352a 745f 362d 3133 2a74 5f36  5*t_5*t_6-13*t_6
│ │ │ │ +000272d0: 5e32 2d33 392a 745f 332a 745f 372d 3239  ^2-39*t_3*t_7-29
│ │ │ │ +000272e0: 2a74 5f34 2a74 5f37 2b39 7c0a 7c2d 2d2d  *t_4*t_7+9|.|---
│ │ │ │  000272f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027330: 2d2d 2d2d 7c0a 7c2a 745f 352a 745f 372b  ----|.|*t_5*t_7+
│ │ │ │ -00027340: 3339 2a74 5f36 2a74 5f37 2b33 362a 745f  39*t_6*t_7+36*t_
│ │ │ │ -00027350: 332a 745f 382b 3133 2a74 5f34 2a74 5f38  3*t_8+13*t_4*t_8
│ │ │ │ -00027360: 2b32 362a 745f 352a 745f 382b 3337 2a74  +26*t_5*t_8+37*t
│ │ │ │ -00027370: 5f36 2a74 5f38 2c74 5f30 2a74 5f34 2d74  _6*t_8,t_0*t_4-t
│ │ │ │ -00027380: 5f30 2a74 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  _0*t|.|---------
│ │ │ │ +00027330: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00027340: 352a 745f 372b 3339 2a74 5f36 2a74 5f37  5*t_7+39*t_6*t_7
│ │ │ │ +00027350: 2b33 362a 745f 332a 745f 382b 3133 2a74  +36*t_3*t_8+13*t
│ │ │ │ +00027360: 5f34 2a74 5f38 2b32 362a 745f 352a 745f  _4*t_8+26*t_5*t_
│ │ │ │ +00027370: 382b 3337 2a74 5f36 2a74 5f38 2c74 5f30  8+37*t_6*t_8,t_0
│ │ │ │ +00027380: 2a74 5f34 2d74 5f30 2a74 7c0a 7c2d 2d2d  *t_4-t_0*t|.|---
│ │ │ │  00027390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000273d0: 2d2d 2d2d 7c0a 7c5f 352d 382a 745f 312a  ----|.|_5-8*t_1*
│ │ │ │ -000273e0: 745f 352d 3335 2a74 5f32 2a74 5f35 2d31  t_5-35*t_2*t_5-1
│ │ │ │ -000273f0: 302a 745f 332a 745f 352d 3333 2a74 5f34  0*t_3*t_5-33*t_4
│ │ │ │ -00027400: 2a74 5f35 2b35 2a74 5f35 5e32 2b31 352a  *t_5+5*t_5^2+15*
│ │ │ │ -00027410: 745f 302a 745f 362b 3135 2a74 5f31 2a74  t_0*t_6+15*t_1*t
│ │ │ │ -00027420: 5f36 2b35 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  _6+5|.|---------
│ │ │ │ +000273d0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f 352d  ----------|.|_5-
│ │ │ │ +000273e0: 382a 745f 312a 745f 352d 3335 2a74 5f32  8*t_1*t_5-35*t_2
│ │ │ │ +000273f0: 2a74 5f35 2d31 302a 745f 332a 745f 352d  *t_5-10*t_3*t_5-
│ │ │ │ +00027400: 3333 2a74 5f34 2a74 5f35 2b35 2a74 5f35  33*t_4*t_5+5*t_5
│ │ │ │ +00027410: 5e32 2b31 352a 745f 302a 745f 362b 3135  ^2+15*t_0*t_6+15
│ │ │ │ +00027420: 2a74 5f31 2a74 5f36 2b35 7c0a 7c2d 2d2d  *t_1*t_6+5|.|---
│ │ │ │  00027430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027470: 2d2d 2d2d 7c0a 7c2a 745f 322a 745f 362b  ----|.|*t_2*t_6+
│ │ │ │ -00027480: 3135 2a74 5f34 2a74 5f36 2d33 382a 745f  15*t_4*t_6-38*t_
│ │ │ │ -00027490: 352a 745f 362d 3232 2a74 5f36 5e32 2b33  5*t_6-22*t_6^2+3
│ │ │ │ -000274a0: 312a 745f 332a 745f 372d 3235 2a74 5f34  1*t_3*t_7-25*t_4
│ │ │ │ -000274b0: 2a74 5f37 2d31 392a 745f 352a 745f 372b  *t_7-19*t_5*t_7+
│ │ │ │ -000274c0: 3437 2a74 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  47*t|.|---------
│ │ │ │ +00027470: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00027480: 322a 745f 362b 3135 2a74 5f34 2a74 5f36  2*t_6+15*t_4*t_6
│ │ │ │ +00027490: 2d33 382a 745f 352a 745f 362d 3232 2a74  -38*t_5*t_6-22*t
│ │ │ │ +000274a0: 5f36 5e32 2b33 312a 745f 332a 745f 372d  _6^2+31*t_3*t_7-
│ │ │ │ +000274b0: 3235 2a74 5f34 2a74 5f37 2d31 392a 745f  25*t_4*t_7-19*t_
│ │ │ │ +000274c0: 352a 745f 372b 3437 2a74 7c0a 7c2d 2d2d  5*t_7+47*t|.|---
│ │ │ │  000274d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000274e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000274f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027510: 2d2d 2d2d 7c0a 7c5f 362a 745f 372b 3436  ----|.|_6*t_7+46
│ │ │ │ -00027520: 2a74 5f33 2a74 5f38 2d33 362a 745f 342a  *t_3*t_8-36*t_4*
│ │ │ │ -00027530: 745f 382d 3335 2a74 5f35 2a74 5f38 2d33  t_8-35*t_5*t_8-3
│ │ │ │ -00027540: 312a 745f 362a 745f 382c 745f 322a 745f  1*t_6*t_8,t_2*t_
│ │ │ │ -00027550: 332d 745f 302a 745f 352d 745f 312a 745f  3-t_0*t_5-t_1*t_
│ │ │ │ -00027560: 352d 3335 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  5-35|.|---------
│ │ │ │ +00027510: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f 362a  ----------|.|_6*
│ │ │ │ +00027520: 745f 372b 3436 2a74 5f33 2a74 5f38 2d33  t_7+46*t_3*t_8-3
│ │ │ │ +00027530: 362a 745f 342a 745f 382d 3335 2a74 5f35  6*t_4*t_8-35*t_5
│ │ │ │ +00027540: 2a74 5f38 2d33 312a 745f 362a 745f 382c  *t_8-31*t_6*t_8,
│ │ │ │ +00027550: 745f 322a 745f 332d 745f 302a 745f 352d  t_2*t_3-t_0*t_5-
│ │ │ │ +00027560: 745f 312a 745f 352d 3335 7c0a 7c2d 2d2d  t_1*t_5-35|.|---
│ │ │ │  00027570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000275a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000275b0: 2d2d 2d2d 7c0a 7c2a 745f 322a 745f 352d  ----|.|*t_2*t_5-
│ │ │ │ -000275c0: 3130 2a74 5f33 2a74 5f35 2d33 332a 745f  10*t_3*t_5-33*t_
│ │ │ │ -000275d0: 342a 745f 352b 352a 745f 355e 322b 3134  4*t_5+5*t_5^2+14
│ │ │ │ -000275e0: 2a74 5f30 2a74 5f36 2b31 342a 745f 312a  *t_0*t_6+14*t_1*
│ │ │ │ -000275f0: 745f 362b 352a 745f 322a 745f 362b 3134  t_6+5*t_2*t_6+14
│ │ │ │ -00027600: 2a74 5f34 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *t_4|.|---------
│ │ │ │ +000275b0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +000275c0: 322a 745f 352d 3130 2a74 5f33 2a74 5f35  2*t_5-10*t_3*t_5
│ │ │ │ +000275d0: 2d33 332a 745f 342a 745f 352b 352a 745f  -33*t_4*t_5+5*t_
│ │ │ │ +000275e0: 355e 322b 3134 2a74 5f30 2a74 5f36 2b31  5^2+14*t_0*t_6+1
│ │ │ │ +000275f0: 342a 745f 312a 745f 362b 352a 745f 322a  4*t_1*t_6+5*t_2*
│ │ │ │ +00027600: 745f 362b 3134 2a74 5f34 7c0a 7c2d 2d2d  t_6+14*t_4|.|---
│ │ │ │  00027610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027650: 2d2d 2d2d 7c0a 7c2a 745f 362d 3331 2a74  ----|.|*t_6-31*t
│ │ │ │ -00027660: 5f35 2a74 5f36 2d32 342a 745f 365e 322b  _5*t_6-24*t_6^2+
│ │ │ │ -00027670: 3332 2a74 5f33 2a74 5f37 2d32 352a 745f  32*t_3*t_7-25*t_
│ │ │ │ -00027680: 342a 745f 372d 3139 2a74 5f35 2a74 5f37  4*t_7-19*t_5*t_7
│ │ │ │ -00027690: 2b34 372a 745f 362a 745f 372b 3436 2a74  +47*t_6*t_7+46*t
│ │ │ │ -000276a0: 5f33 2a74 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  _3*t|.|---------
│ │ │ │ +00027650: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00027660: 362d 3331 2a74 5f35 2a74 5f36 2d32 342a  6-31*t_5*t_6-24*
│ │ │ │ +00027670: 745f 365e 322b 3332 2a74 5f33 2a74 5f37  t_6^2+32*t_3*t_7
│ │ │ │ +00027680: 2d32 352a 745f 342a 745f 372d 3139 2a74  -25*t_4*t_7-19*t
│ │ │ │ +00027690: 5f35 2a74 5f37 2b34 372a 745f 362a 745f  _5*t_7+47*t_6*t_
│ │ │ │ +000276a0: 372b 3436 2a74 5f33 2a74 7c0a 7c2d 2d2d  7+46*t_3*t|.|---
│ │ │ │  000276b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000276c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000276e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000276f0: 2d2d 2d2d 7c0a 7c5f 382d 3336 2a74 5f34  ----|.|_8-36*t_4
│ │ │ │ -00027700: 2a74 5f38 2d33 352a 745f 352a 745f 382d  *t_8-35*t_5*t_8-
│ │ │ │ -00027710: 3331 2a74 5f36 2a74 5f38 2c74 5f31 2a74  31*t_6*t_8,t_1*t
│ │ │ │ -00027720: 5f33 2d37 2a74 5f31 2a74 5f35 2b74 5f31  _3-7*t_1*t_5+t_1
│ │ │ │ -00027730: 2a74 5f36 2b74 5f34 2a74 5f36 2d37 2a74  *t_6+t_4*t_6-7*t
│ │ │ │ -00027740: 5f35 2a74 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  _5*t|.|---------
│ │ │ │ +000276f0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f 382d  ----------|.|_8-
│ │ │ │ +00027700: 3336 2a74 5f34 2a74 5f38 2d33 352a 745f  36*t_4*t_8-35*t_
│ │ │ │ +00027710: 352a 745f 382d 3331 2a74 5f36 2a74 5f38  5*t_8-31*t_6*t_8
│ │ │ │ +00027720: 2c74 5f31 2a74 5f33 2d37 2a74 5f31 2a74  ,t_1*t_3-7*t_1*t
│ │ │ │ +00027730: 5f35 2b74 5f31 2a74 5f36 2b74 5f34 2a74  _5+t_1*t_6+t_4*t
│ │ │ │ +00027740: 5f36 2d37 2a74 5f35 2a74 7c0a 7c2d 2d2d  _6-7*t_5*t|.|---
│ │ │ │  00027750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027790: 2d2d 2d2d 7c0a 7c5f 362b 322a 745f 365e  ----|.|_6+2*t_6^
│ │ │ │ -000277a0: 322d 745f 332a 745f 372c 745f 302a 745f  2-t_3*t_7,t_0*t_
│ │ │ │ -000277b0: 332d 3436 2a74 5f30 2a74 5f35 2d33 392a  3-46*t_0*t_5-39*
│ │ │ │ -000277c0: 745f 312a 745f 352d 3433 2a74 5f32 2a74  t_1*t_5-43*t_2*t
│ │ │ │ -000277d0: 5f35 2d34 312a 745f 332a 745f 352d 3236  _5-41*t_3*t_5-26
│ │ │ │ -000277e0: 2a74 5f34 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *t_4|.|---------
│ │ │ │ +00027790: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f 362b  ----------|.|_6+
│ │ │ │ +000277a0: 322a 745f 365e 322d 745f 332a 745f 372c  2*t_6^2-t_3*t_7,
│ │ │ │ +000277b0: 745f 302a 745f 332d 3436 2a74 5f30 2a74  t_0*t_3-46*t_0*t
│ │ │ │ +000277c0: 5f35 2d33 392a 745f 312a 745f 352d 3433  _5-39*t_1*t_5-43
│ │ │ │ +000277d0: 2a74 5f32 2a74 5f35 2d34 312a 745f 332a  *t_2*t_5-41*t_3*
│ │ │ │ +000277e0: 745f 352d 3236 2a74 5f34 7c0a 7c2d 2d2d  t_5-26*t_4|.|---
│ │ │ │  000277f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027830: 2d2d 2d2d 7c0a 7c2a 745f 352d 3238 2a74  ----|.|*t_5-28*t
│ │ │ │ -00027840: 5f35 5e32 2d33 352a 745f 302a 745f 362d  _5^2-35*t_0*t_6-
│ │ │ │ -00027850: 3336 2a74 5f31 2a74 5f36 2b32 302a 745f  36*t_1*t_6+20*t_
│ │ │ │ -00027860: 322a 745f 362d 3336 2a74 5f34 2a74 5f36  2*t_6-36*t_4*t_6
│ │ │ │ -00027870: 2b39 2a74 5f35 2a74 5f36 2b31 352a 745f  +9*t_5*t_6+15*t_
│ │ │ │ -00027880: 365e 322b 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  6^2+|.|---------
│ │ │ │ +00027830: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00027840: 352d 3238 2a74 5f35 5e32 2d33 352a 745f  5-28*t_5^2-35*t_
│ │ │ │ +00027850: 302a 745f 362d 3336 2a74 5f31 2a74 5f36  0*t_6-36*t_1*t_6
│ │ │ │ +00027860: 2b32 302a 745f 322a 745f 362d 3336 2a74  +20*t_2*t_6-36*t
│ │ │ │ +00027870: 5f34 2a74 5f36 2b39 2a74 5f35 2a74 5f36  _4*t_6+9*t_5*t_6
│ │ │ │ +00027880: 2b31 352a 745f 365e 322b 7c0a 7c2d 2d2d  +15*t_6^2+|.|---
│ │ │ │  00027890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000278a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000278b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000278c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000278d0: 2d2d 2d2d 7c0a 7c32 362a 745f 332a 745f  ----|.|26*t_3*t_
│ │ │ │ -000278e0: 372d 352a 745f 342a 745f 372b 3335 2a74  7-5*t_4*t_7+35*t
│ │ │ │ -000278f0: 5f35 2a74 5f37 2d31 302a 745f 362a 745f  _5*t_7-10*t_6*t_
│ │ │ │ -00027900: 372d 3130 2a74 5f33 2a74 5f38 2d34 362a  7-10*t_3*t_8-46*
│ │ │ │ -00027910: 745f 342a 745f 382b 3437 2a74 5f35 2a74  t_4*t_8+47*t_5*t
│ │ │ │ -00027920: 5f38 2d20 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  _8- |.|---------
│ │ │ │ +000278d0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c32 362a  ----------|.|26*
│ │ │ │ +000278e0: 745f 332a 745f 372d 352a 745f 342a 745f  t_3*t_7-5*t_4*t_
│ │ │ │ +000278f0: 372b 3335 2a74 5f35 2a74 5f37 2d31 302a  7+35*t_5*t_7-10*
│ │ │ │ +00027900: 745f 362a 745f 372d 3130 2a74 5f33 2a74  t_6*t_7-10*t_3*t
│ │ │ │ +00027910: 5f38 2d34 362a 745f 342a 745f 382b 3437  _8-46*t_4*t_8+47
│ │ │ │ +00027920: 2a74 5f35 2a74 5f38 2d20 7c0a 7c2d 2d2d  *t_5*t_8- |.|---
│ │ │ │  00027930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027970: 2d2d 2d2d 7c0a 7c32 352a 745f 362a 745f  ----|.|25*t_6*t_
│ │ │ │ -00027980: 382c 745f 325e 322d 3436 2a74 5f31 2a74  8,t_2^2-46*t_1*t
│ │ │ │ -00027990: 5f34 2d33 332a 745f 302a 745f 352d 3435  _4-33*t_0*t_5-45
│ │ │ │ -000279a0: 2a74 5f31 2a74 5f35 2d33 392a 745f 322a  *t_1*t_5-39*t_2*
│ │ │ │ -000279b0: 745f 352d 3339 2a74 5f33 2a74 5f35 2d34  t_5-39*t_3*t_5-4
│ │ │ │ -000279c0: 362a 745f 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  6*t_|.|---------
│ │ │ │ +00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c32 352a  ----------|.|25*
│ │ │ │ +00027980: 745f 362a 745f 382c 745f 325e 322d 3436  t_6*t_8,t_2^2-46
│ │ │ │ +00027990: 2a74 5f31 2a74 5f34 2d33 332a 745f 302a  *t_1*t_4-33*t_0*
│ │ │ │ +000279a0: 745f 352d 3435 2a74 5f31 2a74 5f35 2d33  t_5-45*t_1*t_5-3
│ │ │ │ +000279b0: 392a 745f 322a 745f 352d 3339 2a74 5f33  9*t_2*t_5-39*t_3
│ │ │ │ +000279c0: 2a74 5f35 2d34 362a 745f 7c0a 7c2d 2d2d  *t_5-46*t_|.|---
│ │ │ │  000279d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000279e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000279f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027a10: 2d2d 2d2d 7c0a 7c34 2a74 5f35 2d32 392a  ----|.|4*t_5-29*
│ │ │ │ -00027a20: 745f 355e 322d 3438 2a74 5f30 2a74 5f36  t_5^2-48*t_0*t_6
│ │ │ │ -00027a30: 2d33 382a 745f 312a 745f 362d 3330 2a74  -38*t_1*t_6-30*t
│ │ │ │ -00027a40: 5f32 2a74 5f36 2b31 392a 745f 342a 745f  _2*t_6+19*t_4*t_
│ │ │ │ -00027a50: 362d 3434 2a74 5f35 2a74 5f36 2d34 372a  6-44*t_5*t_6-47*
│ │ │ │ -00027a60: 745f 365e 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  t_6^|.|---------
│ │ │ │ +00027a10: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34 2a74  ----------|.|4*t
│ │ │ │ +00027a20: 5f35 2d32 392a 745f 355e 322d 3438 2a74  _5-29*t_5^2-48*t
│ │ │ │ +00027a30: 5f30 2a74 5f36 2d33 382a 745f 312a 745f  _0*t_6-38*t_1*t_
│ │ │ │ +00027a40: 362d 3330 2a74 5f32 2a74 5f36 2b31 392a  6-30*t_2*t_6+19*
│ │ │ │ +00027a50: 745f 342a 745f 362d 3434 2a74 5f35 2a74  t_4*t_6-44*t_5*t
│ │ │ │ +00027a60: 5f36 2d34 372a 745f 365e 7c0a 7c2d 2d2d  _6-47*t_6^|.|---
│ │ │ │  00027a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ab0: 2d2d 2d2d 7c0a 7c32 2d33 362a 745f 302a  ----|.|2-36*t_0*
│ │ │ │ -00027ac0: 745f 372d 3436 2a74 5f31 2a74 5f37 2b74  t_7-46*t_1*t_7+t
│ │ │ │ -00027ad0: 5f32 2a74 5f37 2d34 342a 745f 332a 745f  _2*t_7-44*t_3*t_
│ │ │ │ -00027ae0: 372b 3438 2a74 5f34 2a74 5f37 2d31 342a  7+48*t_4*t_7-14*
│ │ │ │ -00027af0: 745f 352a 745f 372b 342a 745f 362a 745f  t_5*t_7+4*t_6*t_
│ │ │ │ -00027b00: 372d 3336 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  7-36|.|---------
│ │ │ │ +00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c32 2d33  ----------|.|2-3
│ │ │ │ +00027ac0: 362a 745f 302a 745f 372d 3436 2a74 5f31  6*t_0*t_7-46*t_1
│ │ │ │ +00027ad0: 2a74 5f37 2b74 5f32 2a74 5f37 2d34 342a  *t_7+t_2*t_7-44*
│ │ │ │ +00027ae0: 745f 332a 745f 372b 3438 2a74 5f34 2a74  t_3*t_7+48*t_4*t
│ │ │ │ +00027af0: 5f37 2d31 342a 745f 352a 745f 372b 342a  _7-14*t_5*t_7+4*
│ │ │ │ +00027b00: 745f 362a 745f 372d 3336 7c0a 7c2d 2d2d  t_6*t_7-36|.|---
│ │ │ │  00027b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027b50: 2d2d 2d2d 7c0a 7c2a 745f 302a 745f 382d  ----|.|*t_0*t_8-
│ │ │ │ -00027b60: 3436 2a74 5f31 2a74 5f38 2b34 372a 745f  46*t_1*t_8+47*t_
│ │ │ │ -00027b70: 322a 745f 382d 3334 2a74 5f33 2a74 5f38  2*t_8-34*t_3*t_8
│ │ │ │ -00027b80: 2d32 342a 745f 342a 745f 382d 3132 2a74  -24*t_4*t_8-12*t
│ │ │ │ -00027b90: 5f35 2a74 5f38 2d34 372a 745f 362a 745f  _5*t_8-47*t_6*t_
│ │ │ │ -00027ba0: 382b 3437 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  8+47|.|---------
│ │ │ │ +00027b50: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00027b60: 302a 745f 382d 3436 2a74 5f31 2a74 5f38  0*t_8-46*t_1*t_8
│ │ │ │ +00027b70: 2b34 372a 745f 322a 745f 382d 3334 2a74  +47*t_2*t_8-34*t
│ │ │ │ +00027b80: 5f33 2a74 5f38 2d32 342a 745f 342a 745f  _3*t_8-24*t_4*t_
│ │ │ │ +00027b90: 382d 3132 2a74 5f35 2a74 5f38 2d34 372a  8-12*t_5*t_8-47*
│ │ │ │ +00027ba0: 745f 362a 745f 382b 3437 7c0a 7c2d 2d2d  t_6*t_8+47|.|---
│ │ │ │  00027bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027bf0: 2d2d 2d2d 7c0a 7c2a 745f 372a 745f 382c  ----|.|*t_7*t_8,
│ │ │ │ -00027c00: 745f 312a 745f 322b 362a 745f 312a 745f  t_1*t_2+6*t_1*t_
│ │ │ │ -00027c10: 352b 352a 745f 302a 745f 362d 322a 745f  5+5*t_0*t_6-2*t_
│ │ │ │ -00027c20: 312a 745f 362d 745f 342a 745f 362d 745f  1*t_6-t_4*t_6-t_
│ │ │ │ -00027c30: 352a 745f 362b 352a 745f 302a 745f 372b  5*t_6+5*t_0*t_7+
│ │ │ │ -00027c40: 745f 312a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  t_1*|.|---------
│ │ │ │ +00027bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00027c00: 372a 745f 382c 745f 312a 745f 322b 362a  7*t_8,t_1*t_2+6*
│ │ │ │ +00027c10: 745f 312a 745f 352b 352a 745f 302a 745f  t_1*t_5+5*t_0*t_
│ │ │ │ +00027c20: 362d 322a 745f 312a 745f 362d 745f 342a  6-2*t_1*t_6-t_4*
│ │ │ │ +00027c30: 745f 362d 745f 352a 745f 362b 352a 745f  t_6-t_5*t_6+5*t_
│ │ │ │ +00027c40: 302a 745f 372b 745f 312a 7c0a 7c2d 2d2d  0*t_7+t_1*|.|---
│ │ │ │  00027c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027c90: 2d2d 2d2d 7c0a 7c74 5f37 2d32 2a74 5f32  ----|.|t_7-2*t_2
│ │ │ │ -00027ca0: 2a74 5f37 2d37 2a74 5f35 2a74 5f37 2b32  *t_7-7*t_5*t_7+2
│ │ │ │ -00027cb0: 2a74 5f36 2a74 5f37 2d32 2a74 5f31 2a74  *t_6*t_7-2*t_1*t
│ │ │ │ -00027cc0: 5f38 2b33 2a74 5f37 2a74 5f38 2c74 5f30  _8+3*t_7*t_8,t_0
│ │ │ │ -00027cd0: 2a74 5f32 2b74 5f31 2a74 5f34 2b35 2a74  *t_2+t_1*t_4+5*t
│ │ │ │ -00027ce0: 5f30 2a74 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  _0*t|.|---------
│ │ │ │ +00027c90: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c74 5f37  ----------|.|t_7
│ │ │ │ +00027ca0: 2d32 2a74 5f32 2a74 5f37 2d37 2a74 5f35  -2*t_2*t_7-7*t_5
│ │ │ │ +00027cb0: 2a74 5f37 2b32 2a74 5f36 2a74 5f37 2d32  *t_7+2*t_6*t_7-2
│ │ │ │ +00027cc0: 2a74 5f31 2a74 5f38 2b33 2a74 5f37 2a74  *t_1*t_8+3*t_7*t
│ │ │ │ +00027cd0: 5f38 2c74 5f30 2a74 5f32 2b74 5f31 2a74  _8,t_0*t_2+t_1*t
│ │ │ │ +00027ce0: 5f34 2b35 2a74 5f30 2a74 7c0a 7c2d 2d2d  _4+5*t_0*t|.|---
│ │ │ │  00027cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027d30: 2d2d 2d2d 7c0a 7c5f 352b 3332 2a74 5f31  ----|.|_5+32*t_1
│ │ │ │ -00027d40: 2a74 5f35 2d32 302a 745f 322a 745f 352d  *t_5-20*t_2*t_5-
│ │ │ │ -00027d50: 3437 2a74 5f33 2a74 5f35 2d33 372a 745f  47*t_3*t_5-37*t_
│ │ │ │ -00027d60: 342a 745f 352d 3235 2a74 5f35 5e32 2b31  4*t_5-25*t_5^2+1
│ │ │ │ -00027d70: 392a 745f 302a 745f 362b 3232 2a74 5f31  9*t_0*t_6+22*t_1
│ │ │ │ -00027d80: 2a74 5f36 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *t_6|.|---------
│ │ │ │ +00027d30: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f 352b  ----------|.|_5+
│ │ │ │ +00027d40: 3332 2a74 5f31 2a74 5f35 2d32 302a 745f  32*t_1*t_5-20*t_
│ │ │ │ +00027d50: 322a 745f 352d 3437 2a74 5f33 2a74 5f35  2*t_5-47*t_3*t_5
│ │ │ │ +00027d60: 2d33 372a 745f 342a 745f 352d 3235 2a74  -37*t_4*t_5-25*t
│ │ │ │ +00027d70: 5f35 5e32 2b31 392a 745f 302a 745f 362b  _5^2+19*t_0*t_6+
│ │ │ │ +00027d80: 3232 2a74 5f31 2a74 5f36 7c0a 7c2d 2d2d  22*t_1*t_6|.|---
│ │ │ │  00027d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 7c0a 7c2d 3235 2a74 5f32 2a74  ----|.|-25*t_2*t
│ │ │ │ -00027de0: 5f36 2b32 352a 745f 342a 745f 362d 352a  _6+25*t_4*t_6-5*
│ │ │ │ -00027df0: 745f 352a 745f 362b 3133 2a74 5f36 5e32  t_5*t_6+13*t_6^2
│ │ │ │ -00027e00: 2b35 2a74 5f30 2a74 5f37 2b74 5f31 2a74  +5*t_0*t_7+t_1*t
│ │ │ │ -00027e10: 5f37 2b33 392a 745f 332a 745f 372b 3238  _7+39*t_3*t_7+28
│ │ │ │ -00027e20: 2a74 5f34 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *t_4|.|---------
│ │ │ │ +00027dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d 3235  ----------|.|-25
│ │ │ │ +00027de0: 2a74 5f32 2a74 5f36 2b32 352a 745f 342a  *t_2*t_6+25*t_4*
│ │ │ │ +00027df0: 745f 362d 352a 745f 352a 745f 362b 3133  t_6-5*t_5*t_6+13
│ │ │ │ +00027e00: 2a74 5f36 5e32 2b35 2a74 5f30 2a74 5f37  *t_6^2+5*t_0*t_7
│ │ │ │ +00027e10: 2b74 5f31 2a74 5f37 2b33 392a 745f 332a  +t_1*t_7+39*t_3*
│ │ │ │ +00027e20: 745f 372b 3238 2a74 5f34 7c0a 7c2d 2d2d  t_7+28*t_4|.|---
│ │ │ │  00027e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e70: 2d2d 2d2d 7c0a 7c2a 745f 372d 392a 745f  ----|.|*t_7-9*t_
│ │ │ │ -00027e80: 352a 745f 372d 3339 2a74 5f36 2a74 5f37  5*t_7-39*t_6*t_7
│ │ │ │ -00027e90: 2b34 2a74 5f30 2a74 5f38 2b74 5f31 2a74  +4*t_0*t_8+t_1*t
│ │ │ │ -00027ea0: 5f38 2d33 362a 745f 332a 745f 382d 3134  _8-36*t_3*t_8-14
│ │ │ │ -00027eb0: 2a74 5f34 2a74 5f38 2d32 362a 745f 352a  *t_4*t_8-26*t_5*
│ │ │ │ -00027ec0: 745f 382d 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  t_8-|.|---------
│ │ │ │ +00027e70: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +00027e80: 372d 392a 745f 352a 745f 372d 3339 2a74  7-9*t_5*t_7-39*t
│ │ │ │ +00027e90: 5f36 2a74 5f37 2b34 2a74 5f30 2a74 5f38  _6*t_7+4*t_0*t_8
│ │ │ │ +00027ea0: 2b74 5f31 2a74 5f38 2d33 362a 745f 332a  +t_1*t_8-36*t_3*
│ │ │ │ +00027eb0: 745f 382d 3134 2a74 5f34 2a74 5f38 2d32  t_8-14*t_4*t_8-2
│ │ │ │ +00027ec0: 362a 745f 352a 745f 382d 7c0a 7c2d 2d2d  6*t_5*t_8-|.|---
│ │ │ │  00027ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027f10: 2d2d 2d2d 7c0a 7c33 372a 745f 362a 745f  ----|.|37*t_6*t_
│ │ │ │ -00027f20: 382c 745f 302a 745f 312d 3339 2a74 5f31  8,t_0*t_1-39*t_1
│ │ │ │ -00027f30: 2a74 5f34 2b34 302a 745f 312a 745f 352d  *t_4+40*t_1*t_5-
│ │ │ │ -00027f40: 3337 2a74 5f30 2a74 5f36 2d33 392a 745f  37*t_0*t_6-39*t_
│ │ │ │ -00027f50: 312a 745f 362b 3139 2a74 5f34 2a74 5f36  1*t_6+19*t_4*t_6
│ │ │ │ -00027f60: 2d33 392a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  -39*|.|---------
│ │ │ │ +00027f10: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c33 372a  ----------|.|37*
│ │ │ │ +00027f20: 745f 362a 745f 382c 745f 302a 745f 312d  t_6*t_8,t_0*t_1-
│ │ │ │ +00027f30: 3339 2a74 5f31 2a74 5f34 2b34 302a 745f  39*t_1*t_4+40*t_
│ │ │ │ +00027f40: 312a 745f 352d 3337 2a74 5f30 2a74 5f36  1*t_5-37*t_0*t_6
│ │ │ │ +00027f50: 2d33 392a 745f 312a 745f 362b 3139 2a74  -39*t_1*t_6+19*t
│ │ │ │ +00027f60: 5f34 2a74 5f36 2d33 392a 7c0a 7c2d 2d2d  _4*t_6-39*|.|---
│ │ │ │  00027f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027fb0: 2d2d 2d2d 7c0a 7c74 5f35 2a74 5f36 2d33  ----|.|t_5*t_6-3
│ │ │ │ -00027fc0: 382a 745f 302a 745f 372b 3339 2a74 5f31  8*t_0*t_7+39*t_1
│ │ │ │ -00027fd0: 2a74 5f37 2b31 392a 745f 322a 745f 372b  *t_7+19*t_2*t_7+
│ │ │ │ -00027fe0: 3138 2a74 5f35 2a74 5f37 2d31 392a 745f  18*t_5*t_7-19*t_
│ │ │ │ -00027ff0: 362a 745f 372b 3139 2a74 5f31 2a74 5f38  6*t_7+19*t_1*t_8
│ │ │ │ -00028000: 2b32 302a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  +20*|.|---------
│ │ │ │ +00027fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c74 5f35  ----------|.|t_5
│ │ │ │ +00027fc0: 2a74 5f36 2d33 382a 745f 302a 745f 372b  *t_6-38*t_0*t_7+
│ │ │ │ +00027fd0: 3339 2a74 5f31 2a74 5f37 2b31 392a 745f  39*t_1*t_7+19*t_
│ │ │ │ +00027fe0: 322a 745f 372b 3138 2a74 5f35 2a74 5f37  2*t_7+18*t_5*t_7
│ │ │ │ +00027ff0: 2d31 392a 745f 362a 745f 372b 3139 2a74  -19*t_6*t_7+19*t
│ │ │ │ +00028000: 5f31 2a74 5f38 2b32 302a 7c0a 7c2d 2d2d  _1*t_8+20*|.|---
│ │ │ │  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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028050: 2d2d 2d2d 7c0a 7c74 5f37 2a74 5f38 2c74  ----|.|t_7*t_8,t
│ │ │ │ -00028060: 5f30 5e32 2b31 322a 745f 312a 745f 342b  _0^2+12*t_1*t_4+
│ │ │ │ -00028070: 3230 2a74 5f30 2a74 5f35 2b32 372a 745f  20*t_0*t_5+27*t_
│ │ │ │ -00028080: 312a 745f 352d 382a 745f 322a 745f 352b  1*t_5-8*t_2*t_5+
│ │ │ │ -00028090: 3337 2a74 5f33 2a74 5f35 2b32 382a 745f  37*t_3*t_5+28*t_
│ │ │ │ -000280a0: 342a 745f 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  4*t_|.|---------
│ │ │ │ +00028050: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c74 5f37  ----------|.|t_7
│ │ │ │ +00028060: 2a74 5f38 2c74 5f30 5e32 2b31 322a 745f  *t_8,t_0^2+12*t_
│ │ │ │ +00028070: 312a 745f 342b 3230 2a74 5f30 2a74 5f35  1*t_4+20*t_0*t_5
│ │ │ │ +00028080: 2b32 372a 745f 312a 745f 352d 382a 745f  +27*t_1*t_5-8*t_
│ │ │ │ +00028090: 322a 745f 352b 3337 2a74 5f33 2a74 5f35  2*t_5+37*t_3*t_5
│ │ │ │ +000280a0: 2b32 382a 745f 342a 745f 7c0a 7c2d 2d2d  +28*t_4*t_|.|---
│ │ │ │  000280b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000280c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000280d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000280e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280f0: 2d2d 2d2d 7c0a 7c35 2b33 302a 745f 355e  ----|.|5+30*t_5^
│ │ │ │ -00028100: 322d 3436 2a74 5f30 2a74 5f36 2b32 342a  2-46*t_0*t_6+24*
│ │ │ │ -00028110: 745f 312a 745f 362d 3430 2a74 5f32 2a74  t_1*t_6-40*t_2*t
│ │ │ │ -00028120: 5f36 2b32 352a 745f 342a 745f 362b 3136  _6+25*t_4*t_6+16
│ │ │ │ -00028130: 2a74 5f35 2a74 5f36 2d33 352a 745f 365e  *t_5*t_6-35*t_6^
│ │ │ │ -00028140: 322b 3239 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  2+29|.|---------
│ │ │ │ +000280f0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c35 2b33  ----------|.|5+3
│ │ │ │ +00028100: 302a 745f 355e 322d 3436 2a74 5f30 2a74  0*t_5^2-46*t_0*t
│ │ │ │ +00028110: 5f36 2b32 342a 745f 312a 745f 362d 3430  _6+24*t_1*t_6-40
│ │ │ │ +00028120: 2a74 5f32 2a74 5f36 2b32 352a 745f 342a  *t_2*t_6+25*t_4*
│ │ │ │ +00028130: 745f 362b 3136 2a74 5f35 2a74 5f36 2d33  t_6+16*t_5*t_6-3
│ │ │ │ +00028140: 352a 745f 365e 322b 3239 7c0a 7c2d 2d2d  5*t_6^2+29|.|---
│ │ │ │  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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028190: 2d2d 2d2d 7c0a 7c2a 745f 302a 745f 372b  ----|.|*t_0*t_7+
│ │ │ │ -000281a0: 3132 2a74 5f31 2a74 5f37 2d33 352a 745f  12*t_1*t_7-35*t_
│ │ │ │ -000281b0: 322a 745f 372d 382a 745f 332a 745f 372d  2*t_7-8*t_3*t_7-
│ │ │ │ -000281c0: 3138 2a74 5f34 2a74 5f37 2b34 322a 745f  18*t_4*t_7+42*t_
│ │ │ │ -000281d0: 352a 745f 372d 3132 2a74 5f36 2a74 5f37  5*t_7-12*t_6*t_7
│ │ │ │ -000281e0: 2d36 2a74 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  -6*t|.|---------
│ │ │ │ +00028190: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 745f  ----------|.|*t_
│ │ │ │ +000281a0: 302a 745f 372b 3132 2a74 5f31 2a74 5f37  0*t_7+12*t_1*t_7
│ │ │ │ +000281b0: 2d33 352a 745f 322a 745f 372d 382a 745f  -35*t_2*t_7-8*t_
│ │ │ │ +000281c0: 332a 745f 372d 3138 2a74 5f34 2a74 5f37  3*t_7-18*t_4*t_7
│ │ │ │ +000281d0: 2b34 322a 745f 352a 745f 372d 3132 2a74  +42*t_5*t_7-12*t
│ │ │ │ +000281e0: 5f36 2a74 5f37 2d36 2a74 7c0a 7c2d 2d2d  _6*t_7-6*t|.|---
│ │ │ │  000281f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028230: 2d2d 2d2d 7c0a 7c5f 302a 745f 382b 3132  ----|.|_0*t_8+12
│ │ │ │ -00028240: 2a74 5f31 2a74 5f38 2d31 352a 745f 332a  *t_1*t_8-15*t_3*
│ │ │ │ -00028250: 745f 382b 392a 745f 342a 745f 382b 3230  t_8+9*t_4*t_8+20
│ │ │ │ -00028260: 2a74 5f35 2a74 5f38 2d33 302a 745f 362a  *t_5*t_8-30*t_6*
│ │ │ │ -00028270: 745f 382b 342a 745f 372a 745f 387d 2920  t_8+4*t_7*t_8}) 
│ │ │ │ -00028280: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00028230: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f 302a  ----------|.|_0*
│ │ │ │ +00028240: 745f 382b 3132 2a74 5f31 2a74 5f38 2d31  t_8+12*t_1*t_8-1
│ │ │ │ +00028250: 352a 745f 332a 745f 382b 392a 745f 342a  5*t_3*t_8+9*t_4*
│ │ │ │ +00028260: 745f 382b 3230 2a74 5f35 2a74 5f38 2d33  t_8+20*t_5*t_8-3
│ │ │ │ +00028270: 302a 745f 362a 745f 382b 342a 745f 372a  0*t_6*t_8+4*t_7*
│ │ │ │ +00028280: 745f 387d 2920 2020 2020 7c0a 2b2d 2d2d  t_8})     |.+---
│ │ │ │  00028290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000282a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000282b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000282c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000282d0: 2d2d 2d2d 2b0a 7c69 3820 3a20 2d2d 2077  ----+.|i8 : -- w
│ │ │ │ -000282e0: 6974 686f 7574 2074 6865 206f 7074 696f  ithout the optio
│ │ │ │ -000282f0: 6e20 2743 6f64 696d 4273 496e 763d 3e34  n 'CodimBsInv=>4
│ │ │ │ -00028300: 272c 2069 7420 7461 6b65 7320 6162 6f75  ', it takes abou
│ │ │ │ -00028310: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00028320: 2020 2020 7c0a 7c20 2020 2020 7469 6d65      |.|     time
│ │ │ │ -00028330: 2070 7369 3d61 7070 726f 7869 6d61 7465   psi=approximate
│ │ │ │ -00028340: 496e 7665 7273 654d 6170 2870 6869 2c43  InverseMap(phi,C
│ │ │ │ -00028350: 6f64 696d 4273 496e 763d 3e34 2920 2020  odimBsInv=>4)   
│ │ │ │ -00028360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028370: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00028380: 322e 3238 3933 3473 2028 6370 7529 3b20  2.28934s (cpu); 
│ │ │ │ -00028390: 312e 3639 3538 7320 2874 6872 6561 6429  1.6958s (thread)
│ │ │ │ -000283a0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +000282d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ +000282e0: 3a20 2d2d 2077 6974 686f 7574 2074 6865  : -- without the
│ │ │ │ +000282f0: 206f 7074 696f 6e20 2743 6f64 696d 4273   option 'CodimBs
│ │ │ │ +00028300: 496e 763d 3e34 272c 2069 7420 7461 6b65  Inv=>4', it take
│ │ │ │ +00028310: 7320 6162 6f75 7420 2020 2020 2020 2020  s about         
│ │ │ │ +00028320: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00028330: 2020 7469 6d65 2070 7369 3d61 7070 726f    time psi=appro
│ │ │ │ +00028340: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ +00028350: 2870 6869 2c43 6f64 696d 4273 496e 763d  (phi,CodimBsInv=
│ │ │ │ +00028360: 3e34 2920 2020 2020 2020 2020 2020 2020  >4)             
│ │ │ │ +00028370: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ +00028380: 2075 7365 6420 322e 3039 3732 3773 2028   used 2.09727s (
│ │ │ │ +00028390: 6370 7529 3b20 312e 3739 3037 3773 2028  cpu); 1.79077s (
│ │ │ │ +000283a0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  000283b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283c0: 2020 2020 7c0a 7c2d 2d20 6170 7072 6f78      |.|-- approx
│ │ │ │ -000283d0: 696d 6174 6549 6e76 6572 7365 4d61 703a  imateInverseMap:
│ │ │ │ -000283e0: 2073 7465 7020 3120 6f66 2033 2020 2020   step 1 of 3    
│ │ │ │ -000283f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000283c0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d20            |.|-- 
│ │ │ │ +000283d0: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +000283e0: 7365 4d61 703a 2073 7465 7020 3120 6f66  seMap: step 1 of
│ │ │ │ +000283f0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  00028400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028410: 2020 2020 7c0a 7c2d 2d20 6170 7072 6f78      |.|-- approx
│ │ │ │ -00028420: 696d 6174 6549 6e76 6572 7365 4d61 703a  imateInverseMap:
│ │ │ │ -00028430: 2073 7465 7020 3220 6f66 2033 2020 2020   step 2 of 3    
│ │ │ │ -00028440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028410: 2020 2020 2020 2020 2020 7c0a 7c2d 2d20            |.|-- 
│ │ │ │ +00028420: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +00028430: 7365 4d61 703a 2073 7465 7020 3220 6f66  seMap: step 2 of
│ │ │ │ +00028440: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  00028450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028460: 2020 2020 7c0a 7c2d 2d20 6170 7072 6f78      |.|-- approx
│ │ │ │ -00028470: 696d 6174 6549 6e76 6572 7365 4d61 703a  imateInverseMap:
│ │ │ │ -00028480: 2073 7465 7020 3320 6f66 2033 2020 2020   step 3 of 3    
│ │ │ │ -00028490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028460: 2020 2020 2020 2020 2020 7c0a 7c2d 2d20            |.|-- 
│ │ │ │ +00028470: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +00028480: 7365 4d61 703a 2073 7465 7020 3320 6f66  seMap: step 3 of
│ │ │ │ +00028490: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  000284a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000284b0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00028500: 2020 2020 7c0a 7c6f 3820 3d20 2d2d 2072      |.|o8 = -- r
│ │ │ │ -00028510: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │ -00028520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028500: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +00028510: 3d20 2d2d 2072 6174 696f 6e61 6c20 6d61  = -- rational ma
│ │ │ │ +00028520: 7020 2d2d 2020 2020 2020 2020 2020 2020  p --            
│ │ │ │  00028530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028550: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00028550: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00028560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028570: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +00028570: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │  00028580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000285a0: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ -000285b0: 6365 3a20 7375 6276 6172 6965 7479 206f  ce: subvariety o
│ │ │ │ -000285c0: 6620 5072 6f6a 282d 2d5b 7820 2c20 7820  f Proj(--[x , x 
│ │ │ │ -000285d0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ -000285e0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -000285f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  000288e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00028910: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  00028920: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00028940: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00028ab0: 6574 3a20 5072 6f6a 282d 2d5b 7420 2c20  et: Proj(--[t , 
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│ │ │ │ -00028b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b40: 2020 2020 7c0a 7c20 2020 2020 6465 6669      |.|     defi
│ │ │ │ -00028b50: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │ -00028b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00028ab0: 2020 7461 7267 6574 3a20 5072 6f6a 282d    target: Proj(-
│ │ │ │ +00028ac0: 2d5b 7420 2c20 7420 2c20 7420 2c20 7420  -[t , t , t , t 
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│ │ │ │ +00028b50: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ +00028b60: 3a20 7b20 2020 2020 2020 2020 2020 2020  : {             
│ │ │ │  00028b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00028ba0: 2020 2020 2020 2020 2020 2020 2078 2078               x x
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│ │ │ │ -00028bf0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -00028c00: 3520 2020 2020 3520 3720 2020 2031 2038  5     5 7    1 8
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│ │ │ │ -00028c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00028b90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028bb0: 2020 2078 2078 2020 2b20 3278 2078 2020     x x  + 2x x  
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│ │ │ │ +00028be0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00028c90: 2020 2020 2020 2020 2020 2020 2078 2078               x x
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│ │ │ │  00028e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00028e70: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ -00028e80: 2020 2d20 3478 2078 2020 2b20 3436 7820    - 4x x  + 46x 
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│ │ │ │ -00028ec0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
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│ │ │ │ +00028e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00028f60: 2020 2020 2020 2020 2020 2020 2078 2078               x x
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│ │ │ │ +00028fd0: 2020 2020 2020 3320 3620 2020 2031 2037        3 6    1 7
│ │ │ │ +00028fe0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00028ff0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029040: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00029050: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00029060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029040: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00029050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029060: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00029070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029090: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000290a0: 2020 2020 2020 2020 2020 2020 2078 2020               x  
│ │ │ │ -000290b0: 2d20 3332 7820 7820 202b 2034 3278 2078  - 32x x  + 42x x
│ │ │ │ -000290c0: 2020 2b20 3237 7820 7820 202d 2034 3278    + 27x x  - 42x
│ │ │ │ -000290d0: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ -000290e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000290f0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -00029100: 2020 2020 2030 2034 2020 2020 2020 3020       0 4      0 
│ │ │ │ -00029110: 3620 2020 2020 2032 2036 2020 2020 2020  6      2 6      
│ │ │ │ -00029120: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00029130: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029090: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000290a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000290b0: 2020 2078 2020 2d20 3332 7820 7820 202b     x  - 32x x  +
│ │ │ │ +000290c0: 2034 3278 2078 2020 2b20 3237 7820 7820   42x x  + 27x x 
│ │ │ │ +000290d0: 202d 2034 3278 2078 2020 2020 2020 2020   - 42x x        
│ │ │ │ +000290e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000290f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029100: 2020 2020 3320 2020 2020 2030 2034 2020      3      0 4  
│ │ │ │ +00029110: 2020 2020 3020 3620 2020 2020 2032 2036      0 6      2 6
│ │ │ │ +00029120: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00029130: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029180: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00029190: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ -000291a0: 2020 2b20 7820 7820 202d 2031 3678 2078    + x x  - 16x x
│ │ │ │ -000291b0: 2020 2d20 3130 7820 7820 202b 2032 7820    - 10x x  + 2x 
│ │ │ │ -000291c0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -000291d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000291e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000291f0: 3320 2020 2030 2034 2020 2020 2020 3220  3    0 4      2 
│ │ │ │ -00029200: 3620 2020 2020 2033 2036 2020 2020 2030  6      3 6     0
│ │ │ │ -00029210: 2037 2020 2020 2020 2020 2020 2020 2020   7              
│ │ │ │ -00029220: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029180: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00029190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000291a0: 2020 2078 2078 2020 2b20 7820 7820 202d     x x  + x x  -
│ │ │ │ +000291b0: 2031 3678 2078 2020 2d20 3130 7820 7820   16x x  - 10x x 
│ │ │ │ +000291c0: 202b 2032 7820 7820 2020 2020 2020 2020   + 2x x         
│ │ │ │ +000291d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000291e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000291f0: 2020 2020 3220 3320 2020 2030 2034 2020      2 3    0 4  
│ │ │ │ +00029200: 2020 2020 3220 3620 2020 2020 2033 2036      2 6      3 6
│ │ │ │ +00029210: 2020 2020 2030 2037 2020 2020 2020 2020       0 7        
│ │ │ │ +00029220: 2020 2020 2020 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029270: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00029280: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00029290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029270: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00029280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029290: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  000292a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000292b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000292c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000292d0: 2020 2020 2020 2020 2020 2020 2078 2020               x  
│ │ │ │ -000292e0: 2b20 3336 7820 7820 202b 2078 2078 2020  + 36x x  + x x  
│ │ │ │ -000292f0: 2b20 3778 2078 2020 2d20 3478 2078 2020  + 7x x  - 4x x  
│ │ │ │ -00029300: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ -00029310: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00029320: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00029330: 2020 2020 2030 2034 2020 2020 3420 3520       0 4    4 5 
│ │ │ │ -00029340: 2020 2020 3020 3620 2020 2020 3120 3620      0 6     1 6 
│ │ │ │ -00029350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000292c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000292d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000292e0: 2020 2078 2020 2b20 3336 7820 7820 202b     x  + 36x x  +
│ │ │ │ +000292f0: 2078 2078 2020 2b20 3778 2078 2020 2d20   x x  + 7x x  - 
│ │ │ │ +00029300: 3478 2078 2020 2d20 2020 2020 2020 2020  4x x  -         
│ │ │ │ +00029310: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00029320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029330: 2020 2020 3120 2020 2020 2030 2034 2020      1      0 4  
│ │ │ │ +00029340: 2020 3420 3520 2020 2020 3020 3620 2020    4 5     0 6   
│ │ │ │ +00029350: 2020 3120 3620 2020 2020 2020 2020 2020    1 6           
│ │ │ │ +00029360: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -000293b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000293c0: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ -000293d0: 2020 2b20 3278 2078 2020 2d20 3578 2078    + 2x x  - 5x x
│ │ │ │ -000293e0: 2020 2d20 3334 7820 7820 202d 2031 3678    - 34x x  - 16x
│ │ │ │ -000293f0: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ -00029400: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00029410: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00029420: 3120 2020 2020 3020 3220 2020 2020 3020  1     0 2     0 
│ │ │ │ -00029430: 3320 2020 2020 2030 2034 2020 2020 2020  3      0 4      
│ │ │ │ -00029440: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00029450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00029460: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ -00029470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000293b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000293c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000293d0: 2020 2078 2078 2020 2b20 3278 2078 2020     x x  + 2x x  
│ │ │ │ +000293e0: 2d20 3578 2078 2020 2d20 3334 7820 7820  - 5x x  - 34x x 
│ │ │ │ +000293f0: 202d 2031 3678 2078 2020 2020 2020 2020   - 16x x        
│ │ │ │ +00029400: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00029410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029420: 2020 2020 3020 3120 2020 2020 3020 3220      0 1     0 2 
│ │ │ │ +00029430: 2020 2020 3020 3320 2020 2020 2030 2034      0 3      0 4
│ │ │ │ +00029440: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00029450: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00029460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029470: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │  00029480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000294a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000294a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000294b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000294c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000294d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000294e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000294f0: 2020 2020 7c0a 7c6f 3820 3a20 5261 7469      |.|o8 : Rati
│ │ │ │ -00029500: 6f6e 616c 4d61 7020 2871 7561 6472 6174  onalMap (quadrat
│ │ │ │ -00029510: 6963 2072 6174 696f 6e61 6c20 6d61 7020  ic rational map 
│ │ │ │ -00029520: 6672 6f6d 2038 2d64 696d 656e 7369 6f6e  from 8-dimension
│ │ │ │ -00029530: 616c 2020 2020 2020 2020 2020 2020 2020  al              
│ │ │ │ -00029540: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +000294f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +00029500: 3a20 5261 7469 6f6e 616c 4d61 7020 2871  : RationalMap (q
│ │ │ │ +00029510: 7561 6472 6174 6963 2072 6174 696f 6e61  uadratic rationa
│ │ │ │ +00029520: 6c20 6d61 7020 6672 6f6d 2038 2d64 696d  l map from 8-dim
│ │ │ │ +00029530: 656e 7369 6f6e 616c 2020 2020 2020 2020  ensional        
│ │ │ │ +00029540: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  00029550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029590: 2d2d 2d2d 7c0a 7c74 7269 706c 6520 7469  ----|.|triple ti
│ │ │ │ -000295a0: 6d65 2020 2020 2020 2020 2020 2020 2020  me              
│ │ │ │ +00029590: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c74 7269  ----------|.|tri
│ │ │ │ +000295a0: 706c 6520 7469 6d65 2020 2020 2020 2020  ple time        
│ │ │ │  000295b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000295c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000295d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000295e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000295e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000295f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029630: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029630: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00029640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029680: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029680: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -000296d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000296d0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029720: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029770: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029770: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -000297c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000297c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029810: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029810: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029860: 2020 2020 7c0a 7c78 202c 2078 202c 2078      |.|x , x , x
│ │ │ │ -00029870: 202c 2078 202c 2078 2020 2c20 7820 205d   , x , x  , x  ]
│ │ │ │ -00029880: 2920 6465 6669 6e65 6420 6279 2020 2020  ) defined by    
│ │ │ │ -00029890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029860: 2020 2020 2020 2020 2020 7c0a 7c78 202c            |.|x ,
│ │ │ │ +00029870: 2078 202c 2078 202c 2078 202c 2078 2020   x , x , x , x  
│ │ │ │ +00029880: 2c20 7820 205d 2920 6465 6669 6e65 6420  , x  ]) defined 
│ │ │ │ +00029890: 6279 2020 2020 2020 2020 2020 2020 2020  by              
│ │ │ │  000298a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298b0: 2020 2020 7c0a 7c20 3620 2020 3720 2020      |.| 6   7   
│ │ │ │ -000298c0: 3820 2020 3920 2020 3130 2020 2031 3120  8   9   10   11 
│ │ │ │ -000298d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000298b0: 2020 2020 2020 2020 2020 7c0a 7c20 3620            |.| 6 
│ │ │ │ +000298c0: 2020 3720 2020 3820 2020 3920 2020 3130    7   8   9   10
│ │ │ │ +000298d0: 2020 2031 3120 2020 2020 2020 2020 2020     11           
│ │ │ │  000298e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000298f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029900: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029900: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029950: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029950: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c2b 2031 3378 2078 2020      |.|+ 13x x  
│ │ │ │ -000299b0: 2b20 3431 7820 7820 202d 2078 2078 2020  + 41x x  - x x  
│ │ │ │ -000299c0: 2b20 3132 7820 7820 202b 2032 3578 2078  + 12x x  + 25x x
│ │ │ │ -000299d0: 2020 2b20 3235 7820 7820 202b 2032 3378    + 25x x  + 23x
│ │ │ │ -000299e0: 2078 2020 2d20 3378 2078 2020 2b20 3278   x  - 3x x  + 2x
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│ │ │ │ -00029a00: 2020 2020 2033 2035 2020 2020 3020 3620       3 5    0 6 
│ │ │ │ -00029a10: 2020 2020 2032 2036 2020 2020 2020 3120       2 6      1 
│ │ │ │ -00029a20: 3720 2020 2020 2033 2037 2020 2020 2020  7      3 7      
│ │ │ │ -00029a30: 3520 3720 2020 2020 3620 3720 2020 2020  5 7     6 7     
│ │ │ │ -00029a40: 3020 2020 7c0a 7c20 2020 2020 2020 2020  0   |.|         
│ │ │ │ +000299a0: 2020 2020 2020 2020 2020 7c0a 7c2b 2031            |.|+ 1
│ │ │ │ +000299b0: 3378 2078 2020 2b20 3431 7820 7820 202d  3x x  + 41x x  -
│ │ │ │ +000299c0: 2078 2078 2020 2b20 3132 7820 7820 202b   x x  + 12x x  +
│ │ │ │ +000299d0: 2032 3578 2078 2020 2b20 3235 7820 7820   25x x  + 25x x 
│ │ │ │ +000299e0: 202b 2032 3378 2078 2020 2d20 3378 2078   + 23x x  - 3x x
│ │ │ │ +000299f0: 2020 2b20 3278 2078 2020 7c0a 7c20 2020    + 2x x  |.|   
│ │ │ │ +00029a00: 2020 3220 3520 2020 2020 2033 2035 2020    2 5      3 5  
│ │ │ │ +00029a10: 2020 3020 3620 2020 2020 2032 2036 2020    0 6      2 6  
│ │ │ │ +00029a20: 2020 2020 3120 3720 2020 2020 2033 2037      1 7      3 7
│ │ │ │ +00029a30: 2020 2020 2020 3520 3720 2020 2020 3620        5 7     6 
│ │ │ │ +00029a40: 3720 2020 2020 3020 2020 7c0a 7c20 2020  7     0   |.|   
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029a90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029a90: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ae0: 2020 2020 7c0a 7c20 2d20 3330 7820 7820      |.| - 30x x 
│ │ │ │ -00029af0: 202b 2032 3778 2078 2020 2b20 3331 7820   + 27x x  + 31x 
│ │ │ │ -00029b00: 7820 202d 2033 3678 2078 2020 2d20 7820  x  - 36x x  - x 
│ │ │ │ -00029b10: 7820 202b 2031 3378 2078 2020 2b20 3878  x  + 13x x  + 8x
│ │ │ │ -00029b20: 2078 2020 2b20 3978 2078 2020 2b20 3436   x  + 9x x  + 46
│ │ │ │ -00029b30: 7820 2020 7c0a 7c20 2020 2020 2032 2035  x   |.|      2 5
│ │ │ │ -00029b40: 2020 2020 2020 3320 3520 2020 2020 2032        3 5      2
│ │ │ │ -00029b50: 2036 2020 2020 2020 3320 3620 2020 2030   6      3 6    0
│ │ │ │ -00029b60: 2037 2020 2020 2020 3120 3720 2020 2020   7      1 7     
│ │ │ │ -00029b70: 3320 3720 2020 2020 3520 3720 2020 2020  3 7     5 7     
│ │ │ │ -00029b80: 2036 2020 7c0a 7c20 2020 2020 2020 2020   6  |.|         
│ │ │ │ +00029ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2d20            |.| - 
│ │ │ │ +00029af0: 3330 7820 7820 202b 2032 3778 2078 2020  30x x  + 27x x  
│ │ │ │ +00029b00: 2b20 3331 7820 7820 202d 2033 3678 2078  + 31x x  - 36x x
│ │ │ │ +00029b10: 2020 2d20 7820 7820 202b 2031 3378 2078    - x x  + 13x x
│ │ │ │ +00029b20: 2020 2b20 3878 2078 2020 2b20 3978 2078    + 8x x  + 9x x
│ │ │ │ +00029b30: 2020 2b20 3436 7820 2020 7c0a 7c20 2020    + 46x   |.|   
│ │ │ │ +00029b40: 2020 2032 2035 2020 2020 2020 3320 3520     2 5      3 5 
│ │ │ │ +00029b50: 2020 2020 2032 2036 2020 2020 2020 3320       2 6      3 
│ │ │ │ +00029b60: 3620 2020 2030 2037 2020 2020 2020 3120  6    0 7      1 
│ │ │ │ +00029b70: 3720 2020 2020 3320 3720 2020 2020 3520  7     3 7     5 
│ │ │ │ +00029b80: 3720 2020 2020 2036 2020 7c0a 7c20 2020  7      6  |.|   
│ │ │ │  00029b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029bd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00029be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029c20: 2020 2020 7c0a 7c2b 2035 7820 7820 202d      |.|+ 5x x  -
│ │ │ │ -00029c30: 2033 3678 2078 2020 2b20 3337 7820 7820   36x x  + 37x x 
│ │ │ │ -00029c40: 202b 2034 3878 2078 2020 2d20 3578 2078   + 48x x  - 5x x
│ │ │ │ -00029c50: 2020 2d20 3578 2078 2020 2b20 7820 7820    - 5x x  + x x 
│ │ │ │ -00029c60: 202b 2032 3078 2078 2020 2b20 3130 7820   + 20x x  + 10x 
│ │ │ │ -00029c70: 7820 2020 7c0a 7c20 2020 2030 2035 2020  x   |.|    0 5  
│ │ │ │ -00029c80: 2020 2020 3220 3520 2020 2020 2033 2035      2 5      3 5
│ │ │ │ -00029c90: 2020 2020 2020 3220 3620 2020 2020 3120        2 6     1 
│ │ │ │ -00029ca0: 3720 2020 2020 3320 3720 2020 2035 2037  7     3 7    5 7
│ │ │ │ -00029cb0: 2020 2020 2020 3620 3720 2020 2020 2030        6 7      0
│ │ │ │ -00029cc0: 2038 2020 7c0a 7c20 2020 2020 2020 2020   8  |.|         
│ │ │ │ +00029c20: 2020 2020 2020 2020 2020 7c0a 7c2b 2035            |.|+ 5
│ │ │ │ +00029c30: 7820 7820 202d 2033 3678 2078 2020 2b20  x x  - 36x x  + 
│ │ │ │ +00029c40: 3337 7820 7820 202b 2034 3878 2078 2020  37x x  + 48x x  
│ │ │ │ +00029c50: 2d20 3578 2078 2020 2d20 3578 2078 2020  - 5x x  - 5x x  
│ │ │ │ +00029c60: 2b20 7820 7820 202b 2032 3078 2078 2020  + x x  + 20x x  
│ │ │ │ +00029c70: 2b20 3130 7820 7820 2020 7c0a 7c20 2020  + 10x x   |.|   
│ │ │ │ +00029c80: 2030 2035 2020 2020 2020 3220 3520 2020   0 5      2 5   
│ │ │ │ +00029c90: 2020 2033 2035 2020 2020 2020 3220 3620     3 5      2 6 
│ │ │ │ +00029ca0: 2020 2020 3120 3720 2020 2020 3320 3720      1 7     3 7 
│ │ │ │ +00029cb0: 2020 2035 2037 2020 2020 2020 3620 3720     5 7      6 7 
│ │ │ │ +00029cc0: 2020 2020 2030 2038 2020 7c0a 7c20 2020       0 8  |.|   
│ │ │ │  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 2020 2020 2020 2020                  
│ │ │ │ -00029d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029d10: 2020 2020 2020 2020 2020 7c0a 7c20 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: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029d60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 2020 2020 2020                  
│ │ │ │ -00029db0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029db0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029e00: 2020 2020 2020 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e50: 2020 2020 7c0a 7c20 3278 2078 2020 2b20      |.| 2x x  + 
│ │ │ │ -00029e60: 7820 7820 202d 2036 7820 7820 202d 2035  x x  - 6x x  - 5
│ │ │ │ -00029e70: 7820 7820 2020 2d20 3130 7820 7820 2020  x x   - 10x x   
│ │ │ │ -00029e80: 2b20 3235 7820 7820 2020 2b20 3578 2078  + 25x x   + 5x x
│ │ │ │ -00029e90: 2020 202d 2035 7820 7820 202c 2020 2020     - 5x x  ,    
│ │ │ │ -00029ea0: 2020 2020 7c0a 7c20 2020 3420 3820 2020      |.|   4 8   
│ │ │ │ -00029eb0: 2035 2038 2020 2020 2035 2039 2020 2020   5 8     5 9    
│ │ │ │ -00029ec0: 2031 2031 3020 2020 2020 2032 2031 3020   1 10      2 10 
│ │ │ │ -00029ed0: 2020 2020 2033 2031 3020 2020 2020 3420       3 10     4 
│ │ │ │ -00029ee0: 3130 2020 2020 2035 2031 3020 2020 2020  10     5 10     
│ │ │ │ -00029ef0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00029e50: 2020 2020 2020 2020 2020 7c0a 7c20 3278            |.| 2x
│ │ │ │ +00029e60: 2078 2020 2b20 7820 7820 202d 2036 7820   x  + x x  - 6x 
│ │ │ │ +00029e70: 7820 202d 2035 7820 7820 2020 2d20 3130  x  - 5x x   - 10
│ │ │ │ +00029e80: 7820 7820 2020 2b20 3235 7820 7820 2020  x x   + 25x x   
│ │ │ │ +00029e90: 2b20 3578 2078 2020 202d 2035 7820 7820  + 5x x   - 5x x 
│ │ │ │ +00029ea0: 202c 2020 2020 2020 2020 7c0a 7c20 2020   ,        |.|   
│ │ │ │ +00029eb0: 3420 3820 2020 2035 2038 2020 2020 2035  4 8    5 8     5
│ │ │ │ +00029ec0: 2039 2020 2020 2031 2031 3020 2020 2020   9     1 10     
│ │ │ │ +00029ed0: 2032 2031 3020 2020 2020 2033 2031 3020   2 10      3 10 
│ │ │ │ +00029ee0: 2020 2020 3420 3130 2020 2020 2035 2031      4 10     5 1
│ │ │ │ +00029ef0: 3020 2020 2020 2020 2020 7c0a 7c20 2020  0         |.|   
│ │ │ │  00029f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f40: 2020 2020 7c0a 7c20 2d20 3678 2078 2020      |.| - 6x x  
│ │ │ │ -00029f50: 2d20 3436 7820 7820 202d 2034 3478 2078  - 46x x  - 44x x
│ │ │ │ -00029f60: 2020 2d20 3232 7820 7820 202b 2033 3678    - 22x x  + 36x
│ │ │ │ -00029f70: 2078 2020 202d 2032 3578 2078 2020 202b   x   - 25x x   +
│ │ │ │ -00029f80: 2031 3478 2078 2020 202b 2031 3578 2078   14x x   + 15x x
│ │ │ │ -00029f90: 2020 2020 7c0a 7c20 2020 2020 3420 3820      |.|     4 8 
│ │ │ │ -00029fa0: 2020 2020 2035 2038 2020 2020 2020 3620       5 8      6 
│ │ │ │ -00029fb0: 3820 2020 2020 2035 2039 2020 2020 2020  8      5 9      
│ │ │ │ -00029fc0: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ -00029fd0: 2020 2020 3320 3130 2020 2020 2020 3420      3 10      4 
│ │ │ │ -00029fe0: 3130 2020 7c0a 7c20 2020 2020 2020 2020  10  |.|         
│ │ │ │ +00029f40: 2020 2020 2020 2020 2020 7c0a 7c20 2d20            |.| - 
│ │ │ │ +00029f50: 3678 2078 2020 2d20 3436 7820 7820 202d  6x x  - 46x x  -
│ │ │ │ +00029f60: 2034 3478 2078 2020 2d20 3232 7820 7820   44x x  - 22x x 
│ │ │ │ +00029f70: 202b 2033 3678 2078 2020 202d 2032 3578   + 36x x   - 25x
│ │ │ │ +00029f80: 2078 2020 202b 2031 3478 2078 2020 202b   x   + 14x x   +
│ │ │ │ +00029f90: 2031 3578 2078 2020 2020 7c0a 7c20 2020   15x x    |.|   
│ │ │ │ +00029fa0: 2020 3420 3820 2020 2020 2035 2038 2020    4 8      5 8  
│ │ │ │ +00029fb0: 2020 2020 3620 3820 2020 2020 2035 2039      6 8      5 9
│ │ │ │ +00029fc0: 2020 2020 2020 3120 3130 2020 2020 2020        1 10      
│ │ │ │ +00029fd0: 3220 3130 2020 2020 2020 3320 3130 2020  2 10      3 10  
│ │ │ │ +00029fe0: 2020 2020 3420 3130 2020 7c0a 7c20 2020      4 10  |.|   
│ │ │ │  00029ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a030: 2020 2020 7c0a 7c2d 2032 3578 2078 2020      |.|- 25x x  
│ │ │ │ -0002a040: 2d20 3878 2078 2020 2b20 3578 2078 2020  - 8x x  + 5x x  
│ │ │ │ -0002a050: 2d20 3235 7820 7820 202d 2032 3078 2078  - 25x x  - 20x x
│ │ │ │ -0002a060: 2020 202d 2034 3578 2078 2020 202b 2032     - 45x x   + 2
│ │ │ │ -0002a070: 3878 2078 2020 202b 2032 3078 2078 2020  8x x   + 20x x  
│ │ │ │ -0002a080: 202d 2020 7c0a 7c20 2020 2020 3320 3820   -  |.|     3 8 
│ │ │ │ -0002a090: 2020 2020 3420 3820 2020 2020 3520 3820      4 8     5 8 
│ │ │ │ -0002a0a0: 2020 2020 2035 2039 2020 2020 2020 3120       5 9      1 
│ │ │ │ -0002a0b0: 3130 2020 2020 2020 3220 3130 2020 2020  10      2 10    
│ │ │ │ -0002a0c0: 2020 3320 3130 2020 2020 2020 3420 3130    3 10      4 10
│ │ │ │ -0002a0d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a030: 2020 2020 2020 2020 2020 7c0a 7c2d 2032            |.|- 2
│ │ │ │ +0002a040: 3578 2078 2020 2d20 3878 2078 2020 2b20  5x x  - 8x x  + 
│ │ │ │ +0002a050: 3578 2078 2020 2d20 3235 7820 7820 202d  5x x  - 25x x  -
│ │ │ │ +0002a060: 2032 3078 2078 2020 202d 2034 3578 2078   20x x   - 45x x
│ │ │ │ +0002a070: 2020 202b 2032 3878 2078 2020 202b 2032     + 28x x   + 2
│ │ │ │ +0002a080: 3078 2078 2020 202d 2020 7c0a 7c20 2020  0x x   -  |.|   
│ │ │ │ +0002a090: 2020 3320 3820 2020 2020 3420 3820 2020    3 8     4 8   
│ │ │ │ +0002a0a0: 2020 3520 3820 2020 2020 2035 2039 2020    5 8      5 9  
│ │ │ │ +0002a0b0: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ +0002a0c0: 3130 2020 2020 2020 3320 3130 2020 2020  10      3 10    
│ │ │ │ +0002a0d0: 2020 3420 3130 2020 2020 7c0a 7c20 2020    4 10    |.|   
│ │ │ │  0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a120: 2020 2020 7c0a 7c78 2020 2d20 3339 7820      |.|x  - 39x 
│ │ │ │ -0002a130: 7820 202d 2034 3178 2078 2020 2b20 3139  x  - 41x x  + 19
│ │ │ │ -0002a140: 7820 7820 202b 2034 3078 2078 2020 2b20  x x  + 40x x  + 
│ │ │ │ -0002a150: 3433 7820 7820 202d 2033 3978 2078 2020  43x x  - 39x x  
│ │ │ │ -0002a160: 2d20 3337 7820 7820 202b 2033 7820 7820  - 37x x  + 3x x 
│ │ │ │ -0002a170: 202b 2020 7c0a 7c20 3720 2020 2020 2036   +  |.| 7      6
│ │ │ │ -0002a180: 2037 2020 2020 2020 3120 3820 2020 2020   7      1 8     
│ │ │ │ -0002a190: 2032 2038 2020 2020 2020 3320 3820 2020   2 8      3 8   
│ │ │ │ -0002a1a0: 2020 2034 2038 2020 2020 2020 3520 3820     4 8      5 8 
│ │ │ │ -0002a1b0: 2020 2020 2036 2038 2020 2020 2031 2039       6 8     1 9
│ │ │ │ -0002a1c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a120: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +0002a130: 2d20 3339 7820 7820 202d 2034 3178 2078  - 39x x  - 41x x
│ │ │ │ +0002a140: 2020 2b20 3139 7820 7820 202b 2034 3078    + 19x x  + 40x
│ │ │ │ +0002a150: 2078 2020 2b20 3433 7820 7820 202d 2033   x  + 43x x  - 3
│ │ │ │ +0002a160: 3978 2078 2020 2d20 3337 7820 7820 202b  9x x  - 37x x  +
│ │ │ │ +0002a170: 2033 7820 7820 202b 2020 7c0a 7c20 3720   3x x  +  |.| 7 
│ │ │ │ +0002a180: 2020 2020 2036 2037 2020 2020 2020 3120       6 7      1 
│ │ │ │ +0002a190: 3820 2020 2020 2032 2038 2020 2020 2020  8      2 8      
│ │ │ │ +0002a1a0: 3320 3820 2020 2020 2034 2038 2020 2020  3 8      4 8    
│ │ │ │ +0002a1b0: 2020 3520 3820 2020 2020 2036 2038 2020    5 8      6 8  
│ │ │ │ +0002a1c0: 2020 2031 2039 2020 2020 7c0a 7c20 2020     1 9    |.|   
│ │ │ │  0002a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a210: 2020 2020 7c0a 7c20 202b 2034 3678 2078      |.|  + 46x x
│ │ │ │ -0002a220: 2020 2d20 3136 7820 7820 202d 2033 3678    - 16x x  - 36x
│ │ │ │ -0002a230: 2078 2020 2b20 3978 2078 2020 2d20 3237   x  + 9x x  - 27
│ │ │ │ -0002a240: 7820 7820 202d 2034 3678 2078 2020 2d20  x x  - 46x x  - 
│ │ │ │ -0002a250: 3434 7820 7820 202d 2032 3978 2078 2020  44x x  - 29x x  
│ │ │ │ -0002a260: 2d20 2020 7c0a 7c37 2020 2020 2020 3520  -   |.|7      5 
│ │ │ │ -0002a270: 3720 2020 2020 2036 2037 2020 2020 2020  7      6 7      
│ │ │ │ -0002a280: 3120 3820 2020 2020 3220 3820 2020 2020  1 8     2 8     
│ │ │ │ -0002a290: 2033 2038 2020 2020 2020 3420 3820 2020   3 8      4 8   
│ │ │ │ -0002a2a0: 2020 2035 2038 2020 2020 2020 3620 3820     5 8      6 8 
│ │ │ │ -0002a2b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a210: 2020 2020 2020 2020 2020 7c0a 7c20 202b            |.|  +
│ │ │ │ +0002a220: 2034 3678 2078 2020 2d20 3136 7820 7820   46x x  - 16x x 
│ │ │ │ +0002a230: 202d 2033 3678 2078 2020 2b20 3978 2078   - 36x x  + 9x x
│ │ │ │ +0002a240: 2020 2d20 3237 7820 7820 202d 2034 3678    - 27x x  - 46x
│ │ │ │ +0002a250: 2078 2020 2d20 3434 7820 7820 202d 2032   x  - 44x x  - 2
│ │ │ │ +0002a260: 3978 2078 2020 2d20 2020 7c0a 7c37 2020  9x x  -   |.|7  
│ │ │ │ +0002a270: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ +0002a280: 2020 2020 2020 3120 3820 2020 2020 3220        1 8     2 
│ │ │ │ +0002a290: 3820 2020 2020 2033 2038 2020 2020 2020  8      3 8      
│ │ │ │ +0002a2a0: 3420 3820 2020 2020 2035 2038 2020 2020  4 8      5 8    
│ │ │ │ +0002a2b0: 2020 3620 3820 2020 2020 7c0a 7c20 2020    6 8     |.|   
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a300: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a300: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  0002a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a350: 2020 2020 7c0a 7c20 202b 2033 3378 2078      |.|  + 33x x
│ │ │ │ -0002a360: 2020 2b20 3333 7820 7820 202d 2032 3178    + 33x x  - 21x
│ │ │ │ -0002a370: 2078 2020 2d20 3235 7820 7820 202b 2034   x  - 25x x  + 4
│ │ │ │ -0002a380: 3578 2078 2020 2b20 3233 7820 7820 202b  5x x  + 23x x  +
│ │ │ │ -0002a390: 2031 3078 2078 2020 2d20 3778 2078 2020   10x x  - 7x x  
│ │ │ │ -0002a3a0: 2b20 2020 7c0a 7c36 2020 2020 2020 3020  +   |.|6      0 
│ │ │ │ -0002a3b0: 3720 2020 2020 2032 2037 2020 2020 2020  7      2 7      
│ │ │ │ -0002a3c0: 3320 3720 2020 2020 2035 2037 2020 2020  3 7      5 7    
│ │ │ │ -0002a3d0: 2020 3620 3720 2020 2020 2030 2038 2020    6 7      0 8  
│ │ │ │ -0002a3e0: 2020 2020 3120 3820 2020 2020 3220 3820      1 8     2 8 
│ │ │ │ -0002a3f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a350: 2020 2020 2020 2020 2020 7c0a 7c20 202b            |.|  +
│ │ │ │ +0002a360: 2033 3378 2078 2020 2b20 3333 7820 7820   33x x  + 33x x 
│ │ │ │ +0002a370: 202d 2032 3178 2078 2020 2d20 3235 7820   - 21x x  - 25x 
│ │ │ │ +0002a380: 7820 202b 2034 3578 2078 2020 2b20 3233  x  + 45x x  + 23
│ │ │ │ +0002a390: 7820 7820 202b 2031 3078 2078 2020 2d20  x x  + 10x x  - 
│ │ │ │ +0002a3a0: 3778 2078 2020 2b20 2020 7c0a 7c36 2020  7x x  +   |.|6  
│ │ │ │ +0002a3b0: 2020 2020 3020 3720 2020 2020 2032 2037      0 7      2 7
│ │ │ │ +0002a3c0: 2020 2020 2020 3320 3720 2020 2020 2035        3 7      5
│ │ │ │ +0002a3d0: 2037 2020 2020 2020 3620 3720 2020 2020   7      6 7     
│ │ │ │ +0002a3e0: 2030 2038 2020 2020 2020 3120 3820 2020   0 8      1 8   
│ │ │ │ +0002a3f0: 2020 3220 3820 2020 2020 7c0a 7c20 2020    2 8     |.|   
│ │ │ │  0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a410: 2020 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 7c0a 7c20 2b20 3278 2078 2020      |.| + 2x x  
│ │ │ │ -0002a450: 2b20 3334 7820 7820 202b 2032 3278 2078  + 34x x  + 22x x
│ │ │ │ -0002a460: 2020 2b20 3338 7820 7820 202b 2078 2078    + 38x x  + x x
│ │ │ │ -0002a470: 2020 2b20 3330 7820 7820 202d 2032 3178    + 30x x  - 21x
│ │ │ │ -0002a480: 2078 2020 2d20 3334 7820 7820 202d 2032   x  - 34x x  - 2
│ │ │ │ -0002a490: 3178 2020 7c0a 7c20 2020 2020 3220 3720  1x  |.|     2 7 
│ │ │ │ -0002a4a0: 2020 2020 2033 2037 2020 2020 2020 3520       3 7      5 
│ │ │ │ -0002a4b0: 3720 2020 2020 2036 2037 2020 2020 3020  7      6 7    0 
│ │ │ │ -0002a4c0: 3820 2020 2020 2031 2038 2020 2020 2020  8      1 8      
│ │ │ │ -0002a4d0: 3220 3820 2020 2020 2033 2038 2020 2020  2 8      3 8    
│ │ │ │ -0002a4e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a440: 2020 2020 2020 2020 2020 7c0a 7c20 2b20            |.| + 
│ │ │ │ +0002a450: 3278 2078 2020 2b20 3334 7820 7820 202b  2x x  + 34x x  +
│ │ │ │ +0002a460: 2032 3278 2078 2020 2b20 3338 7820 7820   22x x  + 38x x 
│ │ │ │ +0002a470: 202b 2078 2078 2020 2b20 3330 7820 7820   + x x  + 30x x 
│ │ │ │ +0002a480: 202d 2032 3178 2078 2020 2d20 3334 7820   - 21x x  - 34x 
│ │ │ │ +0002a490: 7820 202d 2032 3178 2020 7c0a 7c20 2020  x  - 21x  |.|   
│ │ │ │ +0002a4a0: 2020 3220 3720 2020 2020 2033 2037 2020    2 7      3 7  
│ │ │ │ +0002a4b0: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ +0002a4c0: 2020 2020 3020 3820 2020 2020 2031 2038      0 8      1 8
│ │ │ │ +0002a4d0: 2020 2020 2020 3220 3820 2020 2020 2033        2 8      3
│ │ │ │ +0002a4e0: 2038 2020 2020 2020 2020 7c0a 7c20 2020   8        |.|   
│ │ │ │  0002a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a500: 2020 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 7c0a 7c20 2020 2020 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 7c0a 7c32 3178 2078 2020 2b20      |.|21x x  + 
│ │ │ │ -0002a590: 3330 7820 7820 202d 2078 2078 2020 2d20  30x x  - x x  - 
│ │ │ │ -0002a5a0: 3578 2078 2020 2d20 3239 7820 7820 202b  5x x  - 29x x  +
│ │ │ │ -0002a5b0: 2032 7820 7820 202d 2032 3978 2078 2020   2x x  - 29x x  
│ │ │ │ -0002a5c0: 2d20 3237 7820 7820 202d 2035 7820 7820  - 27x x  - 5x x 
│ │ │ │ -0002a5d0: 202b 2020 7c0a 7c20 2020 3220 3620 2020   +  |.|   2 6   
│ │ │ │ -0002a5e0: 2020 2033 2036 2020 2020 3420 3620 2020     3 6    4 6   
│ │ │ │ -0002a5f0: 2020 3520 3620 2020 2020 2030 2037 2020    5 6      0 7  
│ │ │ │ -0002a600: 2020 2031 2037 2020 2020 2020 3220 3720     1 7      2 7 
│ │ │ │ -0002a610: 2020 2020 2033 2037 2020 2020 2035 2037       3 7     5 7
│ │ │ │ -0002a620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a580: 2020 2020 2020 2020 2020 7c0a 7c32 3178            |.|21x
│ │ │ │ +0002a590: 2078 2020 2b20 3330 7820 7820 202d 2078   x  + 30x x  - x
│ │ │ │ +0002a5a0: 2078 2020 2d20 3578 2078 2020 2d20 3239   x  - 5x x  - 29
│ │ │ │ +0002a5b0: 7820 7820 202b 2032 7820 7820 202d 2032  x x  + 2x x  - 2
│ │ │ │ +0002a5c0: 3978 2078 2020 2d20 3237 7820 7820 202d  9x x  - 27x x  -
│ │ │ │ +0002a5d0: 2035 7820 7820 202b 2020 7c0a 7c20 2020   5x x  +  |.|   
│ │ │ │ +0002a5e0: 3220 3620 2020 2020 2033 2036 2020 2020  2 6      3 6    
│ │ │ │ +0002a5f0: 3420 3620 2020 2020 3520 3620 2020 2020  4 6     5 6     
│ │ │ │ +0002a600: 2030 2037 2020 2020 2031 2037 2020 2020   0 7     1 7    
│ │ │ │ +0002a610: 2020 3220 3720 2020 2020 2033 2037 2020    2 7      3 7  
│ │ │ │ +0002a620: 2020 2035 2037 2020 2020 7c0a 7c20 2020     5 7    |.|   
│ │ │ │  0002a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 202b 2033 3178 2078      |.|  + 31x x
│ │ │ │ -0002a680: 2020 2b20 3239 7820 7820 202b 2031 3078    + 29x x  + 10x
│ │ │ │ -0002a690: 2078 2020 2d20 3237 7820 7820 202d 2033   x  - 27x x  - 3
│ │ │ │ -0002a6a0: 3178 2078 2020 2d20 3137 7820 7820 202d  1x x  - 17x x  -
│ │ │ │ -0002a6b0: 2032 3878 2078 2020 2b20 3339 7820 7820   28x x  + 39x x 
│ │ │ │ -0002a6c0: 202d 2020 7c0a 7c36 2020 2020 2020 3020   -  |.|6      0 
│ │ │ │ -0002a6d0: 3720 2020 2020 2032 2037 2020 2020 2020  7      2 7      
│ │ │ │ -0002a6e0: 3320 3720 2020 2020 2035 2037 2020 2020  3 7      5 7    
│ │ │ │ -0002a6f0: 2020 3620 3720 2020 2020 2030 2038 2020    6 7      0 8  
│ │ │ │ -0002a700: 2020 2020 3120 3820 2020 2020 2032 2038      1 8      2 8
│ │ │ │ -0002a710: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a670: 2020 2020 2020 2020 2020 7c0a 7c20 202b            |.|  +
│ │ │ │ +0002a680: 2033 3178 2078 2020 2b20 3239 7820 7820   31x x  + 29x x 
│ │ │ │ +0002a690: 202b 2031 3078 2078 2020 2d20 3237 7820   + 10x x  - 27x 
│ │ │ │ +0002a6a0: 7820 202d 2033 3178 2078 2020 2d20 3137  x  - 31x x  - 17
│ │ │ │ +0002a6b0: 7820 7820 202d 2032 3878 2078 2020 2b20  x x  - 28x x  + 
│ │ │ │ +0002a6c0: 3339 7820 7820 202d 2020 7c0a 7c36 2020  39x x  -  |.|6  
│ │ │ │ +0002a6d0: 2020 2020 3020 3720 2020 2020 2032 2037      0 7      2 7
│ │ │ │ +0002a6e0: 2020 2020 2020 3320 3720 2020 2020 2035        3 7      5
│ │ │ │ +0002a6f0: 2037 2020 2020 2020 3620 3720 2020 2020   7      6 7     
│ │ │ │ +0002a700: 2030 2038 2020 2020 2020 3120 3820 2020   0 8      1 8   
│ │ │ │ +0002a710: 2020 2032 2038 2020 2020 7c0a 7c20 2020     2 8    |.|   
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a760: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a760: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7b0: 2020 2020 7c0a 7c20 7375 6276 6172 6965      |.| subvarie
│ │ │ │ -0002a7c0: 7479 206f 6620 5050 5e31 3120 746f 2050  ty of PP^11 to P
│ │ │ │ -0002a7d0: 505e 3829 2020 2020 2020 2020 2020 2020  P^8)            
│ │ │ │ +0002a7b0: 2020 2020 2020 2020 2020 7c0a 7c20 7375            |.| su
│ │ │ │ +0002a7c0: 6276 6172 6965 7479 206f 6620 5050 5e31  bvariety of PP^1
│ │ │ │ +0002a7d0: 3120 746f 2050 505e 3829 2020 2020 2020  1 to PP^8)      
│ │ │ │  0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a800: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +0002a800: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  0002a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a850: 2d2d 2d2d 7c0a 7c20 202b 2031 3178 2078  ----|.|  + 11x x
│ │ │ │ -0002a860: 2020 2d20 3337 7820 7820 202d 2032 3378    - 37x x  - 23x
│ │ │ │ -0002a870: 2078 2020 2d20 3333 7820 7820 202b 2038   x  - 33x x  + 8
│ │ │ │ -0002a880: 7820 7820 202b 2031 3078 2078 2020 2d20  x x  + 10x x  - 
│ │ │ │ -0002a890: 3235 7820 7820 202d 2039 7820 7820 202b  25x x  - 9x x  +
│ │ │ │ -0002a8a0: 2033 7820 7c0a 7c38 2020 2020 2020 3120   3x |.|8      1 
│ │ │ │ -0002a8b0: 3820 2020 2020 2033 2038 2020 2020 2020  8      3 8      
│ │ │ │ -0002a8c0: 3420 3820 2020 2020 2036 2038 2020 2020  4 8      6 8    
│ │ │ │ -0002a8d0: 2030 2039 2020 2020 2020 3120 3920 2020   0 9      1 9   
│ │ │ │ -0002a8e0: 2020 2032 2039 2020 2020 2033 2039 2020     2 9     3 9  
│ │ │ │ -0002a8f0: 2020 2034 7c0a 7c20 2020 2020 2020 2020     4|.|         
│ │ │ │ +0002a850: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 202b  ----------|.|  +
│ │ │ │ +0002a860: 2031 3178 2078 2020 2d20 3337 7820 7820   11x x  - 37x x 
│ │ │ │ +0002a870: 202d 2032 3378 2078 2020 2d20 3333 7820   - 23x x  - 33x 
│ │ │ │ +0002a880: 7820 202b 2038 7820 7820 202b 2031 3078  x  + 8x x  + 10x
│ │ │ │ +0002a890: 2078 2020 2d20 3235 7820 7820 202d 2039   x  - 25x x  - 9
│ │ │ │ +0002a8a0: 7820 7820 202b 2033 7820 7c0a 7c38 2020  x x  + 3x |.|8  
│ │ │ │ +0002a8b0: 2020 2020 3120 3820 2020 2020 2033 2038      1 8      3 8
│ │ │ │ +0002a8c0: 2020 2020 2020 3420 3820 2020 2020 2036        4 8      6
│ │ │ │ +0002a8d0: 2038 2020 2020 2030 2039 2020 2020 2020   8     0 9      
│ │ │ │ +0002a8e0: 3120 3920 2020 2020 2032 2039 2020 2020  1 9      2 9    
│ │ │ │ +0002a8f0: 2033 2039 2020 2020 2034 7c0a 7c20 2020   3 9     4|.|   
│ │ │ │  0002a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a940: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a940: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002a950: 2020 2020 2020 2020 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 7c0a 7c78 2020 2b20 3431 7820      |.|x  + 41x 
│ │ │ │ -0002a9a0: 7820 202d 2037 7820 7820 202d 2033 3478  x  - 7x x  - 34x
│ │ │ │ -0002a9b0: 2078 2020 2d20 3978 2078 2020 2d20 3436   x  - 9x x  - 46
│ │ │ │ -0002a9c0: 7820 7820 202d 2031 3778 2078 2020 2b20  x x  - 17x x  + 
│ │ │ │ -0002a9d0: 3332 7820 7820 202d 2038 7820 7820 202d  32x x  - 8x x  -
│ │ │ │ -0002a9e0: 2033 3578 7c0a 7c20 3720 2020 2020 2030   35x|.| 7      0
│ │ │ │ -0002a9f0: 2038 2020 2020 2031 2038 2020 2020 2020   8     1 8      
│ │ │ │ -0002aa00: 3320 3820 2020 2020 3420 3820 2020 2020  3 8     4 8     
│ │ │ │ -0002aa10: 2036 2038 2020 2020 2020 3020 3920 2020   6 8      0 9   
│ │ │ │ -0002aa20: 2020 2031 2039 2020 2020 2032 2039 2020     1 9     2 9  
│ │ │ │ -0002aa30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a990: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +0002a9a0: 2b20 3431 7820 7820 202d 2037 7820 7820  + 41x x  - 7x x 
│ │ │ │ +0002a9b0: 202d 2033 3478 2078 2020 2d20 3978 2078   - 34x x  - 9x x
│ │ │ │ +0002a9c0: 2020 2d20 3436 7820 7820 202d 2031 3778    - 46x x  - 17x
│ │ │ │ +0002a9d0: 2078 2020 2b20 3332 7820 7820 202d 2038   x  + 32x x  - 8
│ │ │ │ +0002a9e0: 7820 7820 202d 2033 3578 7c0a 7c20 3720  x x  - 35x|.| 7 
│ │ │ │ +0002a9f0: 2020 2020 2030 2038 2020 2020 2031 2038       0 8     1 8
│ │ │ │ +0002aa00: 2020 2020 2020 3320 3820 2020 2020 3420        3 8     4 
│ │ │ │ +0002aa10: 3820 2020 2020 2036 2038 2020 2020 2020  8      6 8      
│ │ │ │ +0002aa20: 3020 3920 2020 2020 2031 2039 2020 2020  0 9      1 9    
│ │ │ │ +0002aa30: 2032 2039 2020 2020 2020 7c0a 7c20 2020   2 9      |.|   
│ │ │ │  0002aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002aa80: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2b20 3334 7820 7820      |.| + 34x x 
│ │ │ │ -0002aae0: 202b 2034 3178 2078 2020 2d20 7820 7820   + 41x x  - x x 
│ │ │ │ -0002aaf0: 202b 2078 2078 2020 2b20 3430 7820 7820   + x x  + 40x x 
│ │ │ │ -0002ab00: 202d 2033 3278 2078 2020 2b20 3578 2078   - 32x x  + 5x x
│ │ │ │ -0002ab10: 2020 2d20 3131 7820 7820 202d 2032 3078    - 11x x  - 20x
│ │ │ │ -0002ab20: 2078 2020 7c0a 7c20 2020 2020 2031 2038   x  |.|      1 8
│ │ │ │ -0002ab30: 2020 2020 2020 3320 3820 2020 2034 2038        3 8    4 8
│ │ │ │ -0002ab40: 2020 2020 3620 3820 2020 2020 2030 2039      6 8      0 9
│ │ │ │ -0002ab50: 2020 2020 2020 3120 3920 2020 2020 3220        1 9     2 
│ │ │ │ -0002ab60: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ -0002ab70: 3420 3920 7c0a 7c20 2020 2020 2020 2020  4 9 |.|         
│ │ │ │ +0002aad0: 2020 2020 2020 2020 2020 7c0a 7c20 2b20            |.| + 
│ │ │ │ +0002aae0: 3334 7820 7820 202b 2034 3178 2078 2020  34x x  + 41x x  
│ │ │ │ +0002aaf0: 2d20 7820 7820 202b 2078 2078 2020 2b20  - x x  + x x  + 
│ │ │ │ +0002ab00: 3430 7820 7820 202d 2033 3278 2078 2020  40x x  - 32x x  
│ │ │ │ +0002ab10: 2b20 3578 2078 2020 2d20 3131 7820 7820  + 5x x  - 11x x 
│ │ │ │ +0002ab20: 202d 2032 3078 2078 2020 7c0a 7c20 2020   - 20x x  |.|   
│ │ │ │ +0002ab30: 2020 2031 2038 2020 2020 2020 3320 3820     1 8      3 8 
│ │ │ │ +0002ab40: 2020 2034 2038 2020 2020 3620 3820 2020     4 8    6 8   
│ │ │ │ +0002ab50: 2020 2030 2039 2020 2020 2020 3120 3920     0 9      1 9 
│ │ │ │ +0002ab60: 2020 2020 3220 3920 2020 2020 2033 2039      2 9      3 9
│ │ │ │ +0002ab70: 2020 2020 2020 3420 3920 7c0a 7c20 2020        4 9 |.|   
│ │ │ │  0002ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002abc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002abc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ac10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ac60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002acb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002acb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ad00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ad50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ada0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ada0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002adf0: 2020 2020 7c0a 7c20 2b20 3336 7820 7820      |.| + 36x x 
│ │ │ │ -0002ae00: 2020 2b20 3336 7820 7820 2020 2d20 3336    + 36x x   - 36
│ │ │ │ -0002ae10: 7820 7820 202c 2020 2020 2020 2020 2020  x x  ,          
│ │ │ │ +0002adf0: 2020 2020 2020 2020 2020 7c0a 7c20 2b20            |.| + 
│ │ │ │ +0002ae00: 3336 7820 7820 2020 2b20 3336 7820 7820  36x x   + 36x x 
│ │ │ │ +0002ae10: 2020 2d20 3336 7820 7820 202c 2020 2020    - 36x x  ,    
│ │ │ │  0002ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae40: 2020 2020 7c0a 7c20 2020 2020 2035 2031      |.|      5 1
│ │ │ │ -0002ae50: 3020 2020 2020 2036 2031 3020 2020 2020  0      6 10     
│ │ │ │ -0002ae60: 2035 2031 3120 2020 2020 2020 2020 2020   5 11           
│ │ │ │ +0002ae40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002ae50: 2020 2035 2031 3020 2020 2020 2036 2031     5 10      6 1
│ │ │ │ +0002ae60: 3020 2020 2020 2035 2031 3120 2020 2020  0      5 11     
│ │ │ │  0002ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ae90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aee0: 2020 2020 7c0a 7c20 3235 7820 7820 202c      |.| 25x x  ,
│ │ │ │ -0002aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aee0: 2020 2020 2020 2020 2020 7c0a 7c20 3235            |.| 25
│ │ │ │ +0002aef0: 7820 7820 202c 2020 2020 2020 2020 2020  x x  ,          
│ │ │ │  0002af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af30: 2020 2020 7c0a 7c20 2020 2035 2031 3020      |.|    5 10 
│ │ │ │ -0002af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002af30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002af40: 2035 2031 3020 2020 2020 2020 2020 2020   5 10           
│ │ │ │  0002af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002af80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afd0: 2020 2020 7c0a 7c20 3338 7820 7820 202d      |.| 38x x  -
│ │ │ │ -0002afe0: 2032 7820 7820 202b 2033 3978 2078 2020   2x x  + 39x x  
│ │ │ │ -0002aff0: 2d20 3131 7820 7820 202d 2032 3878 2078  - 11x x  - 28x x
│ │ │ │ -0002b000: 2020 202b 2032 3178 2078 2020 202d 2035     + 21x x   - 5
│ │ │ │ -0002b010: 7820 7820 2020 2d20 3130 7820 7820 2020  x x   - 10x x   
│ │ │ │ -0002b020: 2b20 7820 7c0a 7c20 2020 2032 2039 2020  + x |.|    2 9  
│ │ │ │ -0002b030: 2020 2033 2039 2020 2020 2020 3420 3920     3 9      4 9 
│ │ │ │ -0002b040: 2020 2020 2035 2039 2020 2020 2020 3120       5 9      1 
│ │ │ │ -0002b050: 3130 2020 2020 2020 3220 3130 2020 2020  10      2 10    
│ │ │ │ -0002b060: 2033 2031 3020 2020 2020 2034 2031 3020   3 10      4 10 
│ │ │ │ -0002b070: 2020 2035 7c0a 7c20 2020 2020 2020 2020     5|.|         
│ │ │ │ +0002afd0: 2020 2020 2020 2020 2020 7c0a 7c20 3338            |.| 38
│ │ │ │ +0002afe0: 7820 7820 202d 2032 7820 7820 202b 2033  x x  - 2x x  + 3
│ │ │ │ +0002aff0: 3978 2078 2020 2d20 3131 7820 7820 202d  9x x  - 11x x  -
│ │ │ │ +0002b000: 2032 3878 2078 2020 202b 2032 3178 2078   28x x   + 21x x
│ │ │ │ +0002b010: 2020 202d 2035 7820 7820 2020 2d20 3130     - 5x x   - 10
│ │ │ │ +0002b020: 7820 7820 2020 2b20 7820 7c0a 7c20 2020  x x   + x |.|   
│ │ │ │ +0002b030: 2032 2039 2020 2020 2033 2039 2020 2020   2 9     3 9    
│ │ │ │ +0002b040: 2020 3420 3920 2020 2020 2035 2039 2020    4 9      5 9  
│ │ │ │ +0002b050: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ +0002b060: 3130 2020 2020 2033 2031 3020 2020 2020  10     3 10     
│ │ │ │ +0002b070: 2034 2031 3020 2020 2035 7c0a 7c20 2020   4 10    5|.|   
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0c0: 2020 2020 7c0a 7c31 3778 2078 2020 2b20      |.|17x x  + 
│ │ │ │ -0002b0d0: 3138 7820 7820 202b 2034 3578 2078 2020  18x x  + 45x x  
│ │ │ │ -0002b0e0: 2b20 3136 7820 7820 202b 2035 7820 7820  + 16x x  + 5x x 
│ │ │ │ -0002b0f0: 202b 2033 3678 2078 2020 2b20 3339 7820   + 36x x  + 39x 
│ │ │ │ -0002b100: 7820 2020 2d20 3336 7820 7820 2020 2d20  x   - 36x x   - 
│ │ │ │ -0002b110: 3333 7820 7c0a 7c20 2020 3120 3920 2020  33x |.|   1 9   
│ │ │ │ -0002b120: 2020 2032 2039 2020 2020 2020 3320 3920     2 9      3 9 
│ │ │ │ -0002b130: 2020 2020 2034 2039 2020 2020 2035 2039       4 9     5 9
│ │ │ │ -0002b140: 2020 2020 2020 3620 3920 2020 2020 2031        6 9      1
│ │ │ │ -0002b150: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ -0002b160: 2020 2033 7c0a 7c20 2020 2020 2020 2020     3|.|         
│ │ │ │ +0002b0c0: 2020 2020 2020 2020 2020 7c0a 7c31 3778            |.|17x
│ │ │ │ +0002b0d0: 2078 2020 2b20 3138 7820 7820 202b 2034   x  + 18x x  + 4
│ │ │ │ +0002b0e0: 3578 2078 2020 2b20 3136 7820 7820 202b  5x x  + 16x x  +
│ │ │ │ +0002b0f0: 2035 7820 7820 202b 2033 3678 2078 2020   5x x  + 36x x  
│ │ │ │ +0002b100: 2b20 3339 7820 7820 2020 2d20 3336 7820  + 39x x   - 36x 
│ │ │ │ +0002b110: 7820 2020 2d20 3333 7820 7c0a 7c20 2020  x   - 33x |.|   
│ │ │ │ +0002b120: 3120 3920 2020 2020 2032 2039 2020 2020  1 9      2 9    
│ │ │ │ +0002b130: 2020 3320 3920 2020 2020 2034 2039 2020    3 9      4 9  
│ │ │ │ +0002b140: 2020 2035 2039 2020 2020 2020 3620 3920     5 9      6 9 
│ │ │ │ +0002b150: 2020 2020 2031 2031 3020 2020 2020 2032       1 10      2
│ │ │ │ +0002b160: 2031 3020 2020 2020 2033 7c0a 7c20 2020   10      3|.|   
│ │ │ │  0002b170: 2020 2020 2020 2020 2020 2020 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 2020 2020                  
│ │ │ │ -0002b1b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b1b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b200: 2020 2020 7c0a 7c35 7820 7820 202d 2037      |.|5x x  - 7
│ │ │ │ -0002b210: 7820 7820 202b 2034 3578 2078 2020 2b20  x x  + 45x x  + 
│ │ │ │ -0002b220: 3235 7820 7820 202d 2034 3778 2078 2020  25x x  - 47x x  
│ │ │ │ -0002b230: 2b20 3478 2078 2020 2d20 3239 7820 7820  + 4x x  - 29x x 
│ │ │ │ -0002b240: 202b 2033 3278 2078 2020 2d20 3133 7820   + 32x x  - 13x 
│ │ │ │ -0002b250: 7820 202d 7c0a 7c20 2033 2038 2020 2020  x  -|.|  3 8    
│ │ │ │ -0002b260: 2034 2038 2020 2020 2020 3520 3820 2020   4 8      5 8   
│ │ │ │ -0002b270: 2020 2036 2038 2020 2020 2020 3020 3920     6 8      0 9 
│ │ │ │ -0002b280: 2020 2020 3120 3920 2020 2020 2032 2039      1 9      2 9
│ │ │ │ -0002b290: 2020 2020 2020 3320 3920 2020 2020 2034        3 9      4
│ │ │ │ -0002b2a0: 2039 2020 7c0a 7c20 2020 2020 2020 2020   9  |.|         
│ │ │ │ +0002b200: 2020 2020 2020 2020 2020 7c0a 7c35 7820            |.|5x 
│ │ │ │ +0002b210: 7820 202d 2037 7820 7820 202b 2034 3578  x  - 7x x  + 45x
│ │ │ │ +0002b220: 2078 2020 2b20 3235 7820 7820 202d 2034   x  + 25x x  - 4
│ │ │ │ +0002b230: 3778 2078 2020 2b20 3478 2078 2020 2d20  7x x  + 4x x  - 
│ │ │ │ +0002b240: 3239 7820 7820 202b 2033 3278 2078 2020  29x x  + 32x x  
│ │ │ │ +0002b250: 2d20 3133 7820 7820 202d 7c0a 7c20 2033  - 13x x  -|.|  3
│ │ │ │ +0002b260: 2038 2020 2020 2034 2038 2020 2020 2020   8     4 8      
│ │ │ │ +0002b270: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ +0002b280: 2020 3020 3920 2020 2020 3120 3920 2020    0 9     1 9   
│ │ │ │ +0002b290: 2020 2032 2039 2020 2020 2020 3320 3920     2 9      3 9 
│ │ │ │ +0002b2a0: 2020 2020 2034 2039 2020 7c0a 7c20 2020       4 9  |.|   
│ │ │ │  0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2f0: 2020 2020 7c0a 7c20 7820 202b 2033 3878      |.| x  + 38x
│ │ │ │ -0002b300: 2078 2020 2b20 3436 7820 7820 202d 2036   x  + 46x x  - 6
│ │ │ │ -0002b310: 7820 7820 202b 2031 3278 2078 2020 2d20  x x  + 12x x  - 
│ │ │ │ -0002b320: 3438 7820 7820 202b 2031 3778 2078 2020  48x x  + 17x x  
│ │ │ │ -0002b330: 2d20 3339 7820 7820 202d 2034 3478 2078  - 39x x  - 44x x
│ │ │ │ -0002b340: 2020 2d20 7c0a 7c34 2038 2020 2020 2020    - |.|4 8      
│ │ │ │ -0002b350: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ -0002b360: 2030 2039 2020 2020 2020 3120 3920 2020   0 9      1 9   
│ │ │ │ -0002b370: 2020 2032 2039 2020 2020 2020 3320 3920     2 9      3 9 
│ │ │ │ -0002b380: 2020 2020 2034 2039 2020 2020 2020 3520       4 9      5 
│ │ │ │ -0002b390: 3920 2020 7c0a 7c20 2020 2020 2020 2020  9   |.|         
│ │ │ │ +0002b2f0: 2020 2020 2020 2020 2020 7c0a 7c20 7820            |.| x 
│ │ │ │ +0002b300: 202b 2033 3878 2078 2020 2b20 3436 7820   + 38x x  + 46x 
│ │ │ │ +0002b310: 7820 202d 2036 7820 7820 202b 2031 3278  x  - 6x x  + 12x
│ │ │ │ +0002b320: 2078 2020 2d20 3438 7820 7820 202b 2031   x  - 48x x  + 1
│ │ │ │ +0002b330: 3778 2078 2020 2d20 3339 7820 7820 202d  7x x  - 39x x  -
│ │ │ │ +0002b340: 2034 3478 2078 2020 2d20 7c0a 7c34 2038   44x x  - |.|4 8
│ │ │ │ +0002b350: 2020 2020 2020 3520 3820 2020 2020 2036        5 8      6
│ │ │ │ +0002b360: 2038 2020 2020 2030 2039 2020 2020 2020   8     0 9      
│ │ │ │ +0002b370: 3120 3920 2020 2020 2032 2039 2020 2020  1 9      2 9    
│ │ │ │ +0002b380: 2020 3320 3920 2020 2020 2034 2039 2020    3 9      4 9  
│ │ │ │ +0002b390: 2020 2020 3520 3920 2020 7c0a 7c20 2020      5 9   |.|   
│ │ │ │  0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b3e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b430: 2020 2020 7c0a 7c20 3478 2078 2020 2b20      |.| 4x x  + 
│ │ │ │ -0002b440: 3137 7820 7820 202d 2034 3478 2078 2020  17x x  - 44x x  
│ │ │ │ -0002b450: 2b20 3138 7820 7820 202d 2034 3278 2078  + 18x x  - 42x x
│ │ │ │ -0002b460: 2020 2b20 3339 7820 7820 202b 2039 7820    + 39x x  + 9x 
│ │ │ │ -0002b470: 7820 202d 2031 3378 2078 2020 2b20 3438  x  - 13x x  + 48
│ │ │ │ -0002b480: 7820 7820 7c0a 7c20 2020 3620 3720 2020  x x |.|   6 7   
│ │ │ │ -0002b490: 2020 2030 2038 2020 2020 2020 3120 3820     0 8      1 8 
│ │ │ │ -0002b4a0: 2020 2020 2032 2038 2020 2020 2020 3320       2 8      3 
│ │ │ │ -0002b4b0: 3820 2020 2020 2034 2038 2020 2020 2035  8      4 8     5
│ │ │ │ -0002b4c0: 2038 2020 2020 2020 3620 3820 2020 2020   8      6 8     
│ │ │ │ -0002b4d0: 2030 2039 7c0a 7c20 2020 2020 2020 2020   0 9|.|         
│ │ │ │ +0002b430: 2020 2020 2020 2020 2020 7c0a 7c20 3478            |.| 4x
│ │ │ │ +0002b440: 2078 2020 2b20 3137 7820 7820 202d 2034   x  + 17x x  - 4
│ │ │ │ +0002b450: 3478 2078 2020 2b20 3138 7820 7820 202d  4x x  + 18x x  -
│ │ │ │ +0002b460: 2034 3278 2078 2020 2b20 3339 7820 7820   42x x  + 39x x 
│ │ │ │ +0002b470: 202b 2039 7820 7820 202d 2031 3378 2078   + 9x x  - 13x x
│ │ │ │ +0002b480: 2020 2b20 3438 7820 7820 7c0a 7c20 2020    + 48x x |.|   
│ │ │ │ +0002b490: 3620 3720 2020 2020 2030 2038 2020 2020  6 7      0 8    
│ │ │ │ +0002b4a0: 2020 3120 3820 2020 2020 2032 2038 2020    1 8      2 8  
│ │ │ │ +0002b4b0: 2020 2020 3320 3820 2020 2020 2034 2038      3 8      4 8
│ │ │ │ +0002b4c0: 2020 2020 2035 2038 2020 2020 2020 3620       5 8      6 
│ │ │ │ +0002b4d0: 3820 2020 2020 2030 2039 7c0a 7c20 2020  8      0 9|.|   
│ │ │ │  0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b520: 2020 2020 7c0a 7c20 3332 7820 7820 202d      |.| 32x x  -
│ │ │ │ -0002b530: 2037 7820 7820 202d 2032 3978 2078 2020   7x x  - 29x x  
│ │ │ │ -0002b540: 2d20 3437 7820 7820 202b 2033 3578 2078  - 47x x  + 35x x
│ │ │ │ -0002b550: 2020 2b20 3338 7820 7820 202d 2034 3378    + 38x x  - 43x
│ │ │ │ -0002b560: 2078 2020 2d20 3278 2078 2020 2d20 3332   x  - 2x x  - 32
│ │ │ │ -0002b570: 7820 7820 7c0a 7c20 2020 2033 2038 2020  x x |.|    3 8  
│ │ │ │ -0002b580: 2020 2034 2038 2020 2020 2020 3520 3820     4 8      5 8 
│ │ │ │ -0002b590: 2020 2020 2036 2038 2020 2020 2020 3020       6 8      0 
│ │ │ │ -0002b5a0: 3920 2020 2020 2031 2039 2020 2020 2020  9      1 9      
│ │ │ │ -0002b5b0: 3220 3920 2020 2020 3320 3920 2020 2020  2 9     3 9     
│ │ │ │ -0002b5c0: 2034 2039 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d   4 9|.|---------
│ │ │ │ +0002b520: 2020 2020 2020 2020 2020 7c0a 7c20 3332            |.| 32
│ │ │ │ +0002b530: 7820 7820 202d 2037 7820 7820 202d 2032  x x  - 7x x  - 2
│ │ │ │ +0002b540: 3978 2078 2020 2d20 3437 7820 7820 202b  9x x  - 47x x  +
│ │ │ │ +0002b550: 2033 3578 2078 2020 2b20 3338 7820 7820   35x x  + 38x x 
│ │ │ │ +0002b560: 202d 2034 3378 2078 2020 2d20 3278 2078   - 43x x  - 2x x
│ │ │ │ +0002b570: 2020 2d20 3332 7820 7820 7c0a 7c20 2020    - 32x x |.|   
│ │ │ │ +0002b580: 2033 2038 2020 2020 2034 2038 2020 2020   3 8     4 8    
│ │ │ │ +0002b590: 2020 3520 3820 2020 2020 2036 2038 2020    5 8      6 8  
│ │ │ │ +0002b5a0: 2020 2020 3020 3920 2020 2020 2031 2039      0 9      1 9
│ │ │ │ +0002b5b0: 2020 2020 2020 3220 3920 2020 2020 3320        2 9     3 
│ │ │ │ +0002b5c0: 3920 2020 2020 2034 2039 7c0a 7c2d 2d2d  9      4 9|.|---
│ │ │ │  0002b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b610: 2d2d 2d2d 7c0a 7c78 2020 2b20 3234 7820  ----|.|x  + 24x 
│ │ │ │ -0002b620: 7820 202d 2032 3778 2078 2020 2d20 3578  x  - 27x x  - 5x
│ │ │ │ -0002b630: 2078 2020 202b 2032 3878 2078 2020 202b   x   + 28x x   +
│ │ │ │ -0002b640: 2033 3778 2078 2020 202b 2039 7820 7820   37x x   + 9x x 
│ │ │ │ -0002b650: 2020 2b20 3237 7820 7820 2020 2d20 3235    + 27x x   - 25
│ │ │ │ -0002b660: 7820 7820 7c0a 7c20 3920 2020 2020 2035  x x |.| 9      5
│ │ │ │ -0002b670: 2039 2020 2020 2020 3620 3920 2020 2020   9      6 9     
│ │ │ │ -0002b680: 3020 3130 2020 2020 2020 3120 3130 2020  0 10      1 10  
│ │ │ │ -0002b690: 2020 2020 3220 3130 2020 2020 2034 2031      2 10     4 1
│ │ │ │ -0002b6a0: 3020 2020 2020 2036 2031 3020 2020 2020  0      6 10     
│ │ │ │ -0002b6b0: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ +0002b610: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c78 2020  ----------|.|x  
│ │ │ │ +0002b620: 2b20 3234 7820 7820 202d 2032 3778 2078  + 24x x  - 27x x
│ │ │ │ +0002b630: 2020 2d20 3578 2078 2020 202b 2032 3878    - 5x x   + 28x
│ │ │ │ +0002b640: 2078 2020 202b 2033 3778 2078 2020 202b   x   + 37x x   +
│ │ │ │ +0002b650: 2039 7820 7820 2020 2b20 3237 7820 7820   9x x   + 27x x 
│ │ │ │ +0002b660: 2020 2d20 3235 7820 7820 7c0a 7c20 3920    - 25x x |.| 9 
│ │ │ │ +0002b670: 2020 2020 2035 2039 2020 2020 2020 3620       5 9      6 
│ │ │ │ +0002b680: 3920 2020 2020 3020 3130 2020 2020 2020  9     0 10      
│ │ │ │ +0002b690: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ +0002b6a0: 2020 2034 2031 3020 2020 2020 2036 2031     4 10      6 1
│ │ │ │ +0002b6b0: 3020 2020 2020 2030 2020 7c0a 7c20 2020  0      0  |.|   
│ │ │ │  0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b700: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b750: 2020 2020 7c0a 7c20 7820 202d 2034 3678      |.| x  - 46x
│ │ │ │ -0002b760: 2078 2020 2b20 3236 7820 7820 202b 2031   x  + 26x x  + 1
│ │ │ │ -0002b770: 3778 2078 2020 2b20 3135 7820 7820 2020  7x x  + 15x x   
│ │ │ │ -0002b780: 2b20 3335 7820 7820 2020 2b20 3334 7820  + 35x x   + 34x 
│ │ │ │ -0002b790: 7820 2020 2b20 3230 7820 7820 2020 2b20  x   + 20x x   + 
│ │ │ │ -0002b7a0: 3134 7820 7c0a 7c33 2039 2020 2020 2020  14x |.|3 9      
│ │ │ │ -0002b7b0: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ -0002b7c0: 2020 3620 3920 2020 2020 2030 2031 3020    6 9      0 10 
│ │ │ │ -0002b7d0: 2020 2020 2031 2031 3020 2020 2020 2032       1 10      2
│ │ │ │ -0002b7e0: 2031 3020 2020 2020 2034 2031 3020 2020   10      4 10   
│ │ │ │ -0002b7f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b750: 2020 2020 2020 2020 2020 7c0a 7c20 7820            |.| x 
│ │ │ │ +0002b760: 202d 2034 3678 2078 2020 2b20 3236 7820   - 46x x  + 26x 
│ │ │ │ +0002b770: 7820 202b 2031 3778 2078 2020 2b20 3135  x  + 17x x  + 15
│ │ │ │ +0002b780: 7820 7820 2020 2b20 3335 7820 7820 2020  x x   + 35x x   
│ │ │ │ +0002b790: 2b20 3334 7820 7820 2020 2b20 3230 7820  + 34x x   + 20x 
│ │ │ │ +0002b7a0: 7820 2020 2b20 3134 7820 7c0a 7c33 2039  x   + 14x |.|3 9
│ │ │ │ +0002b7b0: 2020 2020 2020 3420 3920 2020 2020 2035        4 9      5
│ │ │ │ +0002b7c0: 2039 2020 2020 2020 3620 3920 2020 2020   9      6 9     
│ │ │ │ +0002b7d0: 2030 2031 3020 2020 2020 2031 2031 3020   0 10      1 10 
│ │ │ │ +0002b7e0: 2020 2020 2032 2031 3020 2020 2020 2034       2 10      4
│ │ │ │ +0002b7f0: 2031 3020 2020 2020 2020 7c0a 7c20 2020   10       |.|   
│ │ │ │  0002b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b840: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b840: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b890: 2020 2020 7c0a 7c2b 2034 3578 2078 2020      |.|+ 45x x  
│ │ │ │ -0002b8a0: 2d20 3134 7820 7820 202d 2032 3578 2078  - 14x x  - 25x x
│ │ │ │ -0002b8b0: 2020 202b 2034 3578 2078 2020 202d 2034     + 45x x   - 4
│ │ │ │ -0002b8c0: 3178 2078 2020 202d 2034 3678 2078 2020  1x x   - 46x x  
│ │ │ │ -0002b8d0: 202b 2038 7820 7820 2020 2d20 3238 7820   + 8x x   - 28x 
│ │ │ │ -0002b8e0: 7820 2020 7c0a 7c20 2020 2020 3520 3920  x   |.|     5 9 
│ │ │ │ -0002b8f0: 2020 2020 2036 2039 2020 2020 2020 3020       6 9      0 
│ │ │ │ -0002b900: 3130 2020 2020 2020 3120 3130 2020 2020  10      1 10    
│ │ │ │ -0002b910: 2020 3220 3130 2020 2020 2020 3420 3130    2 10      4 10
│ │ │ │ -0002b920: 2020 2020 2036 2031 3020 2020 2020 2030       6 10      0
│ │ │ │ -0002b930: 2031 3120 7c0a 7c20 2020 2020 2020 2020   11 |.|         
│ │ │ │ +0002b890: 2020 2020 2020 2020 2020 7c0a 7c2b 2034            |.|+ 4
│ │ │ │ +0002b8a0: 3578 2078 2020 2d20 3134 7820 7820 202d  5x x  - 14x x  -
│ │ │ │ +0002b8b0: 2032 3578 2078 2020 202b 2034 3578 2078   25x x   + 45x x
│ │ │ │ +0002b8c0: 2020 202d 2034 3178 2078 2020 202d 2034     - 41x x   - 4
│ │ │ │ +0002b8d0: 3678 2078 2020 202b 2038 7820 7820 2020  6x x   + 8x x   
│ │ │ │ +0002b8e0: 2d20 3238 7820 7820 2020 7c0a 7c20 2020  - 28x x   |.|   
│ │ │ │ +0002b8f0: 2020 3520 3920 2020 2020 2036 2039 2020    5 9      6 9  
│ │ │ │ +0002b900: 2020 2020 3020 3130 2020 2020 2020 3120      0 10      1 
│ │ │ │ +0002b910: 3130 2020 2020 2020 3220 3130 2020 2020  10      2 10    
│ │ │ │ +0002b920: 2020 3420 3130 2020 2020 2036 2031 3020    4 10     6 10 
│ │ │ │ +0002b930: 2020 2020 2030 2031 3120 7c0a 7c20 2020       0 11 |.|   
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b980: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b9d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ba20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ba70: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bac0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bb10: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bb60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bbb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bc00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bc50: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bcf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bcf0: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bd40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c78 2020 202d 2078 2078      |.|x   - x x
│ │ │ │ -0002bda0: 2020 202d 2034 3678 2078 2020 202b 2032     - 46x x   + 2
│ │ │ │ -0002bdb0: 7820 7820 2020 2b20 7820 7820 202c 2020  x x   + x x  ,  
│ │ │ │ -0002bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bd90: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +0002bda0: 202d 2078 2078 2020 202d 2034 3678 2078   - x x   - 46x x
│ │ │ │ +0002bdb0: 2020 202b 2032 7820 7820 2020 2b20 7820     + 2x x   + x 
│ │ │ │ +0002bdc0: 7820 202c 2020 2020 2020 2020 2020 2020  x  ,            
│ │ │ │  0002bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bde0: 2020 2020 7c0a 7c20 3130 2020 2020 3620      |.| 10    6 
│ │ │ │ -0002bdf0: 3130 2020 2020 2020 3120 3131 2020 2020  10      1 11    
│ │ │ │ -0002be00: 2032 2031 3120 2020 2035 2031 3120 2020   2 11    5 11   
│ │ │ │ -0002be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bde0: 2020 2020 2020 2020 2020 7c0a 7c20 3130            |.| 10
│ │ │ │ +0002bdf0: 2020 2020 3620 3130 2020 2020 2020 3120      6 10      1 
│ │ │ │ +0002be00: 3131 2020 2020 2032 2031 3120 2020 2035  11     2 11    5
│ │ │ │ +0002be10: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │  0002be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002be30: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c78 2020 202d 2032 3978      |.|x   - 29x
│ │ │ │ -0002be90: 2078 2020 202b 2032 3678 2078 2020 202d   x   + 26x x   -
│ │ │ │ -0002bea0: 2032 3678 2078 2020 202d 2034 3178 2078   26x x   - 41x x
│ │ │ │ -0002beb0: 2020 202d 2034 3578 2078 2020 202d 2033     - 45x x   - 3
│ │ │ │ -0002bec0: 3678 2078 2020 202b 2032 3678 2078 2020  6x x   + 26x x  
│ │ │ │ -0002bed0: 2c20 2020 7c0a 7c20 3130 2020 2020 2020  ,   |.| 10      
│ │ │ │ -0002bee0: 3420 3130 2020 2020 2020 3520 3130 2020  4 10      5 10  
│ │ │ │ -0002bef0: 2020 2020 3620 3130 2020 2020 2020 3120      6 10      1 
│ │ │ │ -0002bf00: 3131 2020 2020 2020 3220 3131 2020 2020  11      2 11    
│ │ │ │ -0002bf10: 2020 3420 3131 2020 2020 2020 3520 3131    4 11      5 11
│ │ │ │ -0002bf20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002be80: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +0002be90: 202d 2032 3978 2078 2020 202b 2032 3678   - 29x x   + 26x
│ │ │ │ +0002bea0: 2078 2020 202d 2032 3678 2078 2020 202d   x   - 26x x   -
│ │ │ │ +0002beb0: 2034 3178 2078 2020 202d 2034 3578 2078   41x x   - 45x x
│ │ │ │ +0002bec0: 2020 202d 2033 3678 2078 2020 202b 2032     - 36x x   + 2
│ │ │ │ +0002bed0: 3678 2078 2020 2c20 2020 7c0a 7c20 3130  6x x  ,   |.| 10
│ │ │ │ +0002bee0: 2020 2020 2020 3420 3130 2020 2020 2020        4 10      
│ │ │ │ +0002bef0: 3520 3130 2020 2020 2020 3620 3130 2020  5 10      6 10  
│ │ │ │ +0002bf00: 2020 2020 3120 3131 2020 2020 2020 3220      1 11      2 
│ │ │ │ +0002bf10: 3131 2020 2020 2020 3420 3131 2020 2020  11      4 11    
│ │ │ │ +0002bf20: 2020 3520 3131 2020 2020 7c0a 7c20 2020    5 11    |.|   
│ │ │ │  0002bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bf70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfc0: 2020 2020 7c0a 7c20 3437 7820 7820 202d      |.| 47x x  -
│ │ │ │ -0002bfd0: 2034 3278 2078 2020 2d20 3138 7820 7820   42x x  - 18x x 
│ │ │ │ -0002bfe0: 2020 2d20 3578 2078 2020 202b 2034 3478    - 5x x   + 44x
│ │ │ │ -0002bff0: 2078 2020 202d 2033 3978 2078 2020 202d   x   - 39x x   -
│ │ │ │ -0002c000: 2031 3578 2078 2020 202d 2033 3178 2078   15x x   - 31x x
│ │ │ │ -0002c010: 2020 2020 7c0a 7c20 2020 2035 2039 2020      |.|    5 9  
│ │ │ │ -0002c020: 2020 2020 3620 3920 2020 2020 2030 2031      6 9      0 1
│ │ │ │ -0002c030: 3020 2020 2020 3120 3130 2020 2020 2020  0     1 10      
│ │ │ │ -0002c040: 3220 3130 2020 2020 2020 3320 3130 2020  2 10      3 10  
│ │ │ │ -0002c050: 2020 2020 3420 3130 2020 2020 2020 3520      4 10      5 
│ │ │ │ -0002c060: 3130 2020 7c0a 7c20 2020 2020 2020 2020  10  |.|         
│ │ │ │ +0002bfc0: 2020 2020 2020 2020 2020 7c0a 7c20 3437            |.| 47
│ │ │ │ +0002bfd0: 7820 7820 202d 2034 3278 2078 2020 2d20  x x  - 42x x  - 
│ │ │ │ +0002bfe0: 3138 7820 7820 2020 2d20 3578 2078 2020  18x x   - 5x x  
│ │ │ │ +0002bff0: 202b 2034 3478 2078 2020 202d 2033 3978   + 44x x   - 39x
│ │ │ │ +0002c000: 2078 2020 202d 2031 3578 2078 2020 202d   x   - 15x x   -
│ │ │ │ +0002c010: 2033 3178 2078 2020 2020 7c0a 7c20 2020   31x x    |.|   
│ │ │ │ +0002c020: 2035 2039 2020 2020 2020 3620 3920 2020   5 9      6 9   
│ │ │ │ +0002c030: 2020 2030 2031 3020 2020 2020 3120 3130     0 10     1 10
│ │ │ │ +0002c040: 2020 2020 2020 3220 3130 2020 2020 2020        2 10      
│ │ │ │ +0002c050: 3320 3130 2020 2020 2020 3420 3130 2020  3 10      4 10  
│ │ │ │ +0002c060: 2020 2020 3520 3130 2020 7c0a 7c20 2020      5 10  |.|   
│ │ │ │  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 7c0a 7c35 7820 7820 2020 2d20      |.|5x x   - 
│ │ │ │ -0002c0c0: 3135 7820 7820 2020 2d20 3133 7820 7820  15x x   - 13x x 
│ │ │ │ -0002c0d0: 2020 2d20 3230 7820 7820 2020 2d20 3435    - 20x x   - 45
│ │ │ │ -0002c0e0: 7820 7820 2020 2b20 3478 2078 2020 202d  x x   + 4x x   -
│ │ │ │ -0002c0f0: 2034 7820 7820 2020 2b20 3130 7820 7820   4x x   + 10x x 
│ │ │ │ -0002c100: 2020 2b20 7c0a 7c20 2030 2031 3020 2020    + |.|  0 10   
│ │ │ │ -0002c110: 2020 2031 2031 3020 2020 2020 2032 2031     1 10      2 1
│ │ │ │ -0002c120: 3020 2020 2020 2033 2031 3020 2020 2020  0      3 10     
│ │ │ │ -0002c130: 2034 2031 3020 2020 2020 3520 3130 2020   4 10     5 10  
│ │ │ │ -0002c140: 2020 2036 2031 3020 2020 2020 2031 2031     6 10      1 1
│ │ │ │ -0002c150: 3120 2020 7c0a 7c20 2020 2020 2020 2020  1   |.|         
│ │ │ │ +0002c0b0: 2020 2020 2020 2020 2020 7c0a 7c35 7820            |.|5x 
│ │ │ │ +0002c0c0: 7820 2020 2d20 3135 7820 7820 2020 2d20  x   - 15x x   - 
│ │ │ │ +0002c0d0: 3133 7820 7820 2020 2d20 3230 7820 7820  13x x   - 20x x 
│ │ │ │ +0002c0e0: 2020 2d20 3435 7820 7820 2020 2b20 3478    - 45x x   + 4x
│ │ │ │ +0002c0f0: 2078 2020 202d 2034 7820 7820 2020 2b20   x   - 4x x   + 
│ │ │ │ +0002c100: 3130 7820 7820 2020 2b20 7c0a 7c20 2030  10x x   + |.|  0
│ │ │ │ +0002c110: 2031 3020 2020 2020 2031 2031 3020 2020   10      1 10   
│ │ │ │ +0002c120: 2020 2032 2031 3020 2020 2020 2033 2031     2 10      3 1
│ │ │ │ +0002c130: 3020 2020 2020 2034 2031 3020 2020 2020  0      4 10     
│ │ │ │ +0002c140: 3520 3130 2020 2020 2036 2031 3020 2020  5 10     6 10   
│ │ │ │ +0002c150: 2020 2031 2031 3120 2020 7c0a 7c20 2020     1 11   |.|   
│ │ │ │  0002c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c1a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2d20 3332 7820 7820      |.| - 32x x 
│ │ │ │ -0002c200: 202d 2033 3778 2078 2020 2d20 3478 2078   - 37x x  - 4x x
│ │ │ │ -0002c210: 2020 2d20 3338 7820 7820 202b 2031 3078    - 38x x  + 10x
│ │ │ │ -0002c220: 2078 2020 2b20 3133 7820 7820 202b 2031   x  + 13x x  + 1
│ │ │ │ -0002c230: 3278 2078 2020 202b 2033 3578 2078 2020  2x x   + 35x x  
│ │ │ │ -0002c240: 202b 2020 7c0a 7c20 2020 2020 2031 2039   +  |.|      1 9
│ │ │ │ -0002c250: 2020 2020 2020 3220 3920 2020 2020 3320        2 9     3 
│ │ │ │ -0002c260: 3920 2020 2020 2034 2039 2020 2020 2020  9      4 9      
│ │ │ │ -0002c270: 3520 3920 2020 2020 2036 2039 2020 2020  5 9      6 9    
│ │ │ │ -0002c280: 2020 3020 3130 2020 2020 2020 3120 3130    0 10      1 10
│ │ │ │ -0002c290: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c1f0: 2020 2020 2020 2020 2020 7c0a 7c20 2d20            |.| - 
│ │ │ │ +0002c200: 3332 7820 7820 202d 2033 3778 2078 2020  32x x  - 37x x  
│ │ │ │ +0002c210: 2d20 3478 2078 2020 2d20 3338 7820 7820  - 4x x  - 38x x 
│ │ │ │ +0002c220: 202b 2031 3078 2078 2020 2b20 3133 7820   + 10x x  + 13x 
│ │ │ │ +0002c230: 7820 202b 2031 3278 2078 2020 202b 2033  x  + 12x x   + 3
│ │ │ │ +0002c240: 3578 2078 2020 202b 2020 7c0a 7c20 2020  5x x   +  |.|   
│ │ │ │ +0002c250: 2020 2031 2039 2020 2020 2020 3220 3920     1 9      2 9 
│ │ │ │ +0002c260: 2020 2020 3320 3920 2020 2020 2034 2039      3 9      4 9
│ │ │ │ +0002c270: 2020 2020 2020 3520 3920 2020 2020 2036        5 9      6
│ │ │ │ +0002c280: 2039 2020 2020 2020 3020 3130 2020 2020   9      0 10    
│ │ │ │ +0002c290: 2020 3120 3130 2020 2020 7c0a 7c20 2020    1 10    |.|   
│ │ │ │  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 7c0a 7c20 2d20 3433 7820 7820      |.| - 43x x 
│ │ │ │ -0002c2f0: 202b 2031 3278 2078 2020 2d20 3132 7820   + 12x x  - 12x 
│ │ │ │ -0002c300: 7820 2020 2b20 3134 7820 7820 2020 2d20  x   + 14x x   - 
│ │ │ │ -0002c310: 3437 7820 7820 2020 2d20 3436 7820 7820  47x x   - 46x x 
│ │ │ │ -0002c320: 2020 2d20 3134 7820 7820 2020 2b20 3438    - 14x x   + 48
│ │ │ │ -0002c330: 7820 7820 7c0a 7c20 2020 2020 2035 2039  x x |.|      5 9
│ │ │ │ -0002c340: 2020 2020 2020 3620 3920 2020 2020 2030        6 9      0
│ │ │ │ -0002c350: 2031 3020 2020 2020 2031 2031 3020 2020   10      1 10   
│ │ │ │ -0002c360: 2020 2032 2031 3020 2020 2020 2033 2031     2 10      3 1
│ │ │ │ -0002c370: 3020 2020 2020 2034 2031 3020 2020 2020  0      4 10     
│ │ │ │ -0002c380: 2035 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d   5  |.|---------
│ │ │ │ +0002c2e0: 2020 2020 2020 2020 2020 7c0a 7c20 2d20            |.| - 
│ │ │ │ +0002c2f0: 3433 7820 7820 202b 2031 3278 2078 2020  43x x  + 12x x  
│ │ │ │ +0002c300: 2d20 3132 7820 7820 2020 2b20 3134 7820  - 12x x   + 14x 
│ │ │ │ +0002c310: 7820 2020 2d20 3437 7820 7820 2020 2d20  x   - 47x x   - 
│ │ │ │ +0002c320: 3436 7820 7820 2020 2d20 3134 7820 7820  46x x   - 14x x 
│ │ │ │ +0002c330: 2020 2b20 3438 7820 7820 7c0a 7c20 2020    + 48x x |.|   
│ │ │ │ +0002c340: 2020 2035 2039 2020 2020 2020 3620 3920     5 9      6 9 
│ │ │ │ +0002c350: 2020 2020 2030 2031 3020 2020 2020 2031       0 10      1
│ │ │ │ +0002c360: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ +0002c370: 2020 2033 2031 3020 2020 2020 2034 2031     3 10      4 1
│ │ │ │ +0002c380: 3020 2020 2020 2035 2020 7c0a 7c2d 2d2d  0      5  |.|---
│ │ │ │  0002c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c3d0: 2d2d 2d2d 7c0a 7c20 2020 2b20 3978 2078  ----|.|   + 9x x
│ │ │ │ -0002c3e0: 2020 202b 2032 3778 2078 2020 202d 2032     + 27x x   - 2
│ │ │ │ -0002c3f0: 3778 2078 2020 2c20 2020 2020 2020 2020  7x x  ,         
│ │ │ │ +0002c3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020  ----------|.|   
│ │ │ │ +0002c3e0: 2b20 3978 2078 2020 202b 2032 3778 2078  + 9x x   + 27x x
│ │ │ │ +0002c3f0: 2020 202d 2032 3778 2078 2020 2c20 2020     - 27x x  ,   
│ │ │ │  0002c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c420: 2020 2020 7c0a 7c31 3120 2020 2020 3220      |.|11     2 
│ │ │ │ -0002c430: 3131 2020 2020 2020 3420 3131 2020 2020  11      4 11    
│ │ │ │ -0002c440: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │ +0002c420: 2020 2020 2020 2020 2020 7c0a 7c31 3120            |.|11 
│ │ │ │ +0002c430: 2020 2020 3220 3131 2020 2020 2020 3420      2 11      4 
│ │ │ │ +0002c440: 3131 2020 2020 2020 3520 3131 2020 2020  11      5 11    
│ │ │ │  0002c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c470: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c470: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c4c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c510: 2020 2020 7c0a 7c20 7820 2020 2b20 3336      |.| x   + 36
│ │ │ │ -0002c520: 7820 7820 2020 2b20 3335 7820 7820 2020  x x   + 35x x   
│ │ │ │ -0002c530: 2d20 3137 7820 7820 202c 2020 2020 2020  - 17x x  ,      
│ │ │ │ +0002c510: 2020 2020 2020 2020 2020 7c0a 7c20 7820            |.| x 
│ │ │ │ +0002c520: 2020 2b20 3336 7820 7820 2020 2b20 3335    + 36x x   + 35
│ │ │ │ +0002c530: 7820 7820 2020 2d20 3137 7820 7820 202c  x x   - 17x x  ,
│ │ │ │  0002c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c560: 2020 2020 7c0a 7c30 2031 3120 2020 2020      |.|0 11     
│ │ │ │ -0002c570: 2031 2031 3120 2020 2020 2032 2031 3120   1 11      2 11 
│ │ │ │ -0002c580: 2020 2020 2034 2031 3120 2020 2020 2020       4 11       
│ │ │ │ +0002c560: 2020 2020 2020 2020 2020 7c0a 7c30 2031            |.|0 1
│ │ │ │ +0002c570: 3120 2020 2020 2031 2031 3120 2020 2020  1      1 11     
│ │ │ │ +0002c580: 2032 2031 3120 2020 2020 2034 2031 3120   2 11      4 11 
│ │ │ │  0002c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c5b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c600: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c600: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c20 2b20 3131 7820 7820      |.| + 11x x 
│ │ │ │ -0002c660: 2020 2b20 3134 7820 7820 2020 2d20 3878    + 14x x   - 8x
│ │ │ │ -0002c670: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ +0002c650: 2020 2020 2020 2020 2020 7c0a 7c20 2b20            |.| + 
│ │ │ │ +0002c660: 3131 7820 7820 2020 2b20 3134 7820 7820  11x x   + 14x x 
│ │ │ │ +0002c670: 2020 2d20 3878 2078 2020 2020 2020 2020    - 8x x        
│ │ │ │  0002c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6a0: 2020 2020 7c0a 7c20 2020 2020 2032 2031      |.|      2 1
│ │ │ │ -0002c6b0: 3120 2020 2020 2034 2031 3120 2020 2020  1      4 11     
│ │ │ │ -0002c6c0: 3520 3131 2020 2020 2020 2020 2020 2020  5 11            
│ │ │ │ +0002c6a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002c6b0: 2020 2032 2031 3120 2020 2020 2034 2031     2 11      4 1
│ │ │ │ +0002c6c0: 3120 2020 2020 3520 3131 2020 2020 2020  1     5 11      
│ │ │ │  0002c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c6f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c740: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c790: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c7e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c830: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c830: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c880: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c8d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c920: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c970: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c9c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ca10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ca60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cab0: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cb00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cb50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cba0: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cbf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cc40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cc90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cce0: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cd30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c2b 2031 3578 2078 2020      |.|+ 15x x  
│ │ │ │ -0002cd90: 202d 2031 3778 2078 2020 202d 2032 3978   - 17x x   - 29x
│ │ │ │ -0002cda0: 2078 2020 202b 2033 3578 2078 2020 202d   x   + 35x x   -
│ │ │ │ -0002cdb0: 2031 3778 2078 2020 202d 2033 3178 2078   17x x   - 31x x
│ │ │ │ -0002cdc0: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
│ │ │ │ -0002cdd0: 2020 2020 7c0a 7c20 2020 2020 3620 3130      |.|     6 10
│ │ │ │ -0002cde0: 2020 2020 2020 3020 3131 2020 2020 2020        0 11      
│ │ │ │ -0002cdf0: 3120 3131 2020 2020 2020 3220 3131 2020  1 11      2 11  
│ │ │ │ -0002ce00: 2020 2020 3320 3131 2020 2020 2020 3520      3 11      5 
│ │ │ │ -0002ce10: 3131 2020 2020 2020 2020 2020 2020 2020  11              
│ │ │ │ -0002ce20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cd80: 2020 2020 2020 2020 2020 7c0a 7c2b 2031            |.|+ 1
│ │ │ │ +0002cd90: 3578 2078 2020 202d 2031 3778 2078 2020  5x x   - 17x x  
│ │ │ │ +0002cda0: 202d 2032 3978 2078 2020 202b 2033 3578   - 29x x   + 35x
│ │ │ │ +0002cdb0: 2078 2020 202d 2031 3778 2078 2020 202d   x   - 17x x   -
│ │ │ │ +0002cdc0: 2033 3178 2078 2020 2c20 2020 2020 2020   31x x  ,       
│ │ │ │ +0002cdd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002cde0: 2020 3620 3130 2020 2020 2020 3020 3131    6 10      0 11
│ │ │ │ +0002cdf0: 2020 2020 2020 3120 3131 2020 2020 2020        1 11      
│ │ │ │ +0002ce00: 3220 3131 2020 2020 2020 3320 3131 2020  2 11      3 11  
│ │ │ │ +0002ce10: 2020 2020 3520 3131 2020 2020 2020 2020      5 11        
│ │ │ │ +0002ce20: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c20 3878 2078 2020 202b      |.| 8x x   +
│ │ │ │ -0002ce80: 2034 7820 7820 202c 2020 2020 2020 2020   4x x  ,        
│ │ │ │ +0002ce70: 2020 2020 2020 2020 2020 7c0a 7c20 3878            |.| 8x
│ │ │ │ +0002ce80: 2078 2020 202b 2034 7820 7820 202c 2020   x   + 4x x  ,  
│ │ │ │  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 7c0a 7c20 2020 3220 3131 2020      |.|   2 11  
│ │ │ │ -0002ced0: 2020 2035 2031 3120 2020 2020 2020 2020     5 11         
│ │ │ │ +0002cec0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002ced0: 3220 3131 2020 2020 2035 2031 3120 2020  2 11     5 11   
│ │ │ │  0002cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cf10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cf10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cf60: 2020 2020 2020 2020 2020 7c0a 7c20 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 7c0a 7c31 3378 2078 2020 202b      |.|13x x   +
│ │ │ │ -0002cfc0: 2033 3178 2078 2020 202b 2033 3078 2078   31x x   + 30x x
│ │ │ │ -0002cfd0: 2020 202d 2034 3578 2078 2020 202d 2031     - 45x x   - 1
│ │ │ │ -0002cfe0: 3578 2078 2020 202b 2033 3778 2078 2020  5x x   + 37x x  
│ │ │ │ -0002cff0: 202b 2034 3478 2078 2020 202b 2037 7820   + 44x x   + 7x 
│ │ │ │ -0002d000: 7820 2020 7c0a 7c20 2020 3220 3130 2020  x   |.|   2 10  
│ │ │ │ -0002d010: 2020 2020 3320 3130 2020 2020 2020 3420      3 10      4 
│ │ │ │ -0002d020: 3130 2020 2020 2020 3520 3130 2020 2020  10      5 10    
│ │ │ │ -0002d030: 2020 3620 3130 2020 2020 2020 3020 3131    6 10      0 11
│ │ │ │ -0002d040: 2020 2020 2020 3120 3131 2020 2020 2032        1 11     2
│ │ │ │ -0002d050: 2031 3120 7c0a 7c20 2020 2020 2020 2020   11 |.|         
│ │ │ │ +0002cfb0: 2020 2020 2020 2020 2020 7c0a 7c31 3378            |.|13x
│ │ │ │ +0002cfc0: 2078 2020 202b 2033 3178 2078 2020 202b   x   + 31x x   +
│ │ │ │ +0002cfd0: 2033 3078 2078 2020 202d 2034 3578 2078   30x x   - 45x x
│ │ │ │ +0002cfe0: 2020 202d 2031 3578 2078 2020 202b 2033     - 15x x   + 3
│ │ │ │ +0002cff0: 3778 2078 2020 202b 2034 3478 2078 2020  7x x   + 44x x  
│ │ │ │ +0002d000: 202b 2037 7820 7820 2020 7c0a 7c20 2020   + 7x x   |.|   
│ │ │ │ +0002d010: 3220 3130 2020 2020 2020 3320 3130 2020  2 10      3 10  
│ │ │ │ +0002d020: 2020 2020 3420 3130 2020 2020 2020 3520      4 10      5 
│ │ │ │ +0002d030: 3130 2020 2020 2020 3620 3130 2020 2020  10      6 10    
│ │ │ │ +0002d040: 2020 3020 3131 2020 2020 2020 3120 3131    0 11      1 11
│ │ │ │ +0002d050: 2020 2020 2032 2031 3120 7c0a 7c20 2020       2 11 |.|   
│ │ │ │  0002d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2b20 3132 7820      |.|   + 12x 
│ │ │ │ -0002d0b0: 7820 2020 2d20 3132 7820 7820 2020 2b20  x   - 12x x   + 
│ │ │ │ -0002d0c0: 3438 7820 7820 2020 2d20 7820 7820 2020  48x x   - x x   
│ │ │ │ -0002d0d0: 2d20 3978 2078 2020 202b 2034 3878 2078  - 9x x   + 48x x
│ │ │ │ -0002d0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0f0: 2020 2020 7c0a 7c31 3020 2020 2020 2036      |.|10      6
│ │ │ │ -0002d100: 2031 3020 2020 2020 2030 2031 3120 2020   10      0 11   
│ │ │ │ -0002d110: 2020 2031 2031 3120 2020 2032 2031 3120     1 11    2 11 
│ │ │ │ -0002d120: 2020 2020 3320 3131 2020 2020 2020 3520      3 11      5 
│ │ │ │ -0002d130: 3131 2020 2020 2020 2020 2020 2020 2020  11              
│ │ │ │ -0002d140: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +0002d0a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002d0b0: 2b20 3132 7820 7820 2020 2d20 3132 7820  + 12x x   - 12x 
│ │ │ │ +0002d0c0: 7820 2020 2b20 3438 7820 7820 2020 2d20  x   + 48x x   - 
│ │ │ │ +0002d0d0: 7820 7820 2020 2d20 3978 2078 2020 202b  x x   - 9x x   +
│ │ │ │ +0002d0e0: 2034 3878 2078 2020 2020 2020 2020 2020   48x x          
│ │ │ │ +0002d0f0: 2020 2020 2020 2020 2020 7c0a 7c31 3020            |.|10 
│ │ │ │ +0002d100: 2020 2020 2036 2031 3020 2020 2020 2030       6 10      0
│ │ │ │ +0002d110: 2031 3120 2020 2020 2031 2031 3120 2020   11      1 11   
│ │ │ │ +0002d120: 2032 2031 3120 2020 2020 3320 3131 2020   2 11     3 11  
│ │ │ │ +0002d130: 2020 2020 3520 3131 2020 2020 2020 2020      5 11        
│ │ │ │ +0002d140: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  0002d150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 7c0a 7c2b 2039 7820 7820 2020  ----|.|+ 9x x   
│ │ │ │ -0002d1a0: 2d20 3235 7820 7820 2020 2d20 3435 7820  - 25x x   - 45x 
│ │ │ │ -0002d1b0: 7820 202c 2020 2020 2020 2020 2020 2020  x  ,            
│ │ │ │ +0002d190: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2b 2039  ----------|.|+ 9
│ │ │ │ +0002d1a0: 7820 7820 2020 2d20 3235 7820 7820 2020  x x   - 25x x   
│ │ │ │ +0002d1b0: 2d20 3435 7820 7820 202c 2020 2020 2020  - 45x x  ,      
│ │ │ │  0002d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d1e0: 2020 2020 7c0a 7c20 2020 2033 2031 3120      |.|    3 11 
│ │ │ │ -0002d1f0: 2020 2020 2034 2031 3120 2020 2020 2035       4 11      5
│ │ │ │ -0002d200: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │ +0002d1e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002d1f0: 2033 2031 3120 2020 2020 2034 2031 3120   3 11      4 11 
│ │ │ │ +0002d200: 2020 2020 2035 2031 3120 2020 2020 2020       5 11       
│ │ │ │  0002d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d230: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002d230: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  0002d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d280: 2d2d 2d2d 2b0a 7c69 3920 3a20 2d2d 2062  ----+.|i9 : -- b
│ │ │ │ -0002d290: 7574 2e2e 2e20 2020 2020 2020 2020 2020  ut...           
│ │ │ │ +0002d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920  ----------+.|i9 
│ │ │ │ +0002d290: 3a20 2d2d 2062 7574 2e2e 2e20 2020 2020  : -- but...     
│ │ │ │  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 7c0a 7c20 2020 2020 7068 6920      |.|     phi 
│ │ │ │ -0002d2e0: 2a20 7073 6920 3d3d 2031 2020 2020 2020  * psi == 1      
│ │ │ │ +0002d2d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002d2e0: 2020 7068 6920 2a20 7073 6920 3d3d 2031    phi * psi == 1
│ │ │ │  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                  
│ │ │ │ -0002d320: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002d320: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002d330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d370: 2020 2020 7c0a 7c6f 3920 3d20 6661 6c73      |.|o9 = fals
│ │ │ │ -0002d380: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +0002d370: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ +0002d380: 3d20 6661 6c73 6520 2020 2020 2020 2020  = false         
│ │ │ │  0002d390: 2020 2020 2020 2020 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 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002d3c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  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 2b0a 7c69 3130 203a 202d 2d20  ----+.|i10 : -- 
│ │ │ │ -0002d420: 696e 2074 6869 7320 6361 7365 2077 6520  in this case we 
│ │ │ │ -0002d430: 6361 6e20 7265 6d65 6479 2065 6e61 626c  can remedy enabl
│ │ │ │ -0002d440: 696e 6720 7468 6520 6f70 7469 6f6e 2043  ing the option C
│ │ │ │ -0002d450: 6572 7469 6679 2020 2020 2020 2020 2020  ertify          
│ │ │ │ -0002d460: 2020 2020 7c0a 7c20 2020 2020 2074 696d      |.|      tim
│ │ │ │ -0002d470: 6520 7073 6920 3d20 6170 7072 6f78 696d  e psi = approxim
│ │ │ │ -0002d480: 6174 6549 6e76 6572 7365 4d61 7028 7068  ateInverseMap(ph
│ │ │ │ -0002d490: 692c 436f 6469 6d42 7349 6e76 3d3e 342c  i,CodimBsInv=>4,
│ │ │ │ -0002d4a0: 4365 7274 6966 793d 3e74 7275 6529 2020  Certify=>true)  
│ │ │ │ -0002d4b0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -0002d4c0: 332e 3334 3433 3473 2028 6370 7529 3b20  3.34434s (cpu); 
│ │ │ │ -0002d4d0: 322e 3438 3136 3473 2028 7468 7265 6164  2.48164s (thread
│ │ │ │ -0002d4e0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0002d410: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ +0002d420: 203a 202d 2d20 696e 2074 6869 7320 6361   : -- in this ca
│ │ │ │ +0002d430: 7365 2077 6520 6361 6e20 7265 6d65 6479  se we can remedy
│ │ │ │ +0002d440: 2065 6e61 626c 696e 6720 7468 6520 6f70   enabling the op
│ │ │ │ +0002d450: 7469 6f6e 2043 6572 7469 6679 2020 2020  tion Certify    
│ │ │ │ +0002d460: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002d470: 2020 2074 696d 6520 7073 6920 3d20 6170     time psi = ap
│ │ │ │ +0002d480: 7072 6f78 696d 6174 6549 6e76 6572 7365  proximateInverse
│ │ │ │ +0002d490: 4d61 7028 7068 692c 436f 6469 6d42 7349  Map(phi,CodimBsI
│ │ │ │ +0002d4a0: 6e76 3d3e 342c 4365 7274 6966 793d 3e74  nv=>4,Certify=>t
│ │ │ │ +0002d4b0: 7275 6529 2020 2020 2020 7c0a 7c20 2d2d  rue)      |.| --
│ │ │ │ +0002d4c0: 2075 7365 6420 332e 3233 3232 3173 2028   used 3.23221s (
│ │ │ │ +0002d4d0: 6370 7529 3b20 322e 3736 3535 7320 2874  cpu); 2.7655s (t
│ │ │ │ +0002d4e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  0002d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d500: 2020 2020 7c0a 7c2d 2d20 6170 7072 6f78      |.|-- approx
│ │ │ │ -0002d510: 696d 6174 6549 6e76 6572 7365 4d61 703a  imateInverseMap:
│ │ │ │ -0002d520: 2073 7465 7020 3120 6f66 2033 2020 2020   step 1 of 3    
│ │ │ │ -0002d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d500: 2020 2020 2020 2020 2020 7c0a 7c2d 2d20            |.|-- 
│ │ │ │ +0002d510: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +0002d520: 7365 4d61 703a 2073 7465 7020 3120 6f66  seMap: step 1 of
│ │ │ │ +0002d530: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0002d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d550: 2020 2020 7c0a 7c2d 2d20 6170 7072 6f78      |.|-- approx
│ │ │ │ -0002d560: 696d 6174 6549 6e76 6572 7365 4d61 703a  imateInverseMap:
│ │ │ │ -0002d570: 2073 7465 7020 3220 6f66 2033 2020 2020   step 2 of 3    
│ │ │ │ -0002d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d550: 2020 2020 2020 2020 2020 7c0a 7c2d 2d20            |.|-- 
│ │ │ │ +0002d560: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +0002d570: 7365 4d61 703a 2073 7465 7020 3220 6f66  seMap: step 2 of
│ │ │ │ +0002d580: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0002d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5a0: 2020 2020 7c0a 7c2d 2d20 6170 7072 6f78      |.|-- approx
│ │ │ │ -0002d5b0: 696d 6174 6549 6e76 6572 7365 4d61 703a  imateInverseMap:
│ │ │ │ -0002d5c0: 2073 7465 7020 3320 6f66 2033 2020 2020   step 3 of 3    
│ │ │ │ -0002d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d5a0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d20            |.|-- 
│ │ │ │ +0002d5b0: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +0002d5c0: 7365 4d61 703a 2073 7465 7020 3320 6f66  seMap: step 3 of
│ │ │ │ +0002d5d0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0002d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5f0: 2020 2020 7c0a 7c43 6572 7469 6679 3a20      |.|Certify: 
│ │ │ │ -0002d600: 6f75 7470 7574 2063 6572 7469 6669 6564  output certified
│ │ │ │ -0002d610: 2120 2020 2020 2020 2020 2020 2020 2020  !               
│ │ │ │ +0002d5f0: 2020 2020 2020 2020 2020 7c0a 7c43 6572            |.|Cer
│ │ │ │ +0002d600: 7469 6679 3a20 6f75 7470 7574 2063 6572  tify: output cer
│ │ │ │ +0002d610: 7469 6669 6564 2120 2020 2020 2020 2020  tified!         
│ │ │ │  0002d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d640: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002d640: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d690: 2020 2020 7c0a 7c6f 3130 203d 202d 2d20      |.|o10 = -- 
│ │ │ │ -0002d6a0: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ -0002d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d690: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ +0002d6a0: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +0002d6b0: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  0002d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d6e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002d6e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d700: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +0002d700: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │  0002d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d730: 2020 2020 7c0a 7c20 2020 2020 2073 6f75      |.|      sou
│ │ │ │ -0002d740: 7263 653a 2073 7562 7661 7269 6574 7920  rce: subvariety 
│ │ │ │ -0002d750: 6f66 2050 726f 6a28 2d2d 5b78 202c 2078  of Proj(--[x , x
│ │ │ │ -0002d760: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ -0002d770: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ -0002d780: 202c 2078 7c0a 7c20 2020 2020 2020 2020   , x|.|         
│ │ │ │ +0002d730: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002d740: 2020 2073 6f75 7263 653a 2073 7562 7661     source: subva
│ │ │ │ +0002d750: 7269 6574 7920 6f66 2050 726f 6a28 2d2d  riety of Proj(--
│ │ │ │ +0002d760: 5b78 202c 2078 202c 2078 202c 2078 202c  [x , x , x , x ,
│ │ │ │ +0002d770: 2078 202c 2078 202c 2078 202c 2078 202c   x , x , x , x ,
│ │ │ │ +0002d780: 2078 202c 2078 202c 2078 7c0a 7c20 2020   x , x , x|.|   
│ │ │ │  0002d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d7a0: 2020 2020 2020 2020 3937 2020 3020 2020          97  0   
│ │ │ │ -0002d7b0: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ -0002d7c0: 3520 2020 3620 2020 3720 2020 3820 2020  5   6   7   8   
│ │ │ │ -0002d7d0: 3920 2020 7c0a 7c20 2020 2020 2020 2020  9   |.|         
│ │ │ │ -0002d7e0: 2020 2020 207b 2020 2020 2020 2020 2020       {          
│ │ │ │ +0002d7a0: 2020 2020 2020 2020 2020 2020 2020 3937                97
│ │ │ │ +0002d7b0: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ +0002d7c0: 2020 3420 2020 3520 2020 3620 2020 3720    4   5   6   7 
│ │ │ │ +0002d7d0: 2020 3820 2020 3920 2020 7c0a 7c20 2020    8   9   |.|   
│ │ │ │ +0002d7e0: 2020 2020 2020 2020 2020 207b 2020 2020             {    
│ │ │ │  0002d7f0: 2020 2020 2020 2020 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002d820: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d840: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0002d840: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0002d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d870: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002d880: 2020 2020 2020 7820 7820 202d 2038 7820        x x  - 8x 
│ │ │ │ -0002d890: 7820 202b 2032 3578 2020 2d20 3235 7820  x  + 25x  - 25x 
│ │ │ │ -0002d8a0: 7820 202d 2032 3278 2078 2020 2b20 7820  x  - 22x x  + x 
│ │ │ │ -0002d8b0: 7820 202b 2031 3378 2078 2020 2b20 3431  x  + 13x x  + 41
│ │ │ │ -0002d8c0: 7820 7820 7c0a 7c20 2020 2020 2020 2020  x x |.|         
│ │ │ │ -0002d8d0: 2020 2020 2020 2031 2033 2020 2020 2032         1 3     2
│ │ │ │ -0002d8e0: 2033 2020 2020 2020 3320 2020 2020 2032   3      3      2
│ │ │ │ -0002d8f0: 2034 2020 2020 2020 3320 3420 2020 2030   4      3 4    0
│ │ │ │ -0002d900: 2035 2020 2020 2020 3220 3520 2020 2020   5      2 5     
│ │ │ │ -0002d910: 2033 2035 7c0a 7c20 2020 2020 2020 2020   3 5|.|         
│ │ │ │ +0002d870: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002d880: 2020 2020 2020 2020 2020 2020 7820 7820              x x 
│ │ │ │ +0002d890: 202d 2038 7820 7820 202b 2032 3578 2020   - 8x x  + 25x  
│ │ │ │ +0002d8a0: 2d20 3235 7820 7820 202d 2032 3278 2078  - 25x x  - 22x x
│ │ │ │ +0002d8b0: 2020 2b20 7820 7820 202b 2031 3378 2078    + x x  + 13x x
│ │ │ │ +0002d8c0: 2020 2b20 3431 7820 7820 7c0a 7c20 2020    + 41x x |.|   
│ │ │ │ +0002d8d0: 2020 2020 2020 2020 2020 2020 2031 2033               1 3
│ │ │ │ +0002d8e0: 2020 2020 2032 2033 2020 2020 2020 3320       2 3      3 
│ │ │ │ +0002d8f0: 2020 2020 2032 2034 2020 2020 2020 3320       2 4      3 
│ │ │ │ +0002d900: 3420 2020 2030 2035 2020 2020 2020 3220  4    0 5      2 
│ │ │ │ +0002d910: 3520 2020 2020 2033 2035 7c0a 7c20 2020  5      3 5|.|   
│ │ │ │  0002d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d960: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002d970: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0002d980: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0002d960: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002d970: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0002d980: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  0002d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002d9c0: 2020 2020 2020 7820 202b 2031 3778 2078        x  + 17x x
│ │ │ │ -0002d9d0: 2020 2d20 3134 7820 202d 2031 3378 2078    - 14x  - 13x x
│ │ │ │ -0002d9e0: 2020 2b20 3334 7820 7820 202b 2034 3478    + 34x x  + 44x
│ │ │ │ -0002d9f0: 2078 2020 2d20 3330 7820 7820 202b 2032   x  - 30x x  + 2
│ │ │ │ -0002da00: 3778 2078 7c0a 7c20 2020 2020 2020 2020  7x x|.|         
│ │ │ │ -0002da10: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ -0002da20: 3320 2020 2020 2033 2020 2020 2020 3220  3      3      2 
│ │ │ │ -0002da30: 3420 2020 2020 2033 2034 2020 2020 2020  4      3 4      
│ │ │ │ -0002da40: 3020 3520 2020 2020 2032 2035 2020 2020  0 5      2 5    
│ │ │ │ -0002da50: 2020 3320 7c0a 7c20 2020 2020 2020 2020    3 |.|         
│ │ │ │ +0002d9b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002d9c0: 2020 2020 2020 2020 2020 2020 7820 202b              x  +
│ │ │ │ +0002d9d0: 2031 3778 2078 2020 2d20 3134 7820 202d   17x x  - 14x  -
│ │ │ │ +0002d9e0: 2031 3378 2078 2020 2b20 3334 7820 7820   13x x  + 34x x 
│ │ │ │ +0002d9f0: 202b 2034 3478 2078 2020 2d20 3330 7820   + 44x x  - 30x 
│ │ │ │ +0002da00: 7820 202b 2032 3778 2078 7c0a 7c20 2020  x  + 27x x|.|   
│ │ │ │ +0002da10: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0002da20: 2020 2020 3220 3320 2020 2020 2033 2020      2 3      3  
│ │ │ │ +0002da30: 2020 2020 3220 3420 2020 2020 2033 2034      2 4      3 4
│ │ │ │ +0002da40: 2020 2020 2020 3020 3520 2020 2020 2032        0 5      2
│ │ │ │ +0002da50: 2035 2020 2020 2020 3320 7c0a 7c20 2020   5      3 |.|   
│ │ │ │  0002da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002daa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002daa0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dac0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0002dac0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002daf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002db00: 2020 2020 2020 7820 7820 202d 2034 3078        x x  - 40x
│ │ │ │ -0002db10: 2078 2020 2b20 3238 7820 202d 2078 2078   x  + 28x  - x x
│ │ │ │ -0002db20: 2020 2b20 3578 2078 2020 2d20 3136 7820    + 5x x  - 16x 
│ │ │ │ -0002db30: 7820 202b 2035 7820 7820 202d 2033 3678  x  + 5x x  - 36x
│ │ │ │ -0002db40: 2078 2020 7c0a 7c20 2020 2020 2020 2020   x  |.|         
│ │ │ │ -0002db50: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ -0002db60: 3220 3320 2020 2020 2033 2020 2020 3020  2 3      3    0 
│ │ │ │ -0002db70: 3420 2020 2020 3220 3420 2020 2020 2033  4     2 4      3
│ │ │ │ -0002db80: 2034 2020 2020 2030 2035 2020 2020 2020   4     0 5      
│ │ │ │ -0002db90: 3220 3520 7c0a 7c20 2020 2020 2020 2020  2 5 |.|         
│ │ │ │ -0002dba0: 2020 2020 207d 2020 2020 2020 2020 2020       }          
│ │ │ │ +0002daf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002db00: 2020 2020 2020 2020 2020 2020 7820 7820              x x 
│ │ │ │ +0002db10: 202d 2034 3078 2078 2020 2b20 3238 7820   - 40x x  + 28x 
│ │ │ │ +0002db20: 202d 2078 2078 2020 2b20 3578 2078 2020   - x x  + 5x x  
│ │ │ │ +0002db30: 2d20 3136 7820 7820 202b 2035 7820 7820  - 16x x  + 5x x 
│ │ │ │ +0002db40: 202d 2033 3678 2078 2020 7c0a 7c20 2020   - 36x x  |.|   
│ │ │ │ +0002db50: 2020 2020 2020 2020 2020 2020 2031 2032               1 2
│ │ │ │ +0002db60: 2020 2020 2020 3220 3320 2020 2020 2033        2 3      3
│ │ │ │ +0002db70: 2020 2020 3020 3420 2020 2020 3220 3420      0 4     2 4 
│ │ │ │ +0002db80: 2020 2020 2033 2034 2020 2020 2030 2035       3 4     0 5
│ │ │ │ +0002db90: 2020 2020 2020 3220 3520 7c0a 7c20 2020        2 5 |.|   
│ │ │ │ +0002dba0: 2020 2020 2020 2020 2020 207d 2020 2020             }    
│ │ │ │  0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002dbf0: 2020 2020 2020 2020 2020 5a5a 2020 2020            ZZ    
│ │ │ │ -0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dbe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc00: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │  0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc30: 2020 2020 7c0a 7c20 2020 2020 2074 6172      |.|      tar
│ │ │ │ -0002dc40: 6765 743a 2050 726f 6a28 2d2d 5b74 202c  get: Proj(--[t ,
│ │ │ │ -0002dc50: 2074 202c 2074 202c 2074 202c 2074 202c   t , t , t , t ,
│ │ │ │ -0002dc60: 2074 202c 2074 202c 2074 202c 2074 205d   t , t , t , t ]
│ │ │ │ -0002dc70: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0002dc80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002dc90: 2020 2020 2020 2020 2020 3937 2020 3020            97  0 
│ │ │ │ -0002dca0: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ -0002dcb0: 2020 3520 2020 3620 2020 3720 2020 3820    5   6   7   8 
│ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dcd0: 2020 2020 7c0a 7c20 2020 2020 2064 6566      |.|      def
│ │ │ │ -0002dce0: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ -0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002dc40: 2020 2074 6172 6765 743a 2050 726f 6a28     target: Proj(
│ │ │ │ +0002dc50: 2d2d 5b74 202c 2074 202c 2074 202c 2074  --[t , t , t , t
│ │ │ │ +0002dc60: 202c 2074 202c 2074 202c 2074 202c 2074   , t , t , t , t
│ │ │ │ +0002dc70: 202c 2074 205d 2920 2020 2020 2020 2020   , t ])         
│ │ │ │ +0002dc80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dca0: 3937 2020 3020 2020 3120 2020 3220 2020  97  0   1   2   
│ │ │ │ +0002dcb0: 3320 2020 3420 2020 3520 2020 3620 2020  3   4   5   6   
│ │ │ │ +0002dcc0: 3720 2020 3820 2020 2020 2020 2020 2020  7   8           
│ │ │ │ +0002dcd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002dce0: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
│ │ │ │ +0002dcf0: 733a 207b 2020 2020 2020 2020 2020 2020  s: {            
│ │ │ │  0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002dd20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd40: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0002dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd50: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002dd80: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0002dd90: 3278 2078 2020 2b20 3678 2020 2d20 3234  2x x  + 6x  - 24
│ │ │ │ -0002dda0: 7820 7820 202d 2032 3578 2078 2020 2b20  x x  - 25x x  + 
│ │ │ │ -0002ddb0: 3339 7820 7820 202b 2033 7820 7820 202b  39x x  + 3x x  +
│ │ │ │ -0002ddc0: 2034 3678 7c0a 7c20 2020 2020 2020 2020   46x|.|         
│ │ │ │ +0002dd70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd90: 2020 2020 2d20 3278 2078 2020 2b20 3678      - 2x x  + 6x
│ │ │ │ +0002dda0: 2020 2d20 3234 7820 7820 202d 2032 3578    - 24x x  - 25x
│ │ │ │ +0002ddb0: 2078 2020 2b20 3339 7820 7820 202b 2033   x  + 39x x  + 3
│ │ │ │ +0002ddc0: 7820 7820 202b 2034 3678 7c0a 7c20 2020  x x  + 46x|.|   
│ │ │ │  0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dde0: 2020 3220 3320 2020 2020 3320 2020 2020    2 3     3     
│ │ │ │ -0002ddf0: 2032 2034 2020 2020 2020 3320 3420 2020   2 4      3 4   
│ │ │ │ -0002de00: 2020 2030 2035 2020 2020 2032 2035 2020     0 5     2 5  
│ │ │ │ -0002de10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002dde0: 2020 2020 2020 2020 3220 3320 2020 2020          2 3     
│ │ │ │ +0002ddf0: 3320 2020 2020 2032 2034 2020 2020 2020  3      2 4      
│ │ │ │ +0002de00: 3320 3420 2020 2020 2030 2035 2020 2020  3 4      0 5    
│ │ │ │ +0002de10: 2032 2035 2020 2020 2020 7c0a 7c20 2020   2 5      |.|   
│ │ │ │  0002de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002de70: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0002de80: 3139 7820 7820 202d 2034 3078 2078 2020  19x x  - 40x x  
│ │ │ │ -0002de90: 2b20 3139 7820 7820 202b 2031 3978 2078  + 19x x  + 19x x
│ │ │ │ -0002dea0: 2020 2d20 3338 7820 7820 202d 2033 3978    - 38x x  - 39x
│ │ │ │ -0002deb0: 2078 2020 7c0a 7c20 2020 2020 2020 2020   x  |.|         
│ │ │ │ +0002de60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002de80: 2020 2020 2d20 3139 7820 7820 202d 2034      - 19x x  - 4
│ │ │ │ +0002de90: 3078 2078 2020 2b20 3139 7820 7820 202b  0x x  + 19x x  +
│ │ │ │ +0002dea0: 2031 3978 2078 2020 2d20 3338 7820 7820   19x x  - 38x x 
│ │ │ │ +0002deb0: 202d 2033 3978 2078 2020 7c0a 7c20 2020   - 39x x  |.|   
│ │ │ │  0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ded0: 2020 2032 2035 2020 2020 2020 3320 3520     2 5      3 5 
│ │ │ │ -0002dee0: 2020 2020 2031 2038 2020 2020 2020 3220       1 8      2 
│ │ │ │ -0002def0: 3820 2020 2020 2033 2038 2020 2020 2020  8      3 8      
│ │ │ │ -0002df00: 3520 3820 7c0a 7c20 2020 2020 2020 2020  5 8 |.|         
│ │ │ │ +0002ded0: 2020 2020 2020 2020 2032 2035 2020 2020           2 5    
│ │ │ │ +0002dee0: 2020 3320 3520 2020 2020 2031 2038 2020    3 5      1 8  
│ │ │ │ +0002def0: 2020 2020 3220 3820 2020 2020 2033 2038      2 8      3 8
│ │ │ │ +0002df00: 2020 2020 2020 3520 3820 7c0a 7c20 2020        5 8 |.|   
│ │ │ │  0002df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002df60: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0002df70: 3339 7820 7820 202d 2033 3978 2078 2020  39x x  - 39x x  
│ │ │ │ -0002df80: 2d20 3338 7820 7820 202d 2033 3978 2078  - 38x x  - 39x x
│ │ │ │ -0002df90: 2020 2b20 3139 7820 7820 202b 2032 7820    + 19x x  + 2x 
│ │ │ │ -0002dfa0: 7820 202b 7c0a 7c20 2020 2020 2020 2020  x  +|.|         
│ │ │ │ +0002df50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002df70: 2020 2020 2d20 3339 7820 7820 202d 2033      - 39x x  - 3
│ │ │ │ +0002df80: 3978 2078 2020 2d20 3338 7820 7820 202d  9x x  - 38x x  -
│ │ │ │ +0002df90: 2033 3978 2078 2020 2b20 3139 7820 7820   39x x  + 19x x 
│ │ │ │ +0002dfa0: 202b 2032 7820 7820 202b 7c0a 7c20 2020   + 2x x  +|.|   
│ │ │ │  0002dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfc0: 2020 2032 2033 2020 2020 2020 3020 3420     2 3      0 4 
│ │ │ │ -0002dfd0: 2020 2020 2033 2034 2020 2020 2020 3020       3 4      0 
│ │ │ │ -0002dfe0: 3520 2020 2020 2032 2035 2020 2020 2033  5      2 5     3
│ │ │ │ -0002dff0: 2035 2020 7c0a 7c20 2020 2020 2020 2020   5  |.|         
│ │ │ │ +0002dfc0: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ +0002dfd0: 2020 3020 3420 2020 2020 2033 2034 2020    0 4      3 4  
│ │ │ │ +0002dfe0: 2020 2020 3020 3520 2020 2020 2032 2035      0 5      2 5
│ │ │ │ +0002dff0: 2020 2020 2033 2035 2020 7c0a 7c20 2020       3 5  |.|   
│ │ │ │  0002e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e010: 2020 2020 2020 2020 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e040: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e060: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0002e070: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0002e060: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0002e070: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0002e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e090: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0002e0b0: 3339 7820 202d 2038 7820 7820 202b 2032  39x  - 8x x  + 2
│ │ │ │ -0002e0c0: 3578 2020 2b20 3139 7820 7820 202b 2034  5x  + 19x x  + 4
│ │ │ │ -0002e0d0: 3678 2078 2020 2d20 3232 7820 7820 202b  6x x  - 22x x  +
│ │ │ │ -0002e0e0: 2078 2078 7c0a 7c20 2020 2020 2020 2020   x x|.|         
│ │ │ │ +0002e090: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e0b0: 2020 2020 2d20 3339 7820 202d 2038 7820      - 39x  - 8x 
│ │ │ │ +0002e0c0: 7820 202b 2032 3578 2020 2b20 3139 7820  x  + 25x  + 19x 
│ │ │ │ +0002e0d0: 7820 202b 2034 3678 2078 2020 2d20 3232  x  + 46x x  - 22
│ │ │ │ +0002e0e0: 7820 7820 202b 2078 2078 7c0a 7c20 2020  x x  + x x|.|   
│ │ │ │  0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e100: 2020 2031 2020 2020 2032 2033 2020 2020     1     2 3    
│ │ │ │ -0002e110: 2020 3320 2020 2020 2030 2034 2020 2020    3      0 4    
│ │ │ │ -0002e120: 2020 3220 3420 2020 2020 2033 2034 2020    2 4      3 4  
│ │ │ │ -0002e130: 2020 3020 7c0a 7c20 2020 2020 2020 2020    0 |.|         
│ │ │ │ +0002e100: 2020 2020 2020 2020 2031 2020 2020 2032           1     2
│ │ │ │ +0002e110: 2033 2020 2020 2020 3320 2020 2020 2030   3      3      0
│ │ │ │ +0002e120: 2034 2020 2020 2020 3220 3420 2020 2020   4      2 4     
│ │ │ │ +0002e130: 2033 2034 2020 2020 3020 7c0a 7c20 2020   3 4    0 |.|   
│ │ │ │  0002e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e180: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e180: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e1c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002e1d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002e1e0: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0002e1f0: 3339 7820 7820 202b 2031 3978 2078 2020  39x x  + 19x x  
│ │ │ │ -0002e200: 2b20 7820 7820 202b 2038 7820 7820 202d  + x x  + 8x x  -
│ │ │ │ -0002e210: 2032 3578 2020 2d20 3233 7820 7820 202d   25x  - 23x x  -
│ │ │ │ -0002e220: 2033 3078 7c0a 7c20 2020 2020 2020 2020   30x|.|         
│ │ │ │ +0002e1c0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0002e1d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e1f0: 2020 2020 2d20 3339 7820 7820 202b 2031      - 39x x  + 1
│ │ │ │ +0002e200: 3978 2078 2020 2b20 7820 7820 202b 2038  9x x  + x x  + 8
│ │ │ │ +0002e210: 7820 7820 202d 2032 3578 2020 2d20 3233  x x  - 25x  - 23
│ │ │ │ +0002e220: 7820 7820 202d 2033 3078 7c0a 7c20 2020  x x  - 30x|.|   
│ │ │ │  0002e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e240: 2020 2030 2031 2020 2020 2020 3020 3220     0 1      0 2 
│ │ │ │ -0002e250: 2020 2030 2033 2020 2020 2032 2033 2020     0 3     2 3  
│ │ │ │ -0002e260: 2020 2020 3320 2020 2020 2032 2034 2020      3      2 4  
│ │ │ │ -0002e270: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e240: 2020 2020 2020 2020 2030 2031 2020 2020           0 1    
│ │ │ │ +0002e250: 2020 3020 3220 2020 2030 2033 2020 2020    0 2    0 3    
│ │ │ │ +0002e260: 2032 2033 2020 2020 2020 3320 2020 2020   2 3      3     
│ │ │ │ +0002e270: 2032 2034 2020 2020 2020 7c0a 7c20 2020   2 4      |.|   
│ │ │ │  0002e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0002e2e0: 3234 7820 7820 202d 2032 3578 2078 2020  24x x  - 25x x  
│ │ │ │ -0002e2f0: 2b20 3232 7820 7820 202b 2034 3878 2078  + 22x x  + 48x x
│ │ │ │ -0002e300: 2020 2d20 3137 7820 7820 202b 2032 3478    - 17x x  + 24x
│ │ │ │ -0002e310: 2078 2020 7c0a 7c20 2020 2020 2020 2020   x  |.|         
│ │ │ │ +0002e2c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e2e0: 2020 2020 2d20 3234 7820 7820 202d 2032      - 24x x  - 2
│ │ │ │ +0002e2f0: 3578 2078 2020 2b20 3232 7820 7820 202b  5x x  + 22x x  +
│ │ │ │ +0002e300: 2034 3878 2078 2020 2d20 3137 7820 7820   48x x  - 17x x 
│ │ │ │ +0002e310: 202b 2032 3478 2078 2020 7c0a 7c20 2020   + 24x x  |.|   
│ │ │ │  0002e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e330: 2020 2032 2034 2020 2020 2020 3320 3420     2 4      3 4 
│ │ │ │ -0002e340: 2020 2020 2032 2035 2020 2020 2020 3320       2 5      3 
│ │ │ │ -0002e350: 3520 2020 2020 2032 2036 2020 2020 2020  5      2 6      
│ │ │ │ -0002e360: 3120 3720 7c0a 7c20 2020 2020 2020 2020  1 7 |.|         
│ │ │ │ +0002e330: 2020 2020 2020 2020 2032 2034 2020 2020           2 4    
│ │ │ │ +0002e340: 2020 3320 3420 2020 2020 2032 2035 2020    3 4      2 5  
│ │ │ │ +0002e350: 2020 2020 3320 3520 2020 2020 2032 2036      3 5      2 6
│ │ │ │ +0002e360: 2020 2020 2020 3120 3720 7c0a 7c20 2020        1 7 |.|   
│ │ │ │  0002e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002e3c0: 2020 2020 2020 2020 2020 2020 2020 3139                19
│ │ │ │ -0002e3d0: 7820 7820 202b 2078 2078 2020 2b20 3139  x x  + x x  + 19
│ │ │ │ -0002e3e0: 7820 7820 202b 2033 3978 2078 2020 2d20  x x  + 39x x  - 
│ │ │ │ -0002e3f0: 3139 7820 7820 202b 2033 3778 2078 2020  19x x  + 37x x  
│ │ │ │ -0002e400: 2d20 3336 7c0a 7c20 2020 2020 2020 2020  - 36|.|         
│ │ │ │ +0002e3b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e3d0: 2020 2020 3139 7820 7820 202b 2078 2078      19x x  + x x
│ │ │ │ +0002e3e0: 2020 2b20 3139 7820 7820 202b 2033 3978    + 19x x  + 39x
│ │ │ │ +0002e3f0: 2078 2020 2d20 3139 7820 7820 202b 2033   x  - 19x x  + 3
│ │ │ │ +0002e400: 3778 2078 2020 2d20 3336 7c0a 7c20 2020  7x x  - 36|.|   
│ │ │ │  0002e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e420: 2032 2035 2020 2020 3320 3520 2020 2020   2 5    3 5     
│ │ │ │ -0002e430: 2035 2037 2020 2020 2020 3120 3820 2020   5 7      1 8   
│ │ │ │ -0002e440: 2020 2034 2038 2020 2020 2020 3620 3820     4 8      6 8 
│ │ │ │ -0002e450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e420: 2020 2020 2020 2032 2035 2020 2020 3320         2 5    3 
│ │ │ │ +0002e430: 3520 2020 2020 2035 2037 2020 2020 2020  5      5 7      
│ │ │ │ +0002e440: 3120 3820 2020 2020 2034 2038 2020 2020  1 8      4 8    
│ │ │ │ +0002e450: 2020 3620 3820 2020 2020 7c0a 7c20 2020    6 8     |.|   
│ │ │ │  0002e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002e4b0: 2020 2020 2020 2020 2020 2020 2020 3339                39
│ │ │ │ -0002e4c0: 7820 7820 202d 2031 3978 2078 2020 2d20  x x  - 19x x  - 
│ │ │ │ -0002e4d0: 3278 2078 2020 2b20 3339 7820 7820 202b  2x x  + 39x x  +
│ │ │ │ -0002e4e0: 2031 3978 2078 2020 2d20 7820 7820 202d   19x x  - x x  -
│ │ │ │ -0002e4f0: 2033 3778 7c0a 7c20 2020 2020 2020 2020   37x|.|         
│ │ │ │ +0002e4a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e4c0: 2020 2020 3339 7820 7820 202d 2031 3978      39x x  - 19x
│ │ │ │ +0002e4d0: 2078 2020 2d20 3278 2078 2020 2b20 3339   x  - 2x x  + 39
│ │ │ │ +0002e4e0: 7820 7820 202b 2031 3978 2078 2020 2d20  x x  + 19x x  - 
│ │ │ │ +0002e4f0: 7820 7820 202d 2033 3778 7c0a 7c20 2020  x x  - 37x|.|   
│ │ │ │  0002e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e510: 2030 2035 2020 2020 2020 3220 3520 2020   0 5      2 5   
│ │ │ │ -0002e520: 2020 3320 3520 2020 2020 2030 2038 2020    3 5      0 8  
│ │ │ │ -0002e530: 2020 2020 3120 3820 2020 2033 2038 2020      1 8    3 8  
│ │ │ │ -0002e540: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e510: 2020 2020 2020 2030 2035 2020 2020 2020         0 5      
│ │ │ │ +0002e520: 3220 3520 2020 2020 3320 3520 2020 2020  2 5     3 5     
│ │ │ │ +0002e530: 2030 2038 2020 2020 2020 3120 3820 2020   0 8      1 8   
│ │ │ │ +0002e540: 2033 2038 2020 2020 2020 7c0a 7c20 2020   3 8      |.|   
│ │ │ │  0002e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e590: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002e5a0: 2020 2020 2020 2020 2020 2020 2020 3433                43
│ │ │ │ -0002e5b0: 7820 7820 202d 2033 3078 2078 2020 2d20  x x  - 30x x  - 
│ │ │ │ -0002e5c0: 3139 7820 7820 202d 2034 3078 2078 2020  19x x  - 40x x  
│ │ │ │ -0002e5d0: 2b20 3434 7820 7820 202d 2032 3278 2078  + 44x x  - 22x x
│ │ │ │ -0002e5e0: 2020 2d20 7c0a 7c20 2020 2020 2020 2020    - |.|         
│ │ │ │ +0002e590: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e5b0: 2020 2020 3433 7820 7820 202d 2033 3078      43x x  - 30x
│ │ │ │ +0002e5c0: 2078 2020 2d20 3139 7820 7820 202d 2034   x  - 19x x  - 4
│ │ │ │ +0002e5d0: 3078 2078 2020 2b20 3434 7820 7820 202d  0x x  + 44x x  -
│ │ │ │ +0002e5e0: 2032 3278 2078 2020 2d20 7c0a 7c20 2020   22x x  - |.|   
│ │ │ │  0002e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e600: 2032 2034 2020 2020 2020 3320 3420 2020   2 4      3 4   
│ │ │ │ -0002e610: 2020 2030 2035 2020 2020 2020 3220 3520     0 5      2 5 
│ │ │ │ -0002e620: 2020 2020 2033 2035 2020 2020 2020 3220       3 5      2 
│ │ │ │ -0002e630: 3620 2020 7c0a 7c20 2020 2020 2020 2020  6   |.|         
│ │ │ │ -0002e640: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ -0002e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e600: 2020 2020 2020 2032 2034 2020 2020 2020         2 4      
│ │ │ │ +0002e610: 3320 3420 2020 2020 2030 2035 2020 2020  3 4      0 5    
│ │ │ │ +0002e620: 2020 3220 3520 2020 2020 2033 2035 2020    2 5      3 5  
│ │ │ │ +0002e630: 2020 2020 3220 3620 2020 7c0a 7c20 2020      2 6   |.|   
│ │ │ │ +0002e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e650: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  0002e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e680: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e680: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6d0: 2020 2020 7c0a 7c6f 3130 203a 2052 6174      |.|o10 : Rat
│ │ │ │ -0002e6e0: 696f 6e61 6c4d 6170 2028 7175 6164 7261  ionalMap (quadra
│ │ │ │ -0002e6f0: 7469 6320 6269 7261 7469 6f6e 616c 206d  tic birational m
│ │ │ │ -0002e700: 6170 2066 726f 6d20 382d 6469 6d65 6e73  ap from 8-dimens
│ │ │ │ -0002e710: 696f 6e61 6c20 7375 6276 6172 6965 7479  ional subvariety
│ │ │ │ -0002e720: 206f 6620 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d   of |.|---------
│ │ │ │ +0002e6d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ +0002e6e0: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +0002e6f0: 7175 6164 7261 7469 6320 6269 7261 7469  quadratic birati
│ │ │ │ +0002e700: 6f6e 616c 206d 6170 2066 726f 6d20 382d  onal map from 8-
│ │ │ │ +0002e710: 6469 6d65 6e73 696f 6e61 6c20 7375 6276  dimensional subv
│ │ │ │ +0002e720: 6172 6965 7479 206f 6620 7c0a 7c2d 2d2d  ariety of |.|---
│ │ │ │  0002e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e770: 2d2d 2d2d 7c0a 7c20 202c 2078 2020 5d29  ----|.|  , x  ])
│ │ │ │ -0002e780: 2064 6566 696e 6564 2062 7920 2020 2020   defined by     
│ │ │ │ -0002e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e770: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 202c  ----------|.|  ,
│ │ │ │ +0002e780: 2078 2020 5d29 2064 6566 696e 6564 2062   x  ]) defined b
│ │ │ │ +0002e790: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  0002e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7c0: 2020 2020 7c0a 7c31 3020 2020 3131 2020      |.|10   11  
│ │ │ │ -0002e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e7c0: 2020 2020 2020 2020 2020 7c0a 7c31 3020            |.|10 
│ │ │ │ +0002e7d0: 2020 3131 2020 2020 2020 2020 2020 2020    11            
│ │ │ │  0002e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e810: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e810: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e860: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e860: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e8b0: 2020 2020 7c0a 7c20 2d20 7820 7820 202b      |.| - x x  +
│ │ │ │ -0002e8c0: 2031 3278 2078 2020 2b20 3235 7820 7820   12x x  + 25x x 
│ │ │ │ -0002e8d0: 202b 2032 3578 2078 2020 2b20 3233 7820   + 25x x  + 23x 
│ │ │ │ -0002e8e0: 7820 202d 2033 7820 7820 202b 2032 7820  x  - 3x x  + 2x 
│ │ │ │ -0002e8f0: 7820 202b 2031 3178 2078 2020 2d20 3337  x  + 11x x  - 37
│ │ │ │ -0002e900: 7820 2020 7c0a 7c20 2020 2030 2036 2020  x   |.|    0 6  
│ │ │ │ -0002e910: 2020 2020 3220 3620 2020 2020 2031 2037      2 6      1 7
│ │ │ │ -0002e920: 2020 2020 2020 3320 3720 2020 2020 2035        3 7      5
│ │ │ │ -0002e930: 2037 2020 2020 2036 2037 2020 2020 2030   7     6 7     0
│ │ │ │ -0002e940: 2038 2020 2020 2020 3120 3820 2020 2020   8      1 8     
│ │ │ │ -0002e950: 2033 2020 7c0a 7c20 2020 2020 2020 2020   3  |.|         
│ │ │ │ +0002e8b0: 2020 2020 2020 2020 2020 7c0a 7c20 2d20            |.| - 
│ │ │ │ +0002e8c0: 7820 7820 202b 2031 3278 2078 2020 2b20  x x  + 12x x  + 
│ │ │ │ +0002e8d0: 3235 7820 7820 202b 2032 3578 2078 2020  25x x  + 25x x  
│ │ │ │ +0002e8e0: 2b20 3233 7820 7820 202d 2033 7820 7820  + 23x x  - 3x x 
│ │ │ │ +0002e8f0: 202b 2032 7820 7820 202b 2031 3178 2078   + 2x x  + 11x x
│ │ │ │ +0002e900: 2020 2d20 3337 7820 2020 7c0a 7c20 2020    - 37x   |.|   
│ │ │ │ +0002e910: 2030 2036 2020 2020 2020 3220 3620 2020   0 6      2 6   
│ │ │ │ +0002e920: 2020 2031 2037 2020 2020 2020 3320 3720     1 7      3 7 
│ │ │ │ +0002e930: 2020 2020 2035 2037 2020 2020 2036 2037       5 7     6 7
│ │ │ │ +0002e940: 2020 2020 2030 2038 2020 2020 2020 3120       0 8      1 
│ │ │ │ +0002e950: 3820 2020 2020 2033 2020 7c0a 7c20 2020  8      3  |.|   
│ │ │ │  0002e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e9a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9f0: 2020 2020 7c0a 7c20 202b 2033 3178 2078      |.|  + 31x x
│ │ │ │ -0002ea00: 2020 2d20 3336 7820 7820 202d 2078 2078    - 36x x  - x x
│ │ │ │ -0002ea10: 2020 2b20 3133 7820 7820 202b 2038 7820    + 13x x  + 8x 
│ │ │ │ -0002ea20: 7820 202b 2039 7820 7820 202b 2034 3678  x  + 9x x  + 46x
│ │ │ │ -0002ea30: 2078 2020 2b20 3431 7820 7820 202d 2037   x  + 41x x  - 7
│ │ │ │ -0002ea40: 7820 2020 7c0a 7c35 2020 2020 2020 3220  x   |.|5      2 
│ │ │ │ -0002ea50: 3620 2020 2020 2033 2036 2020 2020 3020  6      3 6    0 
│ │ │ │ -0002ea60: 3720 2020 2020 2031 2037 2020 2020 2033  7      1 7     3
│ │ │ │ -0002ea70: 2037 2020 2020 2035 2037 2020 2020 2020   7     5 7      
│ │ │ │ -0002ea80: 3620 3720 2020 2020 2030 2038 2020 2020  6 7      0 8    
│ │ │ │ -0002ea90: 2031 2020 7c0a 7c20 2020 2020 2020 2020   1  |.|         
│ │ │ │ +0002e9f0: 2020 2020 2020 2020 2020 7c0a 7c20 202b            |.|  +
│ │ │ │ +0002ea00: 2033 3178 2078 2020 2d20 3336 7820 7820   31x x  - 36x x 
│ │ │ │ +0002ea10: 202d 2078 2078 2020 2b20 3133 7820 7820   - x x  + 13x x 
│ │ │ │ +0002ea20: 202b 2038 7820 7820 202b 2039 7820 7820   + 8x x  + 9x x 
│ │ │ │ +0002ea30: 202b 2034 3678 2078 2020 2b20 3431 7820   + 46x x  + 41x 
│ │ │ │ +0002ea40: 7820 202d 2037 7820 2020 7c0a 7c35 2020  x  - 7x   |.|5  
│ │ │ │ +0002ea50: 2020 2020 3220 3620 2020 2020 2033 2036      2 6      3 6
│ │ │ │ +0002ea60: 2020 2020 3020 3720 2020 2020 2031 2037      0 7      1 7
│ │ │ │ +0002ea70: 2020 2020 2033 2037 2020 2020 2035 2037       3 7     5 7
│ │ │ │ +0002ea80: 2020 2020 2020 3620 3720 2020 2020 2030        6 7      0
│ │ │ │ +0002ea90: 2038 2020 2020 2031 2020 7c0a 7c20 2020   8     1  |.|   
│ │ │ │  0002eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002eae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eb30: 2020 2020 7c0a 7c2b 2033 3778 2078 2020      |.|+ 37x x  
│ │ │ │ -0002eb40: 2b20 3438 7820 7820 202d 2035 7820 7820  + 48x x  - 5x x 
│ │ │ │ -0002eb50: 202d 2035 7820 7820 202b 2078 2078 2020   - 5x x  + x x  
│ │ │ │ -0002eb60: 2b20 3230 7820 7820 202b 2031 3078 2078  + 20x x  + 10x x
│ │ │ │ -0002eb70: 2020 2b20 3334 7820 7820 202b 2034 3178    + 34x x  + 41x
│ │ │ │ -0002eb80: 2078 2020 7c0a 7c20 2020 2020 3320 3520   x  |.|     3 5 
│ │ │ │ -0002eb90: 2020 2020 2032 2036 2020 2020 2031 2037       2 6     1 7
│ │ │ │ -0002eba0: 2020 2020 2033 2037 2020 2020 3520 3720       3 7    5 7 
│ │ │ │ -0002ebb0: 2020 2020 2036 2037 2020 2020 2020 3020       6 7      0 
│ │ │ │ -0002ebc0: 3820 2020 2020 2031 2038 2020 2020 2020  8      1 8      
│ │ │ │ -0002ebd0: 3320 2020 7c0a 7c20 2020 2020 2020 2020  3   |.|         
│ │ │ │ +0002eb30: 2020 2020 2020 2020 2020 7c0a 7c2b 2033            |.|+ 3
│ │ │ │ +0002eb40: 3778 2078 2020 2b20 3438 7820 7820 202d  7x x  + 48x x  -
│ │ │ │ +0002eb50: 2035 7820 7820 202d 2035 7820 7820 202b   5x x  - 5x x  +
│ │ │ │ +0002eb60: 2078 2078 2020 2b20 3230 7820 7820 202b   x x  + 20x x  +
│ │ │ │ +0002eb70: 2031 3078 2078 2020 2b20 3334 7820 7820   10x x  + 34x x 
│ │ │ │ +0002eb80: 202b 2034 3178 2078 2020 7c0a 7c20 2020   + 41x x  |.|   
│ │ │ │ +0002eb90: 2020 3320 3520 2020 2020 2032 2036 2020    3 5      2 6  
│ │ │ │ +0002eba0: 2020 2031 2037 2020 2020 2033 2037 2020     1 7     3 7  
│ │ │ │ +0002ebb0: 2020 3520 3720 2020 2020 2036 2037 2020    5 7      6 7  
│ │ │ │ +0002ebc0: 2020 2020 3020 3820 2020 2020 2031 2038      0 8      1 8
│ │ │ │ +0002ebd0: 2020 2020 2020 3320 2020 7c0a 7c20 2020        3   |.|   
│ │ │ │  0002ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ec20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ec70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ecc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ecc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ed10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ed60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002edb0: 2020 2020 7c0a 7c20 7820 202d 2033 3978      |.| x  - 39x
│ │ │ │ -0002edc0: 2078 2020 2d20 3137 7820 7820 202d 2033   x  - 17x x  - 3
│ │ │ │ -0002edd0: 3878 2078 2020 2b20 3234 7820 7820 202b  8x x  + 24x x  +
│ │ │ │ -0002ede0: 2032 3478 2078 2020 2b20 3578 2078 2020   24x x  + 5x x  
│ │ │ │ -0002edf0: 2b20 7820 7820 202d 2031 3978 2078 2020  + x x  - 19x x  
│ │ │ │ -0002ee00: 2d20 2020 7c0a 7c33 2035 2020 2020 2020  -   |.|3 5      
│ │ │ │ -0002ee10: 3020 3620 2020 2020 2032 2036 2020 2020  0 6      2 6    
│ │ │ │ -0002ee20: 2020 3320 3620 2020 2020 2031 2037 2020    3 6      1 7  
│ │ │ │ -0002ee30: 2020 2020 3320 3720 2020 2020 3520 3720      3 7     5 7 
│ │ │ │ -0002ee40: 2020 2036 2037 2020 2020 2020 3020 3820     6 7      0 8 
│ │ │ │ -0002ee50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002edb0: 2020 2020 2020 2020 2020 7c0a 7c20 7820            |.| x 
│ │ │ │ +0002edc0: 202d 2033 3978 2078 2020 2d20 3137 7820   - 39x x  - 17x 
│ │ │ │ +0002edd0: 7820 202d 2033 3878 2078 2020 2b20 3234  x  - 38x x  + 24
│ │ │ │ +0002ede0: 7820 7820 202b 2032 3478 2078 2020 2b20  x x  + 24x x  + 
│ │ │ │ +0002edf0: 3578 2078 2020 2b20 7820 7820 202d 2031  5x x  + x x  - 1
│ │ │ │ +0002ee00: 3978 2078 2020 2d20 2020 7c0a 7c33 2035  9x x  -   |.|3 5
│ │ │ │ +0002ee10: 2020 2020 2020 3020 3620 2020 2020 2032        0 6      2
│ │ │ │ +0002ee20: 2036 2020 2020 2020 3320 3620 2020 2020   6      3 6     
│ │ │ │ +0002ee30: 2031 2037 2020 2020 2020 3320 3720 2020   1 7      3 7   
│ │ │ │ +0002ee40: 2020 3520 3720 2020 2036 2037 2020 2020    5 7    6 7    
│ │ │ │ +0002ee50: 2020 3020 3820 2020 2020 7c0a 7c20 2020    0 8     |.|   
│ │ │ │  0002ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eea0: 2020 2020 7c0a 7c2d 2033 3778 2078 2020      |.|- 37x x  
│ │ │ │ -0002eeb0: 2d20 3231 7820 7820 202b 2078 2078 2020  - 21x x  + x x  
│ │ │ │ -0002eec0: 202b 2032 7820 7820 2020 2d20 3578 2078   + 2x x   - 5x x
│ │ │ │ -0002eed0: 2020 202b 2078 2078 2020 202d 2035 7820     + x x   - 5x 
│ │ │ │ -0002eee0: 7820 2020 2b20 3578 2078 2020 2c20 2020  x   + 5x x  ,   
│ │ │ │ -0002eef0: 2020 2020 7c0a 7c20 2020 2020 3620 3820      |.|     6 8 
│ │ │ │ -0002ef00: 2020 2020 2035 2039 2020 2020 3120 3130       5 9    1 10
│ │ │ │ -0002ef10: 2020 2020 2032 2031 3020 2020 2020 3320       2 10     3 
│ │ │ │ -0002ef20: 3130 2020 2020 3520 3130 2020 2020 2036  10    5 10     6
│ │ │ │ -0002ef30: 2031 3020 2020 2020 3520 3131 2020 2020   10     5 11    
│ │ │ │ -0002ef40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002eea0: 2020 2020 2020 2020 2020 7c0a 7c2d 2033            |.|- 3
│ │ │ │ +0002eeb0: 3778 2078 2020 2d20 3231 7820 7820 202b  7x x  - 21x x  +
│ │ │ │ +0002eec0: 2078 2078 2020 202b 2032 7820 7820 2020   x x   + 2x x   
│ │ │ │ +0002eed0: 2d20 3578 2078 2020 202b 2078 2078 2020  - 5x x   + x x  
│ │ │ │ +0002eee0: 202d 2035 7820 7820 2020 2b20 3578 2078   - 5x x   + 5x x
│ │ │ │ +0002eef0: 2020 2c20 2020 2020 2020 7c0a 7c20 2020    ,       |.|   
│ │ │ │ +0002ef00: 2020 3620 3820 2020 2020 2035 2039 2020    6 8      5 9  
│ │ │ │ +0002ef10: 2020 3120 3130 2020 2020 2032 2031 3020    1 10     2 10 
│ │ │ │ +0002ef20: 2020 2020 3320 3130 2020 2020 3520 3130      3 10    5 10
│ │ │ │ +0002ef30: 2020 2020 2036 2031 3020 2020 2020 3520       6 10     5 
│ │ │ │ +0002ef40: 3131 2020 2020 2020 2020 7c0a 7c20 2020  11        |.|   
│ │ │ │  0002ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ef90: 2020 2020 7c0a 7c20 3139 7820 7820 202b      |.| 19x x  +
│ │ │ │ -0002efa0: 2031 3978 2078 2020 2b20 3231 7820 7820   19x x  + 21x x 
│ │ │ │ -0002efb0: 202b 2031 3978 2078 2020 2d20 3139 7820   + 19x x  - 19x 
│ │ │ │ -0002efc0: 7820 202b 2078 2078 2020 2d20 3339 7820  x  + x x  - 39x 
│ │ │ │ -0002efd0: 7820 202b 2033 3778 2078 2020 2b20 3430  x  + 37x x  + 40
│ │ │ │ -0002efe0: 7820 2020 7c0a 7c20 2020 2030 2037 2020  x   |.|    0 7  
│ │ │ │ -0002eff0: 2020 2020 3220 3720 2020 2020 2033 2037      2 7      3 7
│ │ │ │ -0002f000: 2020 2020 2020 3020 3820 2020 2020 2031        0 8      1
│ │ │ │ -0002f010: 2038 2020 2020 3320 3820 2020 2020 2034   8    3 8      4
│ │ │ │ -0002f020: 2038 2020 2020 2020 3620 3820 2020 2020   8      6 8     
│ │ │ │ -0002f030: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ +0002ef90: 2020 2020 2020 2020 2020 7c0a 7c20 3139            |.| 19
│ │ │ │ +0002efa0: 7820 7820 202b 2031 3978 2078 2020 2b20  x x  + 19x x  + 
│ │ │ │ +0002efb0: 3231 7820 7820 202b 2031 3978 2078 2020  21x x  + 19x x  
│ │ │ │ +0002efc0: 2d20 3139 7820 7820 202b 2078 2078 2020  - 19x x  + x x  
│ │ │ │ +0002efd0: 2d20 3339 7820 7820 202b 2033 3778 2078  - 39x x  + 37x x
│ │ │ │ +0002efe0: 2020 2b20 3430 7820 2020 7c0a 7c20 2020    + 40x   |.|   
│ │ │ │ +0002eff0: 2030 2037 2020 2020 2020 3220 3720 2020   0 7      2 7   
│ │ │ │ +0002f000: 2020 2033 2037 2020 2020 2020 3020 3820     3 7      0 8 
│ │ │ │ +0002f010: 2020 2020 2031 2038 2020 2020 3320 3820       1 8    3 8 
│ │ │ │ +0002f020: 2020 2020 2034 2038 2020 2020 2020 3620       4 8      6 
│ │ │ │ +0002f030: 3820 2020 2020 2030 2020 7c0a 7c20 2020  8      0  |.|   
│ │ │ │  0002f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f080: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f080: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f0d0: 2020 2020 7c0a 7c20 202d 2038 7820 7820      |.|  - 8x x 
│ │ │ │ -0002f0e0: 202b 2033 3278 2078 2020 2d20 3339 7820   + 32x x  - 39x 
│ │ │ │ -0002f0f0: 7820 202b 2078 2078 2020 2d20 3338 7820  x  + x x  - 38x 
│ │ │ │ -0002f100: 7820 202d 2033 3078 2078 2020 2d20 3578  x  - 30x x  - 5x
│ │ │ │ -0002f110: 2078 2020 2b20 3339 7820 7820 202b 2078   x  + 39x x  + x
│ │ │ │ -0002f120: 2078 2020 7c0a 7c35 2020 2020 2032 2035   x  |.|5     2 5
│ │ │ │ -0002f130: 2020 2020 2020 3320 3520 2020 2020 2034        3 5      4
│ │ │ │ -0002f140: 2035 2020 2020 3020 3620 2020 2020 2031   5    0 6      1
│ │ │ │ -0002f150: 2036 2020 2020 2020 3220 3620 2020 2020   6      2 6     
│ │ │ │ -0002f160: 3320 3620 2020 2020 2034 2036 2020 2020  3 6      4 6    
│ │ │ │ -0002f170: 3520 2020 7c0a 7c20 2020 2020 2020 2020  5   |.|         
│ │ │ │ +0002f0d0: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +0002f0e0: 2038 7820 7820 202b 2033 3278 2078 2020   8x x  + 32x x  
│ │ │ │ +0002f0f0: 2d20 3339 7820 7820 202b 2078 2078 2020  - 39x x  + x x  
│ │ │ │ +0002f100: 2d20 3338 7820 7820 202d 2033 3078 2078  - 38x x  - 30x x
│ │ │ │ +0002f110: 2020 2d20 3578 2078 2020 2b20 3339 7820    - 5x x  + 39x 
│ │ │ │ +0002f120: 7820 202b 2078 2078 2020 7c0a 7c35 2020  x  + x x  |.|5  
│ │ │ │ +0002f130: 2020 2032 2035 2020 2020 2020 3320 3520     2 5      3 5 
│ │ │ │ +0002f140: 2020 2020 2034 2035 2020 2020 3020 3620       4 5    0 6 
│ │ │ │ +0002f150: 2020 2020 2031 2036 2020 2020 2020 3220       1 6      2 
│ │ │ │ +0002f160: 3620 2020 2020 3320 3620 2020 2020 2034  6     3 6      4
│ │ │ │ +0002f170: 2036 2020 2020 3520 2020 7c0a 7c20 2020   6    5   |.|   
│ │ │ │  0002f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f1c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f1c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f210: 2020 2020 7c0a 7c20 7820 202d 2032 3178      |.| x  - 21x
│ │ │ │ -0002f220: 2078 2020 2d20 3130 7820 7820 202d 2033   x  - 10x x  - 3
│ │ │ │ -0002f230: 3478 2078 2020 2d20 3338 7820 7820 202d  4x x  - 38x x  -
│ │ │ │ -0002f240: 2037 7820 7820 202b 2035 7820 7820 202b   7x x  + 5x x  +
│ │ │ │ -0002f250: 2031 3978 2078 2020 2b20 3233 7820 7820   19x x  + 23x x 
│ │ │ │ -0002f260: 202b 2020 7c0a 7c33 2034 2020 2020 2020   +  |.|3 4      
│ │ │ │ -0002f270: 3020 3520 2020 2020 2032 2035 2020 2020  0 5      2 5    
│ │ │ │ -0002f280: 2020 3320 3520 2020 2020 2030 2036 2020    3 5      0 6  
│ │ │ │ -0002f290: 2020 2032 2036 2020 2020 2033 2036 2020     2 6     3 6  
│ │ │ │ -0002f2a0: 2020 2020 3020 3720 2020 2020 2031 2037      0 7      1 7
│ │ │ │ -0002f2b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f210: 2020 2020 2020 2020 2020 7c0a 7c20 7820            |.| x 
│ │ │ │ +0002f220: 202d 2032 3178 2078 2020 2d20 3130 7820   - 21x x  - 10x 
│ │ │ │ +0002f230: 7820 202d 2033 3478 2078 2020 2d20 3338  x  - 34x x  - 38
│ │ │ │ +0002f240: 7820 7820 202d 2037 7820 7820 202b 2035  x x  - 7x x  + 5
│ │ │ │ +0002f250: 7820 7820 202b 2031 3978 2078 2020 2b20  x x  + 19x x  + 
│ │ │ │ +0002f260: 3233 7820 7820 202b 2020 7c0a 7c33 2034  23x x  +  |.|3 4
│ │ │ │ +0002f270: 2020 2020 2020 3020 3520 2020 2020 2032        0 5      2
│ │ │ │ +0002f280: 2035 2020 2020 2020 3320 3520 2020 2020   5      3 5     
│ │ │ │ +0002f290: 2030 2036 2020 2020 2032 2036 2020 2020   0 6     2 6    
│ │ │ │ +0002f2a0: 2033 2036 2020 2020 2020 3020 3720 2020   3 6      0 7   
│ │ │ │ +0002f2b0: 2020 2031 2037 2020 2020 7c0a 7c20 2020     1 7    |.|   
│ │ │ │  0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f300: 2020 2020 7c0a 7c2b 2032 3478 2078 2020      |.|+ 24x x  
│ │ │ │ -0002f310: 2b20 3578 2078 2020 2b20 7820 7820 202b  + 5x x  + x x  +
│ │ │ │ -0002f320: 2034 3178 2078 2020 2b20 3132 7820 7820   41x x  + 12x x 
│ │ │ │ -0002f330: 202d 2035 7820 7820 202d 2032 3678 2078   - 5x x  - 26x x
│ │ │ │ -0002f340: 2020 2b20 3432 7820 7820 202d 2032 3478    + 42x x  - 24x
│ │ │ │ -0002f350: 2078 2020 7c0a 7c20 2020 2020 3320 3720   x  |.|     3 7 
│ │ │ │ -0002f360: 2020 2020 3520 3720 2020 2036 2037 2020      5 7    6 7  
│ │ │ │ -0002f370: 2020 2020 3120 3820 2020 2020 2033 2038      1 8      3 8
│ │ │ │ -0002f380: 2020 2020 2034 2038 2020 2020 2020 3620       4 8      6 
│ │ │ │ -0002f390: 3820 2020 2020 2031 2039 2020 2020 2020  8      1 9      
│ │ │ │ -0002f3a0: 3220 2020 7c0a 7c20 2020 2020 2020 2020  2   |.|         
│ │ │ │ +0002f300: 2020 2020 2020 2020 2020 7c0a 7c2b 2032            |.|+ 2
│ │ │ │ +0002f310: 3478 2078 2020 2b20 3578 2078 2020 2b20  4x x  + 5x x  + 
│ │ │ │ +0002f320: 7820 7820 202b 2034 3178 2078 2020 2b20  x x  + 41x x  + 
│ │ │ │ +0002f330: 3132 7820 7820 202d 2035 7820 7820 202d  12x x  - 5x x  -
│ │ │ │ +0002f340: 2032 3678 2078 2020 2b20 3432 7820 7820   26x x  + 42x x 
│ │ │ │ +0002f350: 202d 2032 3478 2078 2020 7c0a 7c20 2020   - 24x x  |.|   
│ │ │ │ +0002f360: 2020 3320 3720 2020 2020 3520 3720 2020    3 7     5 7   
│ │ │ │ +0002f370: 2036 2037 2020 2020 2020 3120 3820 2020   6 7      1 8   
│ │ │ │ +0002f380: 2020 2033 2038 2020 2020 2034 2038 2020     3 8     4 8  
│ │ │ │ +0002f390: 2020 2020 3620 3820 2020 2020 2031 2039      6 8      1 9
│ │ │ │ +0002f3a0: 2020 2020 2020 3220 2020 7c0a 7c20 2020        2   |.|   
│ │ │ │  0002f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3f0: 2020 2020 7c0a 7c78 2078 2020 2d20 7820      |.|x x  - x 
│ │ │ │ -0002f400: 7820 2020 2b20 3578 2078 2020 202d 2035  x   + 5x x   - 5
│ │ │ │ -0002f410: 7820 7820 202c 2020 2020 2020 2020 2020  x x  ,          
│ │ │ │ +0002f3f0: 2020 2020 2020 2020 2020 7c0a 7c78 2078            |.|x x
│ │ │ │ +0002f400: 2020 2d20 7820 7820 2020 2b20 3578 2078    - x x   + 5x x
│ │ │ │ +0002f410: 2020 202d 2035 7820 7820 202c 2020 2020     - 5x x  ,    
│ │ │ │  0002f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f440: 2020 2020 7c0a 7c20 3520 3920 2020 2034      |.| 5 9    4
│ │ │ │ -0002f450: 2031 3020 2020 2020 3620 3130 2020 2020   10     6 10    
│ │ │ │ -0002f460: 2035 2031 3120 2020 2020 2020 2020 2020   5 11           
│ │ │ │ +0002f440: 2020 2020 2020 2020 2020 7c0a 7c20 3520            |.| 5 
│ │ │ │ +0002f450: 3920 2020 2034 2031 3020 2020 2020 3620  9    4 10     6 
│ │ │ │ +0002f460: 3130 2020 2020 2035 2031 3120 2020 2020  10     5 11     
│ │ │ │  0002f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f490: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f490: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f4e0: 2020 2020 7c0a 7c20 7820 202b 2033 3778      |.| x  + 37x
│ │ │ │ -0002f4f0: 2078 2020 2b20 7820 7820 2020 2b20 7820   x  + x x   + x 
│ │ │ │ -0002f500: 7820 2020 2d20 3578 2078 2020 202b 2035  x   - 5x x   + 5
│ │ │ │ -0002f510: 7820 7820 202c 2020 2020 2020 2020 2020  x x  ,          
│ │ │ │ +0002f4e0: 2020 2020 2020 2020 2020 7c0a 7c20 7820            |.| x 
│ │ │ │ +0002f4f0: 202b 2033 3778 2078 2020 2b20 7820 7820   + 37x x  + x x 
│ │ │ │ +0002f500: 2020 2b20 7820 7820 2020 2d20 3578 2078    + x x   - 5x x
│ │ │ │ +0002f510: 2020 202b 2035 7820 7820 202c 2020 2020     + 5x x  ,    
│ │ │ │  0002f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f530: 2020 2020 7c0a 7c36 2038 2020 2020 2020      |.|6 8      
│ │ │ │ -0002f540: 3520 3920 2020 2031 2031 3020 2020 2032  5 9    1 10    2
│ │ │ │ -0002f550: 2031 3020 2020 2020 3620 3130 2020 2020   10     6 10    
│ │ │ │ -0002f560: 2035 2031 3120 2020 2020 2020 2020 2020   5 11           
│ │ │ │ +0002f530: 2020 2020 2020 2020 2020 7c0a 7c36 2038            |.|6 8
│ │ │ │ +0002f540: 2020 2020 2020 3520 3920 2020 2031 2031        5 9    1 1
│ │ │ │ +0002f550: 3020 2020 2032 2031 3020 2020 2020 3620  0    2 10     6 
│ │ │ │ +0002f560: 3130 2020 2020 2035 2031 3120 2020 2020  10     5 11     
│ │ │ │  0002f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f580: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f580: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f5d0: 2020 2020 7c0a 7c35 7820 7820 202d 2034      |.|5x x  - 4
│ │ │ │ -0002f5e0: 3378 2078 2020 2d20 3232 7820 7820 202d  3x x  - 22x x  -
│ │ │ │ -0002f5f0: 2034 3378 2078 2020 2d20 3378 2078 2020   43x x  - 3x x  
│ │ │ │ -0002f600: 2d20 3139 7820 7820 202d 2034 3278 2078  - 19x x  - 42x x
│ │ │ │ -0002f610: 2020 2d20 3133 7820 7820 202b 2034 3378    - 13x x  + 43x
│ │ │ │ -0002f620: 2078 2020 7c0a 7c20 2033 2036 2020 2020   x  |.|  3 6    
│ │ │ │ -0002f630: 2020 3120 3720 2020 2020 2033 2037 2020    1 7      3 7  
│ │ │ │ -0002f640: 2020 2020 3520 3720 2020 2020 3620 3720      5 7     6 7 
│ │ │ │ -0002f650: 2020 2020 2030 2038 2020 2020 2020 3120       0 8      1 
│ │ │ │ -0002f660: 3820 2020 2020 2033 2038 2020 2020 2020  8      3 8      
│ │ │ │ -0002f670: 3420 2020 7c0a 7c20 2020 2020 2020 2020  4   |.|         
│ │ │ │ +0002f5d0: 2020 2020 2020 2020 2020 7c0a 7c35 7820            |.|5x 
│ │ │ │ +0002f5e0: 7820 202d 2034 3378 2078 2020 2d20 3232  x  - 43x x  - 22
│ │ │ │ +0002f5f0: 7820 7820 202d 2034 3378 2078 2020 2d20  x x  - 43x x  - 
│ │ │ │ +0002f600: 3378 2078 2020 2d20 3139 7820 7820 202d  3x x  - 19x x  -
│ │ │ │ +0002f610: 2034 3278 2078 2020 2d20 3133 7820 7820   42x x  - 13x x 
│ │ │ │ +0002f620: 202b 2034 3378 2078 2020 7c0a 7c20 2033   + 43x x  |.|  3
│ │ │ │ +0002f630: 2036 2020 2020 2020 3120 3720 2020 2020   6      1 7     
│ │ │ │ +0002f640: 2033 2037 2020 2020 2020 3520 3720 2020   3 7      5 7   
│ │ │ │ +0002f650: 2020 3620 3720 2020 2020 2030 2038 2020    6 7      0 8  
│ │ │ │ +0002f660: 2020 2020 3120 3820 2020 2020 2033 2038      1 8      3 8
│ │ │ │ +0002f670: 2020 2020 2020 3420 2020 7c0a 7c20 2020        4   |.|   
│ │ │ │  0002f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f6c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f6c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f710: 2020 2020 7c0a 7c50 505e 3131 2074 6f20      |.|PP^11 to 
│ │ │ │ -0002f720: 5050 5e38 2920 2020 2020 2020 2020 2020  PP^8)           
│ │ │ │ +0002f710: 2020 2020 2020 2020 2020 7c0a 7c50 505e            |.|PP^
│ │ │ │ +0002f720: 3131 2074 6f20 5050 5e38 2920 2020 2020  11 to PP^8)     
│ │ │ │  0002f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f760: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +0002f760: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  0002f770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f7b0: 2d2d 2d2d 7c0a 7c78 2020 2d20 3233 7820  ----|.|x  - 23x 
│ │ │ │ -0002f7c0: 7820 202d 2033 3378 2078 2020 2b20 3878  x  - 33x x  + 8x
│ │ │ │ -0002f7d0: 2078 2020 2b20 3130 7820 7820 202d 2032   x  + 10x x  - 2
│ │ │ │ -0002f7e0: 3578 2078 2020 2d20 3978 2078 2020 2b20  5x x  - 9x x  + 
│ │ │ │ -0002f7f0: 3378 2078 2020 2b20 3234 7820 7820 2020  3x x  + 24x x   
│ │ │ │ -0002f800: 2020 2020 7c0a 7c20 3820 2020 2020 2034      |.| 8      4
│ │ │ │ -0002f810: 2038 2020 2020 2020 3620 3820 2020 2020   8      6 8     
│ │ │ │ -0002f820: 3020 3920 2020 2020 2031 2039 2020 2020  0 9      1 9    
│ │ │ │ -0002f830: 2020 3220 3920 2020 2020 3320 3920 2020    2 9     3 9   
│ │ │ │ -0002f840: 2020 3420 3920 2020 2020 2035 2020 2020    4 9      5    
│ │ │ │ -0002f850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c78 2020  ----------|.|x  
│ │ │ │ +0002f7c0: 2d20 3233 7820 7820 202d 2033 3378 2078  - 23x x  - 33x x
│ │ │ │ +0002f7d0: 2020 2b20 3878 2078 2020 2b20 3130 7820    + 8x x  + 10x 
│ │ │ │ +0002f7e0: 7820 202d 2032 3578 2078 2020 2d20 3978  x  - 25x x  - 9x
│ │ │ │ +0002f7f0: 2078 2020 2b20 3378 2078 2020 2b20 3234   x  + 3x x  + 24
│ │ │ │ +0002f800: 7820 7820 2020 2020 2020 7c0a 7c20 3820  x x       |.| 8 
│ │ │ │ +0002f810: 2020 2020 2034 2038 2020 2020 2020 3620       4 8      6 
│ │ │ │ +0002f820: 3820 2020 2020 3020 3920 2020 2020 2031  8     0 9      1
│ │ │ │ +0002f830: 2039 2020 2020 2020 3220 3920 2020 2020   9      2 9     
│ │ │ │ +0002f840: 3320 3920 2020 2020 3420 3920 2020 2020  3 9     4 9     
│ │ │ │ +0002f850: 2035 2020 2020 2020 2020 7c0a 7c20 2020   5        |.|   
│ │ │ │  0002f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f8a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f8a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f8f0: 2020 2020 7c0a 7c78 2020 2d20 3334 7820      |.|x  - 34x 
│ │ │ │ -0002f900: 7820 202d 2039 7820 7820 202d 2034 3678  x  - 9x x  - 46x
│ │ │ │ -0002f910: 2078 2020 2d20 3137 7820 7820 202b 2033   x  - 17x x  + 3
│ │ │ │ -0002f920: 3278 2078 2020 2d20 3878 2078 2020 2d20  2x x  - 8x x  - 
│ │ │ │ -0002f930: 3335 7820 7820 202d 2034 3678 2020 2020  35x x  - 46x    
│ │ │ │ -0002f940: 2020 2020 7c0a 7c20 3820 2020 2020 2033      |.| 8      3
│ │ │ │ -0002f950: 2038 2020 2020 2034 2038 2020 2020 2020   8     4 8      
│ │ │ │ -0002f960: 3620 3820 2020 2020 2030 2039 2020 2020  6 8      0 9    
│ │ │ │ -0002f970: 2020 3120 3920 2020 2020 3220 3920 2020    1 9     2 9   
│ │ │ │ -0002f980: 2020 2033 2039 2020 2020 2020 3420 2020     3 9      4   
│ │ │ │ -0002f990: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f8f0: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +0002f900: 2d20 3334 7820 7820 202d 2039 7820 7820  - 34x x  - 9x x 
│ │ │ │ +0002f910: 202d 2034 3678 2078 2020 2d20 3137 7820   - 46x x  - 17x 
│ │ │ │ +0002f920: 7820 202b 2033 3278 2078 2020 2d20 3878  x  + 32x x  - 8x
│ │ │ │ +0002f930: 2078 2020 2d20 3335 7820 7820 202d 2034   x  - 35x x  - 4
│ │ │ │ +0002f940: 3678 2020 2020 2020 2020 7c0a 7c20 3820  6x        |.| 8 
│ │ │ │ +0002f950: 2020 2020 2033 2038 2020 2020 2034 2038       3 8     4 8
│ │ │ │ +0002f960: 2020 2020 2020 3620 3820 2020 2020 2030        6 8      0
│ │ │ │ +0002f970: 2039 2020 2020 2020 3120 3920 2020 2020   9      1 9     
│ │ │ │ +0002f980: 3220 3920 2020 2020 2033 2039 2020 2020  2 9      3 9    
│ │ │ │ +0002f990: 2020 3420 2020 2020 2020 7c0a 7c20 2020    4       |.|   
│ │ │ │  0002f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f9e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f9e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fa30: 2020 2020 7c0a 7c20 202d 2078 2078 2020      |.|  - x x  
│ │ │ │ -0002fa40: 2b20 7820 7820 202b 2034 3078 2078 2020  + x x  + 40x x  
│ │ │ │ -0002fa50: 2d20 3332 7820 7820 202b 2035 7820 7820  - 32x x  + 5x x 
│ │ │ │ -0002fa60: 202d 2031 3178 2078 2020 2d20 3230 7820   - 11x x  - 20x 
│ │ │ │ -0002fa70: 7820 202b 2034 3578 2078 2020 2d20 2020  x  + 45x x  -   
│ │ │ │ -0002fa80: 2020 2020 7c0a 7c38 2020 2020 3420 3820      |.|8    4 8 
│ │ │ │ -0002fa90: 2020 2036 2038 2020 2020 2020 3020 3920     6 8      0 9 
│ │ │ │ -0002faa0: 2020 2020 2031 2039 2020 2020 2032 2039       1 9     2 9
│ │ │ │ -0002fab0: 2020 2020 2020 3320 3920 2020 2020 2034        3 9      4
│ │ │ │ -0002fac0: 2039 2020 2020 2020 3520 3920 2020 2020   9      5 9     
│ │ │ │ -0002fad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fa30: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +0002fa40: 2078 2078 2020 2b20 7820 7820 202b 2034   x x  + x x  + 4
│ │ │ │ +0002fa50: 3078 2078 2020 2d20 3332 7820 7820 202b  0x x  - 32x x  +
│ │ │ │ +0002fa60: 2035 7820 7820 202d 2031 3178 2078 2020   5x x  - 11x x  
│ │ │ │ +0002fa70: 2d20 3230 7820 7820 202b 2034 3578 2078  - 20x x  + 45x x
│ │ │ │ +0002fa80: 2020 2d20 2020 2020 2020 7c0a 7c38 2020    -       |.|8  
│ │ │ │ +0002fa90: 2020 3420 3820 2020 2036 2038 2020 2020    4 8    6 8    
│ │ │ │ +0002faa0: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ +0002fab0: 2020 2032 2039 2020 2020 2020 3320 3920     2 9      3 9 
│ │ │ │ +0002fac0: 2020 2020 2034 2039 2020 2020 2020 3520       4 9      5 
│ │ │ │ +0002fad0: 3920 2020 2020 2020 2020 7c0a 7c20 2020  9         |.|   
│ │ │ │  0002fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fb20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fb20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fb70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fb70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fbc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fc10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fc10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fc60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fcb0: 2020 2020 7c0a 7c33 3778 2078 2020 2b20      |.|37x x  + 
│ │ │ │ -0002fcc0: 3132 7820 7820 202d 2035 7820 7820 202d  12x x  - 5x x  -
│ │ │ │ -0002fcd0: 2035 7820 7820 202b 2032 3178 2078 2020   5x x  + 21x x  
│ │ │ │ -0002fce0: 2b20 3432 7820 7820 202d 2035 7820 7820  + 42x x  - 5x x 
│ │ │ │ -0002fcf0: 202b 2078 2078 2020 2d20 7820 7820 2020   + x x  - x x   
│ │ │ │ -0002fd00: 2020 2020 7c0a 7c20 2020 3120 3820 2020      |.|   1 8   
│ │ │ │ -0002fd10: 2020 2033 2038 2020 2020 2034 2038 2020     3 8     4 8  
│ │ │ │ -0002fd20: 2020 2036 2038 2020 2020 2020 3020 3920     6 8      0 9 
│ │ │ │ -0002fd30: 2020 2020 2031 2039 2020 2020 2032 2039       1 9     2 9
│ │ │ │ -0002fd40: 2020 2020 3320 3920 2020 2034 2020 2020      3 9    4    
│ │ │ │ -0002fd50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fcb0: 2020 2020 2020 2020 2020 7c0a 7c33 3778            |.|37x
│ │ │ │ +0002fcc0: 2078 2020 2b20 3132 7820 7820 202d 2035   x  + 12x x  - 5
│ │ │ │ +0002fcd0: 7820 7820 202d 2035 7820 7820 202b 2032  x x  - 5x x  + 2
│ │ │ │ +0002fce0: 3178 2078 2020 2b20 3432 7820 7820 202d  1x x  + 42x x  -
│ │ │ │ +0002fcf0: 2035 7820 7820 202b 2078 2078 2020 2d20   5x x  + x x  - 
│ │ │ │ +0002fd00: 7820 7820 2020 2020 2020 7c0a 7c20 2020  x x       |.|   
│ │ │ │ +0002fd10: 3120 3820 2020 2020 2033 2038 2020 2020  1 8      3 8    
│ │ │ │ +0002fd20: 2034 2038 2020 2020 2036 2038 2020 2020   4 8     6 8    
│ │ │ │ +0002fd30: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ +0002fd40: 2020 2032 2039 2020 2020 3320 3920 2020     2 9    3 9   
│ │ │ │ +0002fd50: 2034 2020 2020 2020 2020 7c0a 7c20 2020   4        |.|   
│ │ │ │  0002fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fda0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fdf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fdf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fe40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fe90: 2020 2020 7c0a 7c78 2020 2b20 3430 7820      |.|x  + 40x 
│ │ │ │ -0002fea0: 7820 202d 2035 7820 7820 202b 2033 3978  x  - 5x x  + 39x
│ │ │ │ -0002feb0: 2078 2020 2d20 3337 7820 7820 202b 2078   x  - 37x x  + x
│ │ │ │ -0002fec0: 2078 2020 202d 2078 2078 2020 202d 2078   x   - x x   - x
│ │ │ │ -0002fed0: 2078 2020 202b 2078 2078 2020 2020 2020   x   + x x      
│ │ │ │ -0002fee0: 2020 2020 7c0a 7c20 3920 2020 2020 2032      |.| 9      2
│ │ │ │ -0002fef0: 2039 2020 2020 2033 2039 2020 2020 2020   9     3 9      
│ │ │ │ -0002ff00: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ -0002ff10: 3020 3130 2020 2020 3120 3130 2020 2020  0 10    1 10    
│ │ │ │ -0002ff20: 3220 3130 2020 2020 3420 3130 2020 2020  2 10    4 10    
│ │ │ │ -0002ff30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002fe90: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +0002fea0: 2b20 3430 7820 7820 202d 2035 7820 7820  + 40x x  - 5x x 
│ │ │ │ +0002feb0: 202b 2033 3978 2078 2020 2d20 3337 7820   + 39x x  - 37x 
│ │ │ │ +0002fec0: 7820 202b 2078 2078 2020 202d 2078 2078  x  + x x   - x x
│ │ │ │ +0002fed0: 2020 202d 2078 2078 2020 202b 2078 2078     - x x   + x x
│ │ │ │ +0002fee0: 2020 2020 2020 2020 2020 7c0a 7c20 3920            |.| 9 
│ │ │ │ +0002fef0: 2020 2020 2032 2039 2020 2020 2033 2039       2 9     3 9
│ │ │ │ +0002ff00: 2020 2020 2020 3420 3920 2020 2020 2035        4 9      5
│ │ │ │ +0002ff10: 2039 2020 2020 3020 3130 2020 2020 3120   9    0 10    1 
│ │ │ │ +0002ff20: 3130 2020 2020 3220 3130 2020 2020 3420  10    2 10    4 
│ │ │ │ +0002ff30: 3130 2020 2020 2020 2020 7c0a 7c20 2020  10        |.|   
│ │ │ │  0002ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ff80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ff80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0002ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 202d 2032 3778 2078      |.|  - 27x x
│ │ │ │ -0002ffe0: 2020 2d20 3237 7820 7820 202d 2034 3278    - 27x x  - 42x
│ │ │ │ -0002fff0: 2078 2020 2b20 3131 7820 7820 202b 2032   x  + 11x x  + 2
│ │ │ │ -00030000: 7820 7820 202b 2032 3278 2078 2020 2b20  x x  + 22x x  + 
│ │ │ │ -00030010: 3334 7820 7820 202b 2034 3278 2020 2020  34x x  + 42x    
│ │ │ │ -00030020: 2020 2020 7c0a 7c36 2020 2020 2020 3120      |.|6      1 
│ │ │ │ -00030030: 3720 2020 2020 2033 2037 2020 2020 2020  7      3 7      
│ │ │ │ -00030040: 3520 3720 2020 2020 2036 2037 2020 2020  5 7      6 7    
│ │ │ │ -00030050: 2030 2038 2020 2020 2020 3120 3820 2020   0 8      1 8   
│ │ │ │ -00030060: 2020 2033 2038 2020 2020 2020 3420 2020     3 8      4   
│ │ │ │ -00030070: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ffd0: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +0002ffe0: 2032 3778 2078 2020 2d20 3237 7820 7820   27x x  - 27x x 
│ │ │ │ +0002fff0: 202d 2034 3278 2078 2020 2b20 3131 7820   - 42x x  + 11x 
│ │ │ │ +00030000: 7820 202b 2032 7820 7820 202b 2032 3278  x  + 2x x  + 22x
│ │ │ │ +00030010: 2078 2020 2b20 3334 7820 7820 202b 2034   x  + 34x x  + 4
│ │ │ │ +00030020: 3278 2020 2020 2020 2020 7c0a 7c36 2020  2x        |.|6  
│ │ │ │ +00030030: 2020 2020 3120 3720 2020 2020 2033 2037      1 7      3 7
│ │ │ │ +00030040: 2020 2020 2020 3520 3720 2020 2020 2036        5 7      6
│ │ │ │ +00030050: 2037 2020 2020 2030 2038 2020 2020 2020   7     0 8      
│ │ │ │ +00030060: 3120 3820 2020 2020 2033 2038 2020 2020  1 8      3 8    
│ │ │ │ +00030070: 2020 3420 2020 2020 2020 7c0a 7c20 2020    4       |.|   
│ │ │ │  00030080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000300a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000300b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000300c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000300c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000300d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000300e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000300f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030110: 2020 2020 7c0a 7c20 3231 7820 7820 202b      |.| 21x x  +
│ │ │ │ -00030120: 2034 3478 2078 2020 2d20 3134 7820 7820   44x x  - 14x x 
│ │ │ │ -00030130: 202d 2032 3278 2078 2020 2d20 3578 2078   - 22x x  - 5x x
│ │ │ │ -00030140: 2020 2d20 3334 7820 7820 202d 2034 3478    - 34x x  - 44x
│ │ │ │ -00030150: 2078 2020 2b20 3130 7820 7820 2020 2020   x  + 10x x     
│ │ │ │ -00030160: 2020 2020 7c0a 7c20 2020 2033 2037 2020      |.|    3 7  
│ │ │ │ -00030170: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ -00030180: 2020 2020 2020 3020 3820 2020 2020 3120        0 8     1 
│ │ │ │ -00030190: 3820 2020 2020 2033 2038 2020 2020 2020  8      3 8      
│ │ │ │ -000301a0: 3420 3820 2020 2020 2036 2038 2020 2020  4 8      6 8    
│ │ │ │ -000301b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030110: 2020 2020 2020 2020 2020 7c0a 7c20 3231            |.| 21
│ │ │ │ +00030120: 7820 7820 202b 2034 3478 2078 2020 2d20  x x  + 44x x  - 
│ │ │ │ +00030130: 3134 7820 7820 202d 2032 3278 2078 2020  14x x  - 22x x  
│ │ │ │ +00030140: 2d20 3578 2078 2020 2d20 3334 7820 7820  - 5x x  - 34x x 
│ │ │ │ +00030150: 202d 2034 3478 2078 2020 2b20 3130 7820   - 44x x  + 10x 
│ │ │ │ +00030160: 7820 2020 2020 2020 2020 7c0a 7c20 2020  x         |.|   
│ │ │ │ +00030170: 2033 2037 2020 2020 2020 3520 3720 2020   3 7      5 7   
│ │ │ │ +00030180: 2020 2036 2037 2020 2020 2020 3020 3820     6 7      0 8 
│ │ │ │ +00030190: 2020 2020 3120 3820 2020 2020 2033 2038      1 8      3 8
│ │ │ │ +000301a0: 2020 2020 2020 3420 3820 2020 2020 2036        4 8      6
│ │ │ │ +000301b0: 2038 2020 2020 2020 2020 7c0a 7c20 2020   8        |.|   
│ │ │ │  000301c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000301d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000301e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000301f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030200: 2020 2020 7c0a 7c20 202b 2033 3778 2078      |.|  + 37x x
│ │ │ │ -00030210: 2020 2d20 7820 7820 202d 2033 3578 2078    - x x  - 35x x
│ │ │ │ -00030220: 2020 2b20 3978 2078 2020 2b20 3235 7820    + 9x x  + 25x 
│ │ │ │ -00030230: 7820 2020 2d20 3132 7820 7820 2020 2d20  x   - 12x x   - 
│ │ │ │ -00030240: 3336 7820 7820 2020 2d20 3134 7820 2020  36x x   - 14x   
│ │ │ │ -00030250: 2020 2020 7c0a 7c39 2020 2020 2020 3320      |.|9      3 
│ │ │ │ -00030260: 3920 2020 2034 2039 2020 2020 2020 3520  9    4 9      5 
│ │ │ │ -00030270: 3920 2020 2020 3620 3920 2020 2020 2031  9     6 9      1
│ │ │ │ -00030280: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ -00030290: 2020 2034 2031 3020 2020 2020 2020 2020     4 10         
│ │ │ │ -000302a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030200: 2020 2020 2020 2020 2020 7c0a 7c20 202b            |.|  +
│ │ │ │ +00030210: 2033 3778 2078 2020 2d20 7820 7820 202d   37x x  - x x  -
│ │ │ │ +00030220: 2033 3578 2078 2020 2b20 3978 2078 2020   35x x  + 9x x  
│ │ │ │ +00030230: 2b20 3235 7820 7820 2020 2d20 3132 7820  + 25x x   - 12x 
│ │ │ │ +00030240: 7820 2020 2d20 3336 7820 7820 2020 2d20  x   - 36x x   - 
│ │ │ │ +00030250: 3134 7820 2020 2020 2020 7c0a 7c39 2020  14x       |.|9  
│ │ │ │ +00030260: 2020 2020 3320 3920 2020 2034 2039 2020      3 9    4 9  
│ │ │ │ +00030270: 2020 2020 3520 3920 2020 2020 3620 3920      5 9     6 9 
│ │ │ │ +00030280: 2020 2020 2031 2031 3020 2020 2020 2032       1 10      2
│ │ │ │ +00030290: 2031 3020 2020 2020 2034 2031 3020 2020   10      4 10   
│ │ │ │ +000302a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000302b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000302c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000302d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000302e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000302f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000302f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030340: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030390: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030390: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000303a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000303e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000303f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030430: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030430: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030480: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030480: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000304d0: 2020 2020 7c0a 7c20 202d 2032 3678 2078      |.|  - 26x x
│ │ │ │ -000304e0: 2020 2d20 3435 7820 7820 202b 2032 3278    - 45x x  + 22x
│ │ │ │ -000304f0: 2078 2020 2d20 3432 7820 7820 202b 2033   x  - 42x x  + 3
│ │ │ │ -00030500: 7820 7820 202d 2031 3378 2078 2020 2d20  x x  - 13x x  - 
│ │ │ │ -00030510: 3478 2078 2020 2d20 3234 7820 7820 2020  4x x  - 24x x   
│ │ │ │ -00030520: 2020 2020 7c0a 7c38 2020 2020 2020 3620      |.|8      6 
│ │ │ │ -00030530: 3820 2020 2020 2031 2039 2020 2020 2020  8      1 9      
│ │ │ │ -00030540: 3220 3920 2020 2020 2033 2039 2020 2020  2 9      3 9    
│ │ │ │ -00030550: 2034 2039 2020 2020 2020 3520 3920 2020   4 9      5 9   
│ │ │ │ -00030560: 2020 3620 3920 2020 2020 2031 2020 2020    6 9      1    
│ │ │ │ -00030570: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +000304d0: 2020 2020 2020 2020 2020 7c0a 7c20 202d            |.|  -
│ │ │ │ +000304e0: 2032 3678 2078 2020 2d20 3435 7820 7820   26x x  - 45x x 
│ │ │ │ +000304f0: 202b 2032 3278 2078 2020 2d20 3432 7820   + 22x x  - 42x 
│ │ │ │ +00030500: 7820 202b 2033 7820 7820 202d 2031 3378  x  + 3x x  - 13x
│ │ │ │ +00030510: 2078 2020 2d20 3478 2078 2020 2d20 3234   x  - 4x x  - 24
│ │ │ │ +00030520: 7820 7820 2020 2020 2020 7c0a 7c38 2020  x x       |.|8  
│ │ │ │ +00030530: 2020 2020 3620 3820 2020 2020 2031 2039      6 8      1 9
│ │ │ │ +00030540: 2020 2020 2020 3220 3920 2020 2020 2033        2 9      3
│ │ │ │ +00030550: 2039 2020 2020 2034 2039 2020 2020 2020   9     4 9      
│ │ │ │ +00030560: 3520 3920 2020 2020 3620 3920 2020 2020  5 9     6 9     
│ │ │ │ +00030570: 2031 2020 2020 2020 2020 7c0a 7c2d 2d2d   1        |.|---
│ │ │ │  00030580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000305a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000305b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000305c0: 2d2d 2d2d 7c0a 7c20 202d 2032 3778 2078  ----|.|  - 27x x
│ │ │ │ -000305d0: 2020 2d20 3578 2078 2020 202b 2032 3878    - 5x x   + 28x
│ │ │ │ -000305e0: 2078 2020 202b 2033 3778 2078 2020 202b   x   + 37x x   +
│ │ │ │ -000305f0: 2039 7820 7820 2020 2b20 3237 7820 7820   9x x   + 27x x 
│ │ │ │ -00030600: 2020 2d20 3235 7820 7820 2020 2b20 3978    - 25x x   + 9x
│ │ │ │ -00030610: 2078 2020 7c0a 7c39 2020 2020 2020 3620   x  |.|9      6 
│ │ │ │ -00030620: 3920 2020 2020 3020 3130 2020 2020 2020  9     0 10      
│ │ │ │ -00030630: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ -00030640: 2020 2034 2031 3020 2020 2020 2036 2031     4 10      6 1
│ │ │ │ -00030650: 3020 2020 2020 2030 2031 3120 2020 2020  0      0 11     
│ │ │ │ -00030660: 3220 3131 7c0a 7c20 2020 2020 2020 2020  2 11|.|         
│ │ │ │ +000305c0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 202d  ----------|.|  -
│ │ │ │ +000305d0: 2032 3778 2078 2020 2d20 3578 2078 2020   27x x  - 5x x  
│ │ │ │ +000305e0: 202b 2032 3878 2078 2020 202b 2033 3778   + 28x x   + 37x
│ │ │ │ +000305f0: 2078 2020 202b 2039 7820 7820 2020 2b20   x   + 9x x   + 
│ │ │ │ +00030600: 3237 7820 7820 2020 2d20 3235 7820 7820  27x x   - 25x x 
│ │ │ │ +00030610: 2020 2b20 3978 2078 2020 7c0a 7c39 2020    + 9x x  |.|9  
│ │ │ │ +00030620: 2020 2020 3620 3920 2020 2020 3020 3130      6 9     0 10
│ │ │ │ +00030630: 2020 2020 2020 3120 3130 2020 2020 2020        1 10      
│ │ │ │ +00030640: 3220 3130 2020 2020 2034 2031 3020 2020  2 10     4 10   
│ │ │ │ +00030650: 2020 2036 2031 3020 2020 2020 2030 2031     6 10      0 1
│ │ │ │ +00030660: 3120 2020 2020 3220 3131 7c0a 7c20 2020  1     2 11|.|   
│ │ │ │  00030670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000306a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000306b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000306b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000306c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000306d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000306e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000306f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030700: 2020 2020 7c0a 7c78 2020 2b20 3236 7820      |.|x  + 26x 
│ │ │ │ -00030710: 7820 202b 2031 3778 2078 2020 2b20 3135  x  + 17x x  + 15
│ │ │ │ -00030720: 7820 7820 2020 2b20 3335 7820 7820 2020  x x   + 35x x   
│ │ │ │ -00030730: 2b20 3334 7820 7820 2020 2b20 3230 7820  + 34x x   + 20x 
│ │ │ │ -00030740: 7820 2020 2b20 3134 7820 7820 2020 2b20  x   + 14x x   + 
│ │ │ │ -00030750: 3336 7820 7c0a 7c20 3920 2020 2020 2035  36x |.| 9      5
│ │ │ │ -00030760: 2039 2020 2020 2020 3620 3920 2020 2020   9      6 9     
│ │ │ │ -00030770: 2030 2031 3020 2020 2020 2031 2031 3020   0 10      1 10 
│ │ │ │ -00030780: 2020 2020 2032 2031 3020 2020 2020 2034       2 10      4
│ │ │ │ -00030790: 2031 3020 2020 2020 2030 2031 3120 2020   10      0 11   
│ │ │ │ -000307a0: 2020 2031 7c0a 7c20 2020 2020 2020 2020     1|.|         
│ │ │ │ +00030700: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +00030710: 2b20 3236 7820 7820 202b 2031 3778 2078  + 26x x  + 17x x
│ │ │ │ +00030720: 2020 2b20 3135 7820 7820 2020 2b20 3335    + 15x x   + 35
│ │ │ │ +00030730: 7820 7820 2020 2b20 3334 7820 7820 2020  x x   + 34x x   
│ │ │ │ +00030740: 2b20 3230 7820 7820 2020 2b20 3134 7820  + 20x x   + 14x 
│ │ │ │ +00030750: 7820 2020 2b20 3336 7820 7c0a 7c20 3920  x   + 36x |.| 9 
│ │ │ │ +00030760: 2020 2020 2035 2039 2020 2020 2020 3620       5 9      6 
│ │ │ │ +00030770: 3920 2020 2020 2030 2031 3020 2020 2020  9      0 10     
│ │ │ │ +00030780: 2031 2031 3020 2020 2020 2032 2031 3020   1 10      2 10 
│ │ │ │ +00030790: 2020 2020 2034 2031 3020 2020 2020 2030       4 10      0
│ │ │ │ +000307a0: 2031 3120 2020 2020 2031 7c0a 7c20 2020   11      1|.|   
│ │ │ │  000307b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000307f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000307f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030840: 2020 2020 7c0a 7c20 3134 7820 7820 202d      |.| 14x x  -
│ │ │ │ -00030850: 2032 3578 2078 2020 202b 2034 3578 2078   25x x   + 45x x
│ │ │ │ -00030860: 2020 202d 2034 3178 2078 2020 202d 2034     - 41x x   - 4
│ │ │ │ -00030870: 3678 2078 2020 202b 2038 7820 7820 2020  6x x   + 8x x   
│ │ │ │ -00030880: 2d20 3238 7820 7820 2020 2b20 3131 7820  - 28x x   + 11x 
│ │ │ │ -00030890: 7820 2020 7c0a 7c20 2020 2036 2039 2020  x   |.|    6 9  
│ │ │ │ -000308a0: 2020 2020 3020 3130 2020 2020 2020 3120      0 10      1 
│ │ │ │ -000308b0: 3130 2020 2020 2020 3220 3130 2020 2020  10      2 10    
│ │ │ │ -000308c0: 2020 3420 3130 2020 2020 2036 2031 3020    4 10     6 10 
│ │ │ │ -000308d0: 2020 2020 2030 2031 3120 2020 2020 2032       0 11      2
│ │ │ │ -000308e0: 2031 3120 7c0a 7c20 2020 2020 2020 2020   11 |.|         
│ │ │ │ +00030840: 2020 2020 2020 2020 2020 7c0a 7c20 3134            |.| 14
│ │ │ │ +00030850: 7820 7820 202d 2032 3578 2078 2020 202b  x x  - 25x x   +
│ │ │ │ +00030860: 2034 3578 2078 2020 202d 2034 3178 2078   45x x   - 41x x
│ │ │ │ +00030870: 2020 202d 2034 3678 2078 2020 202b 2038     - 46x x   + 8
│ │ │ │ +00030880: 7820 7820 2020 2d20 3238 7820 7820 2020  x x   - 28x x   
│ │ │ │ +00030890: 2b20 3131 7820 7820 2020 7c0a 7c20 2020  + 11x x   |.|   
│ │ │ │ +000308a0: 2036 2039 2020 2020 2020 3020 3130 2020   6 9      0 10  
│ │ │ │ +000308b0: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ +000308c0: 3130 2020 2020 2020 3420 3130 2020 2020  10      4 10    
│ │ │ │ +000308d0: 2036 2031 3020 2020 2020 2030 2031 3120   6 10      0 11 
│ │ │ │ +000308e0: 2020 2020 2032 2031 3120 7c0a 7c20 2020       2 11 |.|   
│ │ │ │  000308f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030930: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030930: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030980: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030980: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000309d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000309d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000309e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030a20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030a70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ac0: 2020 2020 7c0a 7c20 202b 2032 7820 7820      |.|  + 2x x 
│ │ │ │ -00030ad0: 202b 2034 3878 2078 2020 2d20 7820 7820   + 48x x  - x x 
│ │ │ │ -00030ae0: 2020 2b20 3236 7820 7820 2020 2d20 3132    + 26x x   - 12
│ │ │ │ -00030af0: 7820 7820 2020 2d20 3336 7820 7820 2020  x x   - 36x x   
│ │ │ │ -00030b00: 2d20 3138 7820 7820 2020 2d20 3578 2078  - 18x x   - 5x x
│ │ │ │ -00030b10: 2020 202d 7c0a 7c39 2020 2020 2035 2039     -|.|9     5 9
│ │ │ │ -00030b20: 2020 2020 2020 3620 3920 2020 2030 2031        6 9    0 1
│ │ │ │ -00030b30: 3020 2020 2020 2031 2031 3020 2020 2020  0      1 10     
│ │ │ │ -00030b40: 2032 2031 3020 2020 2020 2034 2031 3020   2 10      4 10 
│ │ │ │ -00030b50: 2020 2020 2036 2031 3020 2020 2020 3020       6 10     0 
│ │ │ │ -00030b60: 3131 2020 7c0a 7c20 2020 2020 2020 2020  11  |.|         
│ │ │ │ +00030ac0: 2020 2020 2020 2020 2020 7c0a 7c20 202b            |.|  +
│ │ │ │ +00030ad0: 2032 7820 7820 202b 2034 3878 2078 2020   2x x  + 48x x  
│ │ │ │ +00030ae0: 2d20 7820 7820 2020 2b20 3236 7820 7820  - x x   + 26x x 
│ │ │ │ +00030af0: 2020 2d20 3132 7820 7820 2020 2d20 3336    - 12x x   - 36
│ │ │ │ +00030b00: 7820 7820 2020 2d20 3138 7820 7820 2020  x x   - 18x x   
│ │ │ │ +00030b10: 2d20 3578 2078 2020 202d 7c0a 7c39 2020  - 5x x   -|.|9  
│ │ │ │ +00030b20: 2020 2035 2039 2020 2020 2020 3620 3920     5 9      6 9 
│ │ │ │ +00030b30: 2020 2030 2031 3020 2020 2020 2031 2031     0 10      1 1
│ │ │ │ +00030b40: 3020 2020 2020 2032 2031 3020 2020 2020  0      2 10     
│ │ │ │ +00030b50: 2034 2031 3020 2020 2020 2036 2031 3020   4 10      6 10 
│ │ │ │ +00030b60: 2020 2020 3020 3131 2020 7c0a 7c20 2020      0 11  |.|   
│ │ │ │  00030b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030bb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030bb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030c00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030c50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030c50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c2b 2035 7820 7820 2020      |.|+ 5x x   
│ │ │ │ -00030cb0: 2d20 3578 2078 2020 2c20 2020 2020 2020  - 5x x  ,       
│ │ │ │ +00030ca0: 2020 2020 2020 2020 2020 7c0a 7c2b 2035            |.|+ 5
│ │ │ │ +00030cb0: 7820 7820 2020 2d20 3578 2078 2020 2c20  x x   - 5x x  , 
│ │ │ │  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 7c0a 7c20 2020 2036 2031 3020      |.|    6 10 
│ │ │ │ -00030d00: 2020 2020 3520 3131 2020 2020 2020 2020      5 11        
│ │ │ │ +00030cf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00030d00: 2036 2031 3020 2020 2020 3520 3131 2020   6 10     5 11  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030d40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030d90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030de0: 2020 2020 7c0a 7c78 2020 2b20 3134 7820      |.|x  + 14x 
│ │ │ │ -00030df0: 7820 202b 2038 7820 7820 202b 2031 3678  x  + 8x x  + 16x
│ │ │ │ -00030e00: 2078 2020 2b20 3237 7820 7820 202b 2032   x  + 27x x  + 2
│ │ │ │ -00030e10: 3478 2078 2020 2d20 3131 7820 7820 202d  4x x  - 11x x  -
│ │ │ │ -00030e20: 2036 7820 7820 202b 2032 7820 7820 202d   6x x  + 2x x  -
│ │ │ │ -00030e30: 2035 7820 7c0a 7c20 3820 2020 2020 2036   5x |.| 8      6
│ │ │ │ -00030e40: 2038 2020 2020 2030 2039 2020 2020 2020   8     0 9      
│ │ │ │ -00030e50: 3120 3920 2020 2020 2032 2039 2020 2020  1 9      2 9    
│ │ │ │ -00030e60: 2020 3320 3920 2020 2020 2034 2039 2020    3 9      4 9  
│ │ │ │ -00030e70: 2020 2035 2039 2020 2020 2036 2039 2020     5 9     6 9  
│ │ │ │ -00030e80: 2020 2030 7c0a 7c20 2020 2020 2020 2020     0|.|         
│ │ │ │ +00030de0: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +00030df0: 2b20 3134 7820 7820 202b 2038 7820 7820  + 14x x  + 8x x 
│ │ │ │ +00030e00: 202b 2031 3678 2078 2020 2b20 3237 7820   + 16x x  + 27x 
│ │ │ │ +00030e10: 7820 202b 2032 3478 2078 2020 2d20 3131  x  + 24x x  - 11
│ │ │ │ +00030e20: 7820 7820 202d 2036 7820 7820 202b 2032  x x  - 6x x  + 2
│ │ │ │ +00030e30: 7820 7820 202d 2035 7820 7c0a 7c20 3820  x x  - 5x |.| 8 
│ │ │ │ +00030e40: 2020 2020 2036 2038 2020 2020 2030 2039       6 8     0 9
│ │ │ │ +00030e50: 2020 2020 2020 3120 3920 2020 2020 2032        1 9      2
│ │ │ │ +00030e60: 2039 2020 2020 2020 3320 3920 2020 2020   9      3 9     
│ │ │ │ +00030e70: 2034 2039 2020 2020 2035 2039 2020 2020   4 9     5 9    
│ │ │ │ +00030e80: 2036 2039 2020 2020 2030 7c0a 7c20 2020   6 9     0|.|   
│ │ │ │  00030e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00030ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00030ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f20: 2020 2020 7c0a 7c2d 2034 3478 2078 2020      |.|- 44x x  
│ │ │ │ -00030f30: 2d20 3231 7820 7820 202d 2032 3178 2078  - 21x x  - 21x x
│ │ │ │ -00030f40: 2020 2d20 3978 2078 2020 2b20 3134 7820    - 9x x  + 14x 
│ │ │ │ -00030f50: 7820 202d 2032 3278 2078 2020 2b20 3435  x  - 22x x  + 45
│ │ │ │ -00030f60: 7820 7820 202b 2035 7820 7820 2020 2b20  x x  + 5x x   + 
│ │ │ │ -00030f70: 3230 7820 7c0a 7c20 2020 2020 3020 3920  20x |.|     0 9 
│ │ │ │ -00030f80: 2020 2020 2031 2039 2020 2020 2020 3220       1 9      2 
│ │ │ │ -00030f90: 3920 2020 2020 3320 3920 2020 2020 2034  9     3 9      4
│ │ │ │ -00030fa0: 2039 2020 2020 2020 3520 3920 2020 2020   9      5 9     
│ │ │ │ -00030fb0: 2036 2039 2020 2020 2030 2031 3020 2020   6 9     0 10   
│ │ │ │ -00030fc0: 2020 2031 7c0a 7c20 2020 2020 2020 2020     1|.|         
│ │ │ │ +00030f20: 2020 2020 2020 2020 2020 7c0a 7c2d 2034            |.|- 4
│ │ │ │ +00030f30: 3478 2078 2020 2d20 3231 7820 7820 202d  4x x  - 21x x  -
│ │ │ │ +00030f40: 2032 3178 2078 2020 2d20 3978 2078 2020   21x x  - 9x x  
│ │ │ │ +00030f50: 2b20 3134 7820 7820 202d 2032 3278 2078  + 14x x  - 22x x
│ │ │ │ +00030f60: 2020 2b20 3435 7820 7820 202b 2035 7820    + 45x x  + 5x 
│ │ │ │ +00030f70: 7820 2020 2b20 3230 7820 7c0a 7c20 2020  x   + 20x |.|   
│ │ │ │ +00030f80: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ +00030f90: 2020 2020 3220 3920 2020 2020 3320 3920      2 9     3 9 
│ │ │ │ +00030fa0: 2020 2020 2034 2039 2020 2020 2020 3520       4 9      5 
│ │ │ │ +00030fb0: 3920 2020 2020 2036 2039 2020 2020 2030  9      6 9     0
│ │ │ │ +00030fc0: 2031 3020 2020 2020 2031 7c0a 7c20 2020   10      1|.|   
│ │ │ │  00030fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031010: 2020 2020 7c0a 7c20 7820 2020 2d20 3337      |.| x   - 37
│ │ │ │ -00031020: 7820 7820 2020 2d20 3978 2078 2020 202b  x x   - 9x x   +
│ │ │ │ -00031030: 2031 3478 2078 2020 2c20 2020 2020 2020   14x x  ,       
│ │ │ │ +00031010: 2020 2020 2020 2020 2020 7c0a 7c20 7820            |.| x 
│ │ │ │ +00031020: 2020 2d20 3337 7820 7820 2020 2d20 3978    - 37x x   - 9x
│ │ │ │ +00031030: 2078 2020 202b 2031 3478 2078 2020 2c20   x   + 14x x  , 
│ │ │ │  00031040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031060: 2020 2020 7c0a 7c36 2031 3020 2020 2020      |.|6 10     
│ │ │ │ -00031070: 2032 2031 3120 2020 2020 3420 3131 2020   2 11     4 11  
│ │ │ │ -00031080: 2020 2020 3520 3131 2020 2020 2020 2020      5 11        
│ │ │ │ +00031060: 2020 2020 2020 2020 2020 7c0a 7c36 2031            |.|6 1
│ │ │ │ +00031070: 3020 2020 2020 2032 2031 3120 2020 2020  0      2 11     
│ │ │ │ +00031080: 3420 3131 2020 2020 2020 3520 3131 2020  4 11      5 11  
│ │ │ │  00031090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000310a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000310b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000310b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000310c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031100: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031150: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031150: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000311a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000311a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000311b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000311c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000311d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000311e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000311f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000311f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031240: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031290: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031290: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000312a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000312b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000312c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000312d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000312e0: 2020 2020 7c0a 7c20 2020 2b20 3133 7820      |.|   + 13x 
│ │ │ │ -000312f0: 7820 2020 2b20 3339 7820 7820 2020 2d20  x   + 39x x   - 
│ │ │ │ -00031300: 3678 2078 2020 202b 2035 7820 7820 2020  6x x   + 5x x   
│ │ │ │ -00031310: 2b20 3432 7820 7820 2020 2b20 3478 2078  + 42x x   + 4x x
│ │ │ │ -00031320: 2020 202b 2036 7820 7820 2020 2020 2020     + 6x x       
│ │ │ │ -00031330: 2020 2020 7c0a 7c31 3020 2020 2020 2032      |.|10      2
│ │ │ │ -00031340: 2031 3020 2020 2020 2034 2031 3020 2020   10      4 10   
│ │ │ │ -00031350: 2020 3620 3130 2020 2020 2031 2031 3120    6 10     1 11 
│ │ │ │ -00031360: 2020 2020 2032 2031 3120 2020 2020 3420       2 11     4 
│ │ │ │ -00031370: 3131 2020 2020 2035 2031 3120 2020 2020  11     5 11     
│ │ │ │ -00031380: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +000312e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000312f0: 2b20 3133 7820 7820 2020 2b20 3339 7820  + 13x x   + 39x 
│ │ │ │ +00031300: 7820 2020 2d20 3678 2078 2020 202b 2035  x   - 6x x   + 5
│ │ │ │ +00031310: 7820 7820 2020 2b20 3432 7820 7820 2020  x x   + 42x x   
│ │ │ │ +00031320: 2b20 3478 2078 2020 202b 2036 7820 7820  + 4x x   + 6x x 
│ │ │ │ +00031330: 2020 2020 2020 2020 2020 7c0a 7c31 3020            |.|10 
│ │ │ │ +00031340: 2020 2020 2032 2031 3020 2020 2020 2034       2 10      4
│ │ │ │ +00031350: 2031 3020 2020 2020 3620 3130 2020 2020   10     6 10    
│ │ │ │ +00031360: 2031 2031 3120 2020 2020 2032 2031 3120   1 11      2 11 
│ │ │ │ +00031370: 2020 2020 3420 3131 2020 2020 2035 2031      4 11     5 1
│ │ │ │ +00031380: 3120 2020 2020 2020 2020 7c0a 7c2d 2d2d  1         |.|---
│ │ │ │  00031390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000313a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000313b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000313c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000313d0: 2d2d 2d2d 7c0a 7c20 2b20 3237 7820 7820  ----|.| + 27x x 
│ │ │ │ -000313e0: 2020 2d20 3237 7820 7820 202c 2020 2020    - 27x x  ,    
│ │ │ │ -000313f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000313d0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2b20  ----------|.| + 
│ │ │ │ +000313e0: 3237 7820 7820 2020 2d20 3237 7820 7820  27x x   - 27x x 
│ │ │ │ +000313f0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  00031400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031420: 2020 2020 7c0a 7c20 2020 2020 2034 2031      |.|      4 1
│ │ │ │ -00031430: 3120 2020 2020 2035 2031 3120 2020 2020  1      5 11     
│ │ │ │ -00031440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031420: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00031430: 2020 2034 2031 3120 2020 2020 2035 2031     4 11      5 1
│ │ │ │ +00031440: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00031450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031470: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031470: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000314c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000314c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000314d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031510: 2020 2020 7c0a 7c78 2020 202b 2033 3578      |.|x   + 35x
│ │ │ │ -00031520: 2078 2020 202d 2031 3778 2078 2020 2c20   x   - 17x x  , 
│ │ │ │ -00031530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031510: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +00031520: 202b 2033 3578 2078 2020 202d 2031 3778   + 35x x   - 17x
│ │ │ │ +00031530: 2078 2020 2c20 2020 2020 2020 2020 2020   x  ,           
│ │ │ │  00031540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031560: 2020 2020 7c0a 7c20 3131 2020 2020 2020      |.| 11      
│ │ │ │ -00031570: 3220 3131 2020 2020 2020 3420 3131 2020  2 11      4 11  
│ │ │ │ -00031580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031560: 2020 2020 2020 2020 2020 7c0a 7c20 3131            |.| 11
│ │ │ │ +00031570: 2020 2020 2020 3220 3131 2020 2020 2020        2 11      
│ │ │ │ +00031580: 3420 3131 2020 2020 2020 2020 2020 2020  4 11            
│ │ │ │  00031590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000315a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000315b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000315b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000315c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000315d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000315e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000315f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031600: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031600: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031650: 2020 2020 7c0a 7c2b 2031 3478 2078 2020      |.|+ 14x x  
│ │ │ │ -00031660: 202d 2038 7820 7820 2020 2020 2020 2020   - 8x x         
│ │ │ │ +00031650: 2020 2020 2020 2020 2020 7c0a 7c2b 2031            |.|+ 1
│ │ │ │ +00031660: 3478 2078 2020 202d 2038 7820 7820 2020  4x x   - 8x x   
│ │ │ │  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 7c0a 7c20 2020 2020 3420 3131      |.|     4 11
│ │ │ │ -000316b0: 2020 2020 2035 2031 3120 2020 2020 2020       5 11       
│ │ │ │ +000316a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000316b0: 2020 3420 3131 2020 2020 2035 2031 3120    4 11     5 11 
│ │ │ │  000316c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000316d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000316e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000316f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000316f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031740: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031790: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031790: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000317a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000317b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000317c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000317d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000317e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000317e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000317f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031830: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031830: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031880: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000318a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000318b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000318c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000318d0: 2020 2020 7c0a 7c20 3337 7820 7820 2020      |.| 37x x   
│ │ │ │ -000318e0: 2d20 3578 2078 2020 202d 2039 7820 7820  - 5x x   - 9x x 
│ │ │ │ -000318f0: 2020 2b20 3139 7820 7820 202c 2020 2020    + 19x x  ,    
│ │ │ │ -00031900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000318d0: 2020 2020 2020 2020 2020 7c0a 7c20 3337            |.| 37
│ │ │ │ +000318e0: 7820 7820 2020 2d20 3578 2078 2020 202d  x x   - 5x x   -
│ │ │ │ +000318f0: 2039 7820 7820 2020 2b20 3139 7820 7820   9x x   + 19x x 
│ │ │ │ +00031900: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  00031910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031920: 2020 2020 7c0a 7c20 2020 2032 2031 3120      |.|    2 11 
│ │ │ │ -00031930: 2020 2020 3320 3131 2020 2020 2034 2031      3 11     4 1
│ │ │ │ -00031940: 3120 2020 2020 2035 2031 3120 2020 2020  1      5 11     
│ │ │ │ -00031950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031920: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00031930: 2032 2031 3120 2020 2020 3320 3131 2020   2 11     3 11  
│ │ │ │ +00031940: 2020 2034 2031 3120 2020 2020 2035 2031     4 11      5 1
│ │ │ │ +00031950: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00031960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031970: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031970: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000319a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000319b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000319c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000319c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000319d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000319e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000319f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031a10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031a10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031a60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031a60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031ab0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031b00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031b50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031ba0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c78 2020 202d 2031 3078      |.|x   - 10x
│ │ │ │ -00031c00: 2078 2020 202d 2033 3478 2078 2020 202d   x   - 34x x   -
│ │ │ │ -00031c10: 2038 7820 7820 2020 2b20 3135 7820 7820   8x x   + 15x x 
│ │ │ │ -00031c20: 2020 2d20 3235 7820 7820 2020 2d20 3234    - 25x x   - 24
│ │ │ │ -00031c30: 7820 7820 2020 2d20 3278 2078 2020 202d  x x   - 2x x   -
│ │ │ │ -00031c40: 2020 2020 7c0a 7c20 3130 2020 2020 2020      |.| 10      
│ │ │ │ -00031c50: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ -00031c60: 2020 2034 2031 3020 2020 2020 2036 2031     4 10      6 1
│ │ │ │ -00031c70: 3020 2020 2020 2030 2031 3120 2020 2020  0      0 11     
│ │ │ │ -00031c80: 2032 2031 3120 2020 2020 3420 3131 2020   2 11     4 11  
│ │ │ │ -00031c90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031bf0: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +00031c00: 202d 2031 3078 2078 2020 202d 2033 3478   - 10x x   - 34x
│ │ │ │ +00031c10: 2078 2020 202d 2038 7820 7820 2020 2b20   x   - 8x x   + 
│ │ │ │ +00031c20: 3135 7820 7820 2020 2d20 3235 7820 7820  15x x   - 25x x 
│ │ │ │ +00031c30: 2020 2d20 3234 7820 7820 2020 2d20 3278    - 24x x   - 2x
│ │ │ │ +00031c40: 2078 2020 202d 2020 2020 7c0a 7c20 3130   x   -    |.| 10
│ │ │ │ +00031c50: 2020 2020 2020 3120 3130 2020 2020 2020        1 10      
│ │ │ │ +00031c60: 3220 3130 2020 2020 2034 2031 3020 2020  2 10     4 10   
│ │ │ │ +00031c70: 2020 2036 2031 3020 2020 2020 2030 2031     6 10      0 1
│ │ │ │ +00031c80: 3120 2020 2020 2032 2031 3120 2020 2020  1      2 11     
│ │ │ │ +00031c90: 3420 3131 2020 2020 2020 7c0a 7c20 2020  4 11      |.|   
│ │ │ │  00031ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031ce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c78 2020 202b 2033 3478      |.|x   + 34x
│ │ │ │ -00031d40: 2078 2020 202b 2031 3278 2078 2020 202d   x   + 12x x   -
│ │ │ │ -00031d50: 2032 3578 2078 2020 202b 2032 3078 2078   25x x   + 20x x
│ │ │ │ -00031d60: 2020 202d 2035 7820 7820 2020 2b20 3978     - 5x x   + 9x
│ │ │ │ -00031d70: 2078 2020 202d 2034 3578 2078 2020 202b   x   - 45x x   +
│ │ │ │ -00031d80: 2020 2020 7c0a 7c20 3130 2020 2020 2020      |.| 10      
│ │ │ │ -00031d90: 3220 3130 2020 2020 2020 3420 3130 2020  2 10      4 10  
│ │ │ │ -00031da0: 2020 2020 3620 3130 2020 2020 2020 3020      6 10      0 
│ │ │ │ -00031db0: 3131 2020 2020 2031 2031 3120 2020 2020  11     1 11     
│ │ │ │ -00031dc0: 3220 3131 2020 2020 2020 3420 3131 2020  2 11      4 11  
│ │ │ │ -00031dd0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00031d30: 2020 2020 2020 2020 2020 7c0a 7c78 2020            |.|x  
│ │ │ │ +00031d40: 202b 2033 3478 2078 2020 202b 2031 3278   + 34x x   + 12x
│ │ │ │ +00031d50: 2078 2020 202d 2032 3578 2078 2020 202b   x   - 25x x   +
│ │ │ │ +00031d60: 2032 3078 2078 2020 202d 2035 7820 7820   20x x   - 5x x 
│ │ │ │ +00031d70: 2020 2b20 3978 2078 2020 202d 2034 3578    + 9x x   - 45x
│ │ │ │ +00031d80: 2078 2020 202b 2020 2020 7c0a 7c20 3130   x   +    |.| 10
│ │ │ │ +00031d90: 2020 2020 2020 3220 3130 2020 2020 2020        2 10      
│ │ │ │ +00031da0: 3420 3130 2020 2020 2020 3620 3130 2020  4 10      6 10  
│ │ │ │ +00031db0: 2020 2020 3020 3131 2020 2020 2031 2031      0 11     1 1
│ │ │ │ +00031dc0: 3120 2020 2020 3220 3131 2020 2020 2020  1     2 11      
│ │ │ │ +00031dd0: 3420 3131 2020 2020 2020 7c0a 7c2d 2d2d  4 11      |.|---
│ │ │ │  00031de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031e20: 2d2d 2d2d 7c0a 7c31 3578 2078 2020 2c20  ----|.|15x x  , 
│ │ │ │ -00031e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031e20: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c31 3578  ----------|.|15x
│ │ │ │ +00031e30: 2078 2020 2c20 2020 2020 2020 2020 2020   x  ,           
│ │ │ │  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 7c0a 7c20 2020 3520 3131 2020      |.|   5 11  
│ │ │ │ -00031e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031e70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00031e80: 3520 3131 2020 2020 2020 2020 2020 2020  5 11            
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031ec0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00031f10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00031f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031f60: 2020 2020 7c0a 7c32 3578 2078 2020 2c20      |.|25x x  , 
│ │ │ │ -00031f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031f60: 2020 2020 2020 2020 2020 7c0a 7c32 3578            |.|25x
│ │ │ │ +00031f70: 2078 2020 2c20 2020 2020 2020 2020 2020   x  ,           
│ │ │ │  00031f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031fb0: 2020 2020 7c0a 7c20 2020 3520 3131 2020      |.|   5 11  
│ │ │ │ -00031fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031fb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00031fc0: 3520 3131 2020 2020 2020 2020 2020 2020  5 11            
│ │ │ │  00031fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032000: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00032000: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00032010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032050: 2d2d 2d2d 2b0a 7c69 3131 203a 2061 7373  ----+.|i11 : ass
│ │ │ │ -00032060: 6572 7428 7068 6920 2a20 7073 6920 3d3d  ert(phi * psi ==
│ │ │ │ -00032070: 2031 2920 2020 2020 2020 2020 2020 2020   1)             
│ │ │ │ +00032050: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ +00032060: 203a 2061 7373 6572 7428 7068 6920 2a20   : assert(phi * 
│ │ │ │ +00032070: 7073 6920 3d3d 2031 2920 2020 2020 2020  psi == 1)       
│ │ │ │  00032080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000320a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000320a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
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│ │ │ │  00032700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00032750: 202a 2022 6465 6772 6565 4d61 7028 2e2e   * "degreeMap(..
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│ │ │ │ +00032800: 2020 2a20 2269 7342 6972 6174 696f 6e61    * "isBirationa
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│ │ │ │ +00032860: 2022 5365 6772 6543 6c61 7373 282e 2e2e   "SegreClass(...
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│ │ │ │ +00032890: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
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│ │ │ │  000329b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │  00032d50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00032d60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00032e90: 2e2e 2922 0a20 202a 2022 6973 446f 6d69  ..)".  * "isDomi
│ │ │ │ -00032ea0: 6e61 6e74 282e 2e2e 2c43 6572 7469 6679  nant(...,Certify
│ │ │ │ -00032eb0: 3d3e 2e2e 2e29 220a 2020 2a20 2270 726f  =>...)".  * "pro
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│ │ │ │ -00032ed0: 2e2e 2c43 6572 7469 6679 3d3e 2e2e 2e29  ..,Certify=>...)
│ │ │ │ -00032ee0: 220a 2020 2a20 2253 6567 7265 436c 6173  ".  * "SegreClas
│ │ │ │ -00032ef0: 7328 2e2e 2e2c 4365 7274 6966 793d 3e2e  s(...,Certify=>.
│ │ │ │ -00032f00: 2e2e 2922 0a0a 466f 7220 7468 6520 7072  ..)"..For the pr
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│ │ │ │ -00032f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
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│ │ │ │ -00032f50: 6973 2061 202a 6e6f 7465 2073 796d 626f  is a *note symbo
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│ │ │ │ -00032fa0: 7274 7a4d 6163 5068 6572 736f 6e2c 204e  rtzMacPherson, N
│ │ │ │ -00032fb0: 6578 743a 2043 6f64 696d 4273 496e 762c  ext: CodimBsInv,
│ │ │ │ -00032fc0: 2050 7265 763a 2043 6572 7469 6679 2c20   Prev: Certify, 
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│ │ │ │ -00032fe0: 6877 6172 747a 4d61 6350 6865 7273 6f6e  hwartzMacPherson
│ │ │ │ -00032ff0: 202d 2d20 4368 6572 6e2d 5363 6877 6172   -- Chern-Schwar
│ │ │ │ -00033000: 747a 2d4d 6163 5068 6572 736f 6e20 636c  tz-MacPherson cl
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│ │ │ │ +00032dd0: 7469 6679 3d3e 2e2e 2e29 220a 2020 2a20  tify=>...)".  * 
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│ │ │ │  00033050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00033060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00033070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -00033080: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -00033090: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -000330a0: 2020 2020 2043 6865 726e 5363 6877 6172       ChernSchwar
│ │ │ │ -000330b0: 747a 4d61 6350 6865 7273 6f6e 2049 0a20  tzMacPherson I. 
│ │ │ │ -000330c0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -000330d0: 202a 2049 2c20 616e 202a 6e6f 7465 2069   * I, an *note i
│ │ │ │ -000330e0: 6465 616c 3a20 284d 6163 6175 6c61 7932  deal: (Macaulay2
│ │ │ │ -000330f0: 446f 6329 4964 6561 6c2c 2c20 6120 686f  Doc)Ideal,, a ho
│ │ │ │ -00033100: 6d6f 6765 6e65 6f75 7320 6964 6561 6c20  mogeneous ideal 
│ │ │ │ -00033110: 6465 6669 6e69 6e67 2061 0a20 2020 2020  defining a.     
│ │ │ │ -00033120: 2020 2063 6c6f 7365 6420 7375 6273 6368     closed subsch
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│ │ │ │ -00033140: 7468 6262 7b50 7d5e 6e24 0a20 202a 202a  thbb{P}^n$.  * *
│ │ │ │ -00033150: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ -00033160: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ -00033170: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ -00033180: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ -00033190: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ -000331a0: 202a 202a 6e6f 7465 2042 6c6f 7755 7053   * *note BlowUpS
│ │ │ │ -000331b0: 7472 6174 6567 793a 2042 6c6f 7755 7053  trategy: BlowUpS
│ │ │ │ -000331c0: 7472 6174 6567 792c 203d 3e20 2e2e 2e2c  trategy, => ...,
│ │ │ │ -000331d0: 2064 6566 6175 6c74 2076 616c 7565 0a20   default value. 
│ │ │ │ -000331e0: 2020 2020 2020 2022 456c 696d 696e 6174         "Eliminat
│ │ │ │ -000331f0: 6522 2c0a 2020 2020 2020 2a20 2a6e 6f74  e",.      * *not
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│ │ │ │ -00033230: 2077 6865 7468 6572 2074 6f20 656e 7375   whether to ensu
│ │ │ │ -00033240: 7265 0a20 2020 2020 2020 2063 6f72 7265  re.        corre
│ │ │ │ -00033250: 6374 6e65 7373 206f 6620 6f75 7470 7574  ctness of output
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│ │ │ │ -000332a0: 5f70 645f 7270 2c20 3d3e 202e 2e2e 2c0a  _pd_rp, => ...,.
│ │ │ │ -000332b0: 2020 2020 2020 2020 6465 6661 756c 7420          default 
│ │ │ │ -000332c0: 7661 6c75 6520 7472 7565 2c0a 2020 2a20  value true,.  * 
│ │ │ │ -000332d0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -000332e0: 2061 202a 6e6f 7465 2072 696e 6720 656c   a *note ring el
│ │ │ │ -000332f0: 656d 656e 743a 2028 4d61 6361 756c 6179  ement: (Macaulay
│ │ │ │ -00033300: 3244 6f63 2952 696e 6745 6c65 6d65 6e74  2Doc)RingElement
│ │ │ │ -00033310: 2c2c 2074 6865 2070 7573 682d 666f 7277  ,, the push-forw
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│ │ │ │ -00033330: 6865 2043 686f 7720 7269 6e67 206f 6620  he Chow ring of 
│ │ │ │ -00033340: 245c 6d61 7468 6262 7b50 7d5e 6e24 206f  $\mathbb{P}^n$ o
│ │ │ │ -00033350: 6620 7468 6520 4368 6572 6e2d 5363 6877  f the Chern-Schw
│ │ │ │ -00033360: 6172 747a 2d4d 6163 5068 6572 736f 6e20  artz-MacPherson 
│ │ │ │ -00033370: 636c 6173 730a 2020 2020 2020 2020 2463  class.        $c
│ │ │ │ -00033380: 5f7b 534d 7d28 5829 2420 6f66 2024 5824  _{SM}(X)$ of $X$
│ │ │ │ -00033390: 2e20 496e 2070 6172 7469 6375 6c61 722c  . In particular,
│ │ │ │ -000333a0: 2074 6865 2063 6f65 6666 6963 6965 6e74   the coefficient
│ │ │ │ -000333b0: 206f 6620 2448 5e6e 2420 6769 7665 7320   of $H^n$ gives 
│ │ │ │ -000333c0: 7468 650a 2020 2020 2020 2020 4575 6c65  the.        Eule
│ │ │ │ -000333d0: 7220 6368 6172 6163 7465 7269 7374 6963  r characteristic
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│ │ │ │ -000333f0: 6f66 2024 5824 2c20 7768 6572 6520 2448  of $X$, where $H
│ │ │ │ -00033400: 2420 6465 6e6f 7465 7320 7468 650a 2020  $ denotes the.  
│ │ │ │ -00033410: 2020 2020 2020 6879 7065 7270 6c61 6e65        hyperplane
│ │ │ │ -00033420: 2063 6c61 7373 2e0a 0a44 6573 6372 6970   class...Descrip
│ │ │ │ -00033430: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00033440: 0a0a 5468 6973 2069 7320 616e 2065 7861  ..This is an exa
│ │ │ │ -00033450: 6d70 6c65 206f 6620 6170 706c 6963 6174  mple of applicat
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│ │ │ │ -00033480: 7665 4465 6772 6565 733a 0a70 726f 6a65  veDegrees:.proje
│ │ │ │ -00033490: 6374 6976 6544 6567 7265 6573 2c2c 2064  ctiveDegrees,, d
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│ │ │ │ -000334f0: 6565 2068 7474 703a 2f2f 7777 772e 7363  ee http://www.sc
│ │ │ │ -00033500: 6965 6e63 6564 6972 6563 742e 636f 6d2f  iencedirect.com/
│ │ │ │ -00033510: 7363 6965 6e63 652f 6172 7469 636c 652f  science/article/
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│ │ │ │ -00033530: 3030 3038 3935 2029 2c20 6279 2050 2e20  000895 ), by P. 
│ │ │ │ -00033540: 416c 7566 6669 2e20 5365 6520 616c 736f  Aluffi. See also
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│ │ │ │ -00033560: 6e67 206d 6574 686f 6473 2069 6e20 7468  ng methods in th
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│ │ │ │ -00033580: 2028 7365 6520 6874 7470 3a2f 2f77 7777   (see http://www
│ │ │ │ -00033590: 2e6d 6174 682e 6673 752e 6564 752f 7e61  .math.fsu.edu/~a
│ │ │ │ -000335a0: 6c75 6666 692f 4353 4d2f 4353 4d2e 6874  luffi/CSM/CSM.ht
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│ │ │ │ -000335f0: 7468 2e75 6975 632e 6564 752f 4d61 6361  th.uiuc.edu/Maca
│ │ │ │ -00033600: 756c 6179 322f 646f 632f 4d61 6361 756c  ulay2/doc/Macaul
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│ │ │ │ -00033750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00033090: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ +000330a0: 3a20 0a20 2020 2020 2020 2043 6865 726e  : .        Chern
│ │ │ │ +000330b0: 5363 6877 6172 747a 4d61 6350 6865 7273  SchwartzMacPhers
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│ │ │ │ +000330e0: 6e6f 7465 2069 6465 616c 3a20 284d 6163  note ideal: (Mac
│ │ │ │ +000330f0: 6175 6c61 7932 446f 6329 4964 6561 6c2c  aulay2Doc)Ideal,
│ │ │ │ +00033100: 2c20 6120 686f 6d6f 6765 6e65 6f75 7320  , a homogeneous 
│ │ │ │ +00033110: 6964 6561 6c20 6465 6669 6e69 6e67 2061  ideal defining a
│ │ │ │ +00033120: 0a20 2020 2020 2020 2063 6c6f 7365 6420  .        closed 
│ │ │ │ +00033130: 7375 6273 6368 656d 6520 2458 5c73 7562  subscheme $X\sub
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│ │ │ │ +00033150: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
│ │ │ │ +00033160: 6e61 6c20 696e 7075 7473 3a20 284d 6163  nal inputs: (Mac
│ │ │ │ +00033170: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ +00033180: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
│ │ │ │ +00033190: 7074 696f 6e61 6c20 696e 7075 7473 2c3a  ptional inputs,:
│ │ │ │ +000331a0: 0a20 2020 2020 202a 202a 6e6f 7465 2042  .      * *note B
│ │ │ │ +000331b0: 6c6f 7755 7053 7472 6174 6567 793a 2042  lowUpStrategy: B
│ │ │ │ +000331c0: 6c6f 7755 7053 7472 6174 6567 792c 203d  lowUpStrategy, =
│ │ │ │ +000331d0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +000331e0: 616c 7565 0a20 2020 2020 2020 2022 456c  alue.        "El
│ │ │ │ +000331f0: 696d 696e 6174 6522 2c0a 2020 2020 2020  iminate",.      
│ │ │ │ +00033200: 2a20 2a6e 6f74 6520 4365 7274 6966 793a  * *note Certify:
│ │ │ │ +00033210: 2043 6572 7469 6679 2c20 3d3e 202e 2e2e   Certify, => ...
│ │ │ │ +00033220: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00033230: 6661 6c73 652c 2077 6865 7468 6572 2074  false, whether t
│ │ │ │ +00033240: 6f20 656e 7375 7265 0a20 2020 2020 2020  o ensure.       
│ │ │ │ +00033250: 2063 6f72 7265 6374 6e65 7373 206f 6620   correctness of 
│ │ │ │ +00033260: 6f75 7470 7574 0a20 2020 2020 202a 202a  output.      * *
│ │ │ │ +00033270: 6e6f 7465 2056 6572 626f 7365 3a20 696e  note Verbose: in
│ │ │ │ +00033280: 7665 7273 654d 6170 5f6c 705f 7064 5f70  verseMap_lp_pd_p
│ │ │ │ +00033290: 645f 7064 5f63 6d56 6572 626f 7365 3d3e  d_pd_cmVerbose=>
│ │ │ │ +000332a0: 5f70 645f 7064 5f70 645f 7270 2c20 3d3e  _pd_pd_pd_rp, =>
│ │ │ │ +000332b0: 202e 2e2e 2c0a 2020 2020 2020 2020 6465   ...,.        de
│ │ │ │ +000332c0: 6661 756c 7420 7661 6c75 6520 7472 7565  fault value true
│ │ │ │ +000332d0: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ +000332e0: 2020 2020 202a 2061 202a 6e6f 7465 2072       * a *note r
│ │ │ │ +000332f0: 696e 6720 656c 656d 656e 743a 2028 4d61  ing element: (Ma
│ │ │ │ +00033300: 6361 756c 6179 3244 6f63 2952 696e 6745  caulay2Doc)RingE
│ │ │ │ +00033310: 6c65 6d65 6e74 2c2c 2074 6865 2070 7573  lement,, the pus
│ │ │ │ +00033320: 682d 666f 7277 6172 6420 746f 0a20 2020  h-forward to.   
│ │ │ │ +00033330: 2020 2020 2074 6865 2043 686f 7720 7269       the Chow ri
│ │ │ │ +00033340: 6e67 206f 6620 245c 6d61 7468 6262 7b50  ng of $\mathbb{P
│ │ │ │ +00033350: 7d5e 6e24 206f 6620 7468 6520 4368 6572  }^n$ of the Cher
│ │ │ │ +00033360: 6e2d 5363 6877 6172 747a 2d4d 6163 5068  n-Schwartz-MacPh
│ │ │ │ +00033370: 6572 736f 6e20 636c 6173 730a 2020 2020  erson class.    
│ │ │ │ +00033380: 2020 2020 2463 5f7b 534d 7d28 5829 2420      $c_{SM}(X)$ 
│ │ │ │ +00033390: 6f66 2024 5824 2e20 496e 2070 6172 7469  of $X$. In parti
│ │ │ │ +000333a0: 6375 6c61 722c 2074 6865 2063 6f65 6666  cular, the coeff
│ │ │ │ +000333b0: 6963 6965 6e74 206f 6620 2448 5e6e 2420  icient of $H^n$ 
│ │ │ │ +000333c0: 6769 7665 7320 7468 650a 2020 2020 2020  gives the.      
│ │ │ │ +000333d0: 2020 4575 6c65 7220 6368 6172 6163 7465    Euler characte
│ │ │ │ +000333e0: 7269 7374 6963 206f 6620 7468 6520 7375  ristic of the su
│ │ │ │ +000333f0: 7070 6f72 7420 6f66 2024 5824 2c20 7768  pport of $X$, wh
│ │ │ │ +00033400: 6572 6520 2448 2420 6465 6e6f 7465 7320  ere $H$ denotes 
│ │ │ │ +00033410: 7468 650a 2020 2020 2020 2020 6879 7065  the.        hype
│ │ │ │ +00033420: 7270 6c61 6e65 2063 6c61 7373 2e0a 0a44  rplane class...D
│ │ │ │ +00033430: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00033440: 3d3d 3d3d 3d3d 0a0a 5468 6973 2069 7320  ======..This is 
│ │ │ │ +00033450: 616e 2065 7861 6d70 6c65 206f 6620 6170  an example of ap
│ │ │ │ +00033460: 706c 6963 6174 696f 6e20 6f66 2074 6865  plication of the
│ │ │ │ +00033470: 206d 6574 686f 6420 2a6e 6f74 6520 7072   method *note pr
│ │ │ │ +00033480: 6f6a 6563 7469 7665 4465 6772 6565 733a  ojectiveDegrees:
│ │ │ │ +00033490: 0a70 726f 6a65 6374 6976 6544 6567 7265  .projectiveDegre
│ │ │ │ +000334a0: 6573 2c2c 2064 7565 2074 6f20 7265 7375  es,, due to resu
│ │ │ │ +000334b0: 6c74 7320 7368 6f77 6e20 696e 2043 6f6d  lts shown in Com
│ │ │ │ +000334c0: 7075 7469 6e67 2063 6861 7261 6374 6572  puting character
│ │ │ │ +000334d0: 6973 7469 6320 636c 6173 7365 7320 6f66  istic classes of
│ │ │ │ +000334e0: 0a70 726f 6a65 6374 6976 6520 7363 6865  .projective sche
│ │ │ │ +000334f0: 6d65 7320 2873 6565 2068 7474 703a 2f2f  mes (see http://
│ │ │ │ +00033500: 7777 772e 7363 6965 6e63 6564 6972 6563  www.sciencedirec
│ │ │ │ +00033510: 742e 636f 6d2f 7363 6965 6e63 652f 6172  t.com/science/ar
│ │ │ │ +00033520: 7469 636c 652f 7069 692f 0a53 3037 3437  ticle/pii/.S0747
│ │ │ │ +00033530: 3731 3731 3032 3030 3038 3935 2029 2c20  717102000895 ), 
│ │ │ │ +00033540: 6279 2050 2e20 416c 7566 6669 2e20 5365  by P. Aluffi. Se
│ │ │ │ +00033550: 6520 616c 736f 2074 6865 2063 6f72 7265  e also the corre
│ │ │ │ +00033560: 7370 6f6e 6469 6e67 206d 6574 686f 6473  sponding methods
│ │ │ │ +00033570: 2069 6e20 7468 650a 7061 636b 6167 6573   in the.packages
│ │ │ │ +00033580: 2043 534d 2d41 2028 7365 6520 6874 7470   CSM-A (see http
│ │ │ │ +00033590: 3a2f 2f77 7777 2e6d 6174 682e 6673 752e  ://www.math.fsu.
│ │ │ │ +000335a0: 6564 752f 7e61 6c75 6666 692f 4353 4d2f  edu/~aluffi/CSM/
│ │ │ │ +000335b0: 4353 4d2e 6874 6d6c 2029 2c20 6279 2050  CSM.html ), by P
│ │ │ │ +000335c0: 2e0a 416c 7566 6669 2c20 616e 6420 4368  ..Aluffi, and Ch
│ │ │ │ +000335d0: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ +000335e0: 7365 7320 2873 6565 0a68 7474 703a 2f2f  ses (see.http://
│ │ │ │ +000335f0: 7777 772e 6d61 7468 2e75 6975 632e 6564  www.math.uiuc.ed
│ │ │ │ +00033600: 752f 4d61 6361 756c 6179 322f 646f 632f  u/Macaulay2/doc/
│ │ │ │ +00033610: 4d61 6361 756c 6179 322d 312e 3136 2f73  Macaulay2-1.16/s
│ │ │ │ +00033620: 6861 7265 2f64 6f63 2f4d 6163 6175 6c61  hare/doc/Macaula
│ │ │ │ +00033630: 7932 2f0a 4368 6172 6163 7465 7269 7374  y2/.Characterist
│ │ │ │ +00033640: 6963 436c 6173 7365 732f 6874 6d6c 2f20  icClasses/html/ 
│ │ │ │ +00033650: 292c 2062 7920 4d2e 2048 656c 6d65 7220  ), by M. Helmer 
│ │ │ │ +00033660: 616e 6420 432e 204a 6f73 742e 0a0a 496e  and C. Jost...In
│ │ │ │ +00033670: 2074 6865 2065 7861 6d70 6c65 2062 656c   the example bel
│ │ │ │ +00033680: 6f77 2c20 7765 2063 6f6d 7075 7465 2074  ow, we compute t
│ │ │ │ +00033690: 6865 2070 7573 682d 666f 7277 6172 6420  he push-forward 
│ │ │ │ +000336a0: 746f 2074 6865 2043 686f 7720 7269 6e67  to the Chow ring
│ │ │ │ +000336b0: 206f 660a 245c 6d61 7468 6262 7b50 7d5e   of.$\mathbb{P}^
│ │ │ │ +000336c0: 3424 206f 6620 7468 6520 4368 6572 6e2d  4$ of the Chern-
│ │ │ │ +000336d0: 5363 6877 6172 747a 2d4d 6163 5068 6572  Schwartz-MacPher
│ │ │ │ +000336e0: 736f 6e20 636c 6173 7320 6f66 2074 6865  son class of the
│ │ │ │ +000336f0: 2063 6f6e 6520 6f76 6572 2074 6865 0a74   cone over the.t
│ │ │ │ +00033700: 7769 7374 6564 2063 7562 6963 2063 7572  wisted cubic cur
│ │ │ │ +00033710: 7665 2c20 7573 696e 6720 626f 7468 2061  ve, using both a
│ │ │ │ +00033720: 2070 726f 6261 6269 6c69 7374 6963 2061   probabilistic a
│ │ │ │ +00033730: 6e64 2061 206e 6f6e 2d70 726f 6261 6269  nd a non-probabi
│ │ │ │ +00033740: 6c69 7374 6963 0a61 7070 726f 6163 682e  listic.approach.
│ │ │ │ +00033750: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  00033760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033780: 2d2d 2d2b 0a7c 6931 203a 2047 4628 355e  ---+.|i1 : GF(5^
│ │ │ │ -00033790: 3729 5b78 5f30 2e2e 785f 345d 2020 2020  7)[x_0..x_4]    
│ │ │ │ -000337a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000337b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000337c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033780: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +00033790: 2047 4628 355e 3729 5b78 5f30 2e2e 785f   GF(5^7)[x_0..x_
│ │ │ │ +000337a0: 345d 2020 2020 2020 2020 2020 2020 2020  4]              
│ │ │ │ +000337b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000337c0: 2020 7c0a 7c20 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 207c 0a7c 6f31 203d 2047 4620       |.|o1 = GF 
│ │ │ │ -00033800: 3738 3132 355b 7820 2e2e 7820 5d20 2020  78125[x ..x ]   
│ │ │ │ -00033810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033820: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00033830: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00033840: 3020 2020 3420 2020 2020 2020 2020 2020  0   4           
│ │ │ │ +000337f0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00033800: 203d 2047 4620 3738 3132 355b 7820 2e2e   = GF 78125[x ..
│ │ │ │ +00033810: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │ +00033820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033830: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00033840: 2020 2020 2020 3020 2020 3420 2020 2020        0   4     
│ │ │ │  00033850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033860: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00033870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000338a0: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69  |.|o1 : Polynomi
│ │ │ │ -000338b0: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ +000338a0: 2020 2020 2020 7c0a 7c6f 3120 3a20 506f        |.|o1 : Po
│ │ │ │ +000338b0: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │  000338c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000338d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000338e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000338d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000338e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000338f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033910: 2d2d 2b0a 7c69 3220 3a20 4320 3d20 6d69  --+.|i2 : C = mi
│ │ │ │ -00033920: 6e6f 7273 2832 2c6d 6174 7269 787b 7b78  nors(2,matrix{{x
│ │ │ │ -00033930: 5f30 2c78 5f31 2c78 5f32 7d2c 7b78 5f31  _0,x_1,x_2},{x_1
│ │ │ │ -00033940: 2c78 5f32 2c78 5f33 7d7d 297c 0a7c 2020  ,x_2,x_3}})|.|  
│ │ │ │ -00033950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033910: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +00033920: 4320 3d20 6d69 6e6f 7273 2832 2c6d 6174  C = minors(2,mat
│ │ │ │ +00033930: 7269 787b 7b78 5f30 2c78 5f31 2c78 5f32  rix{{x_0,x_1,x_2
│ │ │ │ +00033940: 7d2c 7b78 5f31 2c78 5f32 2c78 5f33 7d7d  },{x_1,x_2,x_3}}
│ │ │ │ +00033950: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │  00033960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033980: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00033990: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00033980: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00033990: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  000339a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000339b0: 2020 3220 2020 2020 2020 2020 207c 0a7c    2          |.|
│ │ │ │ -000339c0: 6f32 203d 2069 6465 616c 2028 2d20 7820  o2 = ideal (- x 
│ │ │ │ -000339d0: 202b 2078 2078 202c 202d 2078 2078 2020   + x x , - x x  
│ │ │ │ -000339e0: 2b20 7820 7820 2c20 2d20 7820 202b 2078  + x x , - x  + x
│ │ │ │ -000339f0: 2078 2029 2020 7c0a 7c20 2020 2020 2020   x )  |.|       
│ │ │ │ -00033a00: 2020 2020 2020 2020 3120 2020 2030 2032          1    0 2
│ │ │ │ -00033a10: 2020 2020 2031 2032 2020 2020 3020 3320       1 2    0 3 
│ │ │ │ -00033a20: 2020 2020 3220 2020 2031 2033 2020 207c      2    1 3   |
│ │ │ │ -00033a30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000339b0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000339c0: 2020 207c 0a7c 6f32 203d 2069 6465 616c     |.|o2 = ideal
│ │ │ │ +000339d0: 2028 2d20 7820 202b 2078 2078 202c 202d   (- x  + x x , -
│ │ │ │ +000339e0: 2078 2078 2020 2b20 7820 7820 2c20 2d20   x x  + x x , - 
│ │ │ │ +000339f0: 7820 202b 2078 2078 2029 2020 7c0a 7c20  x  + x x )  |.| 
│ │ │ │ +00033a00: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00033a10: 2020 2030 2032 2020 2020 2031 2032 2020     0 2     1 2  
│ │ │ │ +00033a20: 2020 3020 3320 2020 2020 3220 2020 2031    0 3     2    1
│ │ │ │ +00033a30: 2033 2020 207c 0a7c 2020 2020 2020 2020   3   |.|        
│ │ │ │  00033a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a60: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -00033a70: 4964 6561 6c20 6f66 2047 4620 3738 3132  Ideal of GF 7812
│ │ │ │ -00033a80: 355b 7820 2e2e 7820 5d20 2020 2020 2020  5[x ..x ]       
│ │ │ │ +00033a60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00033a70: 7c6f 3220 3a20 4964 6561 6c20 6f66 2047  |o2 : Ideal of G
│ │ │ │ +00033a80: 4620 3738 3132 355b 7820 2e2e 7820 5d20  F 78125[x ..x ] 
│ │ │ │  00033a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033aa0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00033ab0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -00033ac0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00033ad0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00033ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033aa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ac0: 2020 3020 2020 3420 2020 2020 2020 2020    0   4         
│ │ │ │ +00033ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ae0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00033af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033b10: 2d2d 2d2b 0a7c 6933 203a 2074 696d 6520  ---+.|i3 : time 
│ │ │ │ -00033b20: 4368 6572 6e53 6368 7761 7274 7a4d 6163  ChernSchwartzMac
│ │ │ │ -00033b30: 5068 6572 736f 6e20 4320 2020 2020 2020  Pherson C       
│ │ │ │ -00033b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00033b50: 2d2d 2075 7365 6420 312e 3539 3336 3173  -- used 1.59361s
│ │ │ │ -00033b60: 2028 6370 7529 3b20 302e 3935 3336 3737   (cpu); 0.953677
│ │ │ │ -00033b70: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00033b80: 6763 2920 207c 0a7c 2020 2020 2020 2020  gc)  |.|        
│ │ │ │ +00033b10: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +00033b20: 2074 696d 6520 4368 6572 6e53 6368 7761   time ChernSchwa
│ │ │ │ +00033b30: 7274 7a4d 6163 5068 6572 736f 6e20 4320  rtzMacPherson C 
│ │ │ │ +00033b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033b50: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e    |.| -- used 1.
│ │ │ │ +00033b60: 3637 3635 3373 2028 6370 7529 3b20 312e  67653s (cpu); 1.
│ │ │ │ +00033b70: 3031 3131 3373 2028 7468 7265 6164 293b  01113s (thread);
│ │ │ │ +00033b80: 2030 7320 2867 6329 2020 207c 0a7c 2020   0s (gc)   |.|  
│ │ │ │  00033b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033bb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00033bc0: 7c20 2020 2020 2020 3420 2020 2020 3320  |       4     3 
│ │ │ │ -00033bd0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00033bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033bc0: 2020 2020 7c0a 7c20 2020 2020 2020 3420      |.|       4 
│ │ │ │ +00033bd0: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │  00033be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033bf0: 2020 2020 2020 207c 0a7c 6f33 203d 2033         |.|o3 = 3
│ │ │ │ -00033c00: 4820 202b 2035 4820 202b 2033 4820 2020  H  + 5H  + 3H   
│ │ │ │ -00033c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033c00: 6f33 203d 2033 4820 202b 2035 4820 202b  o3 = 3H  + 5H  +
│ │ │ │ +00033c10: 2033 4820 2020 2020 2020 2020 2020 2020   3H             
│ │ │ │  00033c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033c30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00033c30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00033c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033c60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00033c70: 205a 5a5b 485d 2020 2020 2020 2020 2020   ZZ[H]          
│ │ │ │ +00033c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00033c70: 0a7c 2020 2020 205a 5a5b 485d 2020 2020  .|     ZZ[H]    
│ │ │ │  00033c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ca0: 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d 2d20    |.|o3 : ----- 
│ │ │ │ -00033cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ca0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +00033cb0: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │  00033cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033cd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00033ce0: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ +00033cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ce0: 207c 0a7c 2020 2020 2020 2020 3520 2020   |.|        5   
│ │ │ │  00033cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d10: 2020 2020 7c0a 7c20 2020 2020 2020 4820      |.|       H 
│ │ │ │ -00033d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00033d20: 2020 2020 4820 2020 2020 2020 2020 2020      H           
│ │ │ │  00033d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d40: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00033d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00033d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033d80: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7469  ------+.|i4 : ti
│ │ │ │ -00033d90: 6d65 2043 6865 726e 5363 6877 6172 747a  me ChernSchwartz
│ │ │ │ -00033da0: 4d61 6350 6865 7273 6f6e 2843 2c43 6572  MacPherson(C,Cer
│ │ │ │ -00033db0: 7469 6679 3d3e 7472 7565 2920 2020 207c  tify=>true)    |
│ │ │ │ -00033dc0: 0a7c 202d 2d20 7573 6564 2031 2e31 3431  .| -- used 1.141
│ │ │ │ -00033dd0: 3132 7320 2863 7075 293b 2030 2e38 3434  12s (cpu); 0.844
│ │ │ │ -00033de0: 3231 3973 2028 7468 7265 6164 293b 2030  219s (thread); 0
│ │ │ │ -00033df0: 7320 2867 6329 2020 7c0a 7c43 6572 7469  s (gc)  |.|Certi
│ │ │ │ -00033e00: 6679 3a20 6f75 7470 7574 2063 6572 7469  fy: output certi
│ │ │ │ -00033e10: 6669 6564 2120 2020 2020 2020 2020 2020  fied!           
│ │ │ │ +00033d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00033d90: 3420 3a20 7469 6d65 2043 6865 726e 5363  4 : time ChernSc
│ │ │ │ +00033da0: 6877 6172 747a 4d61 6350 6865 7273 6f6e  hwartzMacPherson
│ │ │ │ +00033db0: 2843 2c43 6572 7469 6679 3d3e 7472 7565  (C,Certify=>true
│ │ │ │ +00033dc0: 2920 2020 207c 0a7c 202d 2d20 7573 6564  )    |.| -- used
│ │ │ │ +00033dd0: 2031 2e32 3336 3732 7320 2863 7075 293b   1.23672s (cpu);
│ │ │ │ +00033de0: 2030 2e38 3935 3839 3773 2028 7468 7265   0.895897s (thre
│ │ │ │ +00033df0: 6164 293b 2030 7320 2867 6329 2020 7c0a  ad); 0s (gc)  |.
│ │ │ │ +00033e00: 7c43 6572 7469 6679 3a20 6f75 7470 7574  |Certify: output
│ │ │ │ +00033e10: 2063 6572 7469 6669 6564 2120 2020 2020   certified!     
│ │ │ │  00033e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033e30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00033e30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00033e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033e60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00033e70: 2020 2020 3420 2020 2020 3320 2020 2020      4     3     
│ │ │ │ -00033e80: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00033e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033e70: 7c0a 7c20 2020 2020 2020 3420 2020 2020  |.|       4     
│ │ │ │ +00033e80: 3320 2020 2020 3220 2020 2020 2020 2020  3     2         
│ │ │ │  00033e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ea0: 2020 207c 0a7c 6f34 203d 2033 4820 202b     |.|o4 = 3H  +
│ │ │ │ -00033eb0: 2035 4820 202b 2033 4820 2020 2020 2020   5H  + 3H       
│ │ │ │ +00033ea0: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ +00033eb0: 2033 4820 202b 2035 4820 202b 2033 4820   3H  + 5H  + 3H 
│ │ │ │  00033ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00033ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ee0: 2020 7c0a 7c20 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 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ -00033f20: 485d 2020 2020 2020 2020 2020 2020 2020  H]              
│ │ │ │ +00033f10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00033f20: 2020 205a 5a5b 485d 2020 2020 2020 2020     ZZ[H]        
│ │ │ │  00033f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00033f50: 7c6f 3420 3a20 2d2d 2d2d 2d20 2020 2020  |o4 : -----     
│ │ │ │ -00033f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033f50: 2020 2020 7c0a 7c6f 3420 3a20 2d2d 2d2d      |.|o4 : ----
│ │ │ │ +00033f60: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │  00033f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00033f90: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │ +00033f80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00033f90: 2020 2020 2020 2020 3520 2020 2020 2020          5       
│ │ │ │  00033fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033fc0: 7c0a 7c20 2020 2020 2020 4820 2020 2020  |.|       H     
│ │ │ │ -00033fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033fc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00033fd0: 4820 2020 2020 2020 2020 2020 2020 2020  H               
│ │ │ │  00033fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ff0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00034000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00034000: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00034010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034030: 2d2d 2b0a 7c69 3520 3a20 6f6f 203d 3d20  --+.|i5 : oo == 
│ │ │ │ -00034040: 6f6f 6f20 2020 2020 2020 2020 2020 2020  ooo             
│ │ │ │ +00034030: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +00034040: 6f6f 203d 3d20 6f6f 6f20 2020 2020 2020  oo == ooo       
│ │ │ │  00034050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034060: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00034070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034070: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00034080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340a0: 2020 2020 7c0a 7c6f 3520 3d20 7472 7565      |.|o5 = true
│ │ │ │ -000340b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000340a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +000340b0: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │  000340c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000340e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000340d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000340e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  000340f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034110: 2d2d 2d2d 2d2d 2b0a 0a49 6e20 7468 6520  ------+..In the 
│ │ │ │ -00034120: 6361 7365 2077 6865 6e20 7468 6520 696e  case when the in
│ │ │ │ -00034130: 7075 7420 6964 6561 6c20 4920 6465 6669  put ideal I defi
│ │ │ │ -00034140: 6e65 7320 6120 736d 6f6f 7468 2070 726f  nes a smooth pro
│ │ │ │ -00034150: 6a65 6374 6976 6520 7661 7269 6574 7920  jective variety 
│ │ │ │ -00034160: 2458 242c 2074 6865 0a70 7573 682d 666f  $X$, the.push-fo
│ │ │ │ -00034170: 7277 6172 6420 6f66 2024 635f 7b53 4d7d  rward of $c_{SM}
│ │ │ │ -00034180: 2858 2924 2063 616e 2062 6520 636f 6d70  (X)$ can be comp
│ │ │ │ -00034190: 7574 6564 206d 7563 6820 6d6f 7265 2065  uted much more e
│ │ │ │ -000341a0: 6666 6963 6965 6e74 6c79 2075 7369 6e67  fficiently using
│ │ │ │ -000341b0: 202a 6e6f 7465 0a53 6567 7265 436c 6173   *note.SegreClas
│ │ │ │ -000341c0: 733a 2053 6567 7265 436c 6173 732c 2e20  s: SegreClass,. 
│ │ │ │ -000341d0: 496e 6465 6564 2c20 696e 2074 6869 7320  Indeed, in this 
│ │ │ │ -000341e0: 6361 7365 2c20 2463 5f7b 534d 7d28 5829  case, $c_{SM}(X)
│ │ │ │ -000341f0: 2420 636f 696e 6369 6465 7320 7769 7468  $ coincides with
│ │ │ │ -00034200: 2074 6865 0a28 746f 7461 6c29 2043 6865   the.(total) Che
│ │ │ │ -00034210: 726e 2063 6c61 7373 206f 6620 7468 6520  rn class of the 
│ │ │ │ -00034220: 7461 6e67 656e 7420 6275 6e64 6c65 206f  tangent bundle o
│ │ │ │ -00034230: 6620 2458 2420 616e 6420 6361 6e20 6265  f $X$ and can be
│ │ │ │ -00034240: 206f 6274 6169 6e65 6420 6173 2066 6f6c   obtained as fol
│ │ │ │ -00034250: 6c6f 7773 0a28 696e 2067 656e 6572 616c  lows.(in general
│ │ │ │ -00034260: 2074 6865 206d 6574 686f 6420 6265 6c6f   the method belo
│ │ │ │ -00034270: 7720 6769 7665 7320 7468 6520 7075 7368  w gives the push
│ │ │ │ -00034280: 2d66 6f72 7761 7264 206f 6620 7468 6520  -forward of the 
│ │ │ │ -00034290: 736f 2d63 616c 6c65 640a 4368 6572 6e2d  so-called.Chern-
│ │ │ │ -000342a0: 4675 6c74 6f6e 2063 6c61 7373 292e 0a0a  Fulton class)...
│ │ │ │ -000342b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00034110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49  ------------+..I
│ │ │ │ +00034120: 6e20 7468 6520 6361 7365 2077 6865 6e20  n the case when 
│ │ │ │ +00034130: 7468 6520 696e 7075 7420 6964 6561 6c20  the input ideal 
│ │ │ │ +00034140: 4920 6465 6669 6e65 7320 6120 736d 6f6f  I defines a smoo
│ │ │ │ +00034150: 7468 2070 726f 6a65 6374 6976 6520 7661  th projective va
│ │ │ │ +00034160: 7269 6574 7920 2458 242c 2074 6865 0a70  riety $X$, the.p
│ │ │ │ +00034170: 7573 682d 666f 7277 6172 6420 6f66 2024  ush-forward of $
│ │ │ │ +00034180: 635f 7b53 4d7d 2858 2924 2063 616e 2062  c_{SM}(X)$ can b
│ │ │ │ +00034190: 6520 636f 6d70 7574 6564 206d 7563 6820  e computed much 
│ │ │ │ +000341a0: 6d6f 7265 2065 6666 6963 6965 6e74 6c79  more efficiently
│ │ │ │ +000341b0: 2075 7369 6e67 202a 6e6f 7465 0a53 6567   using *note.Seg
│ │ │ │ +000341c0: 7265 436c 6173 733a 2053 6567 7265 436c  reClass: SegreCl
│ │ │ │ +000341d0: 6173 732c 2e20 496e 6465 6564 2c20 696e  ass,. Indeed, in
│ │ │ │ +000341e0: 2074 6869 7320 6361 7365 2c20 2463 5f7b   this case, $c_{
│ │ │ │ +000341f0: 534d 7d28 5829 2420 636f 696e 6369 6465  SM}(X)$ coincide
│ │ │ │ +00034200: 7320 7769 7468 2074 6865 0a28 746f 7461  s with the.(tota
│ │ │ │ +00034210: 6c29 2043 6865 726e 2063 6c61 7373 206f  l) Chern class o
│ │ │ │ +00034220: 6620 7468 6520 7461 6e67 656e 7420 6275  f the tangent bu
│ │ │ │ +00034230: 6e64 6c65 206f 6620 2458 2420 616e 6420  ndle of $X$ and 
│ │ │ │ +00034240: 6361 6e20 6265 206f 6274 6169 6e65 6420  can be obtained 
│ │ │ │ +00034250: 6173 2066 6f6c 6c6f 7773 0a28 696e 2067  as follows.(in g
│ │ │ │ +00034260: 656e 6572 616c 2074 6865 206d 6574 686f  eneral the metho
│ │ │ │ +00034270: 6420 6265 6c6f 7720 6769 7665 7320 7468  d below gives th
│ │ │ │ +00034280: 6520 7075 7368 2d66 6f72 7761 7264 206f  e push-forward o
│ │ │ │ +00034290: 6620 7468 6520 736f 2d63 616c 6c65 640a  f the so-called.
│ │ │ │ +000342a0: 4368 6572 6e2d 4675 6c74 6f6e 2063 6c61  Chern-Fulton cla
│ │ │ │ +000342b0: 7373 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ss)...+---------
│ │ │ │  000342c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000342d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000342e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000342f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00034300: 7c69 3620 3a20 4368 6572 6e43 6c61 7373  |i6 : ChernClass
│ │ │ │ -00034310: 203d 206d 6574 686f 6428 4f70 7469 6f6e   = method(Option
│ │ │ │ -00034320: 733d 3e7b 4365 7274 6966 793d 3e66 616c  s=>{Certify=>fal
│ │ │ │ -00034330: 7365 7d29 3b20 2020 2020 2020 2020 2020  se});           
│ │ │ │ -00034340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034350: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000342f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034300: 2d2d 2d2d 2b0a 7c69 3620 3a20 4368 6572  ----+.|i6 : Cher
│ │ │ │ +00034310: 6e43 6c61 7373 203d 206d 6574 686f 6428  nClass = method(
│ │ │ │ +00034320: 4f70 7469 6f6e 733d 3e7b 4365 7274 6966  Options=>{Certif
│ │ │ │ +00034330: 793d 3e66 616c 7365 7d29 3b20 2020 2020  y=>false});     
│ │ │ │ +00034340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034350: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00034360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000343a0: 7c69 3720 3a20 4368 6572 6e43 6c61 7373  |i7 : ChernClass
│ │ │ │ -000343b0: 2028 4964 6561 6c29 203a 3d20 6f20 2d3e   (Ideal) := o ->
│ │ │ │ -000343c0: 2028 4929 202d 3e20 2820 2020 2020 2020   (I) -> (       
│ │ │ │ +00034390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000343a0: 2d2d 2d2d 2b0a 7c69 3720 3a20 4368 6572  ----+.|i7 : Cher
│ │ │ │ +000343b0: 6e43 6c61 7373 2028 4964 6561 6c29 203a  nClass (Ideal) :
│ │ │ │ +000343c0: 3d20 6f20 2d3e 2028 4929 202d 3e20 2820  = o -> (I) -> ( 
│ │ │ │  000343d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000343e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000343f0: 7c20 2020 2020 2020 2073 203a 3d20 5365  |        s := Se
│ │ │ │ -00034400: 6772 6543 6c61 7373 2849 2c43 6572 7469  greClass(I,Certi
│ │ │ │ -00034410: 6679 3d3e 6f2e 4365 7274 6966 7929 3b20  fy=>o.Certify); 
│ │ │ │ -00034420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034440: 7c20 2020 2020 2020 2073 2a28 312b 6669  |        s*(1+fi
│ │ │ │ -00034450: 7273 7420 6765 6e73 2072 696e 6720 7329  rst gens ring s)
│ │ │ │ -00034460: 5e28 6e75 6d67 656e 7320 7269 6e67 2049  ^(numgens ring I
│ │ │ │ -00034470: 2929 3b20 2020 2020 2020 2020 2020 2020  ));             
│ │ │ │ -00034480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034490: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000343e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000343f0: 2020 2020 7c0a 7c20 2020 2020 2020 2073      |.|        s
│ │ │ │ +00034400: 203a 3d20 5365 6772 6543 6c61 7373 2849   := SegreClass(I
│ │ │ │ +00034410: 2c43 6572 7469 6679 3d3e 6f2e 4365 7274  ,Certify=>o.Cert
│ │ │ │ +00034420: 6966 7929 3b20 2020 2020 2020 2020 2020  ify);           
│ │ │ │ +00034430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034440: 2020 2020 7c0a 7c20 2020 2020 2020 2073      |.|        s
│ │ │ │ +00034450: 2a28 312b 6669 7273 7420 6765 6e73 2072  *(1+first gens r
│ │ │ │ +00034460: 696e 6720 7329 5e28 6e75 6d67 656e 7320  ing s)^(numgens 
│ │ │ │ +00034470: 7269 6e67 2049 2929 3b20 2020 2020 2020  ring I));       
│ │ │ │ +00034480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034490: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000344a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000344b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000344c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000344d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000344e0: 7c69 3820 3a20 2d2d 2065 7861 6d70 6c65  |i8 : -- example
│ │ │ │ -000344f0: 3a20 4368 6572 6e20 636c 6173 7320 6f66  : Chern class of
│ │ │ │ -00034500: 2047 2831 2c34 2920 2020 2020 2020 2020   G(1,4)         
│ │ │ │ +000344d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000344e0: 2d2d 2d2d 2b0a 7c69 3820 3a20 2d2d 2065  ----+.|i8 : -- e
│ │ │ │ +000344f0: 7861 6d70 6c65 3a20 4368 6572 6e20 636c  xample: Chern cl
│ │ │ │ +00034500: 6173 7320 6f66 2047 2831 2c34 2920 2020  ass of G(1,4)   
│ │ │ │  00034510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034530: 7c20 2020 2020 4720 3d20 4772 6173 736d  |     G = Grassm
│ │ │ │ -00034540: 616e 6e69 616e 2831 2c34 2c43 6f65 6666  annian(1,4,Coeff
│ │ │ │ -00034550: 6963 6965 6e74 5269 6e67 3d3e 5a5a 2f31  icientRing=>ZZ/1
│ │ │ │ -00034560: 3930 3138 3129 2020 2020 2020 2020 2020  90181)          
│ │ │ │ -00034570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034580: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034530: 2020 2020 7c0a 7c20 2020 2020 4720 3d20      |.|     G = 
│ │ │ │ +00034540: 4772 6173 736d 616e 6e69 616e 2831 2c34  Grassmannian(1,4
│ │ │ │ +00034550: 2c43 6f65 6666 6963 6965 6e74 5269 6e67  ,CoefficientRing
│ │ │ │ +00034560: 3d3e 5a5a 2f31 3930 3138 3129 2020 2020  =>ZZ/190181)    
│ │ │ │ +00034570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034580: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00034590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000345a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000345b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000345c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000345d0: 7c6f 3820 3d20 6964 6561 6c20 2870 2020  |o8 = ideal (p  
│ │ │ │ -000345e0: 2070 2020 2020 2d20 7020 2020 7020 2020   p    - p   p   
│ │ │ │ -000345f0: 202b 2070 2020 2070 2020 202c 2070 2020   + p   p   , p  
│ │ │ │ -00034600: 2070 2020 2020 2d20 7020 2020 7020 2020   p    - p   p   
│ │ │ │ -00034610: 202b 2070 2020 2070 2020 202c 2020 7c0a   + p   p   ,  |.
│ │ │ │ -00034620: 7c20 2020 2020 2020 2020 2020 2020 322c  |             2,
│ │ │ │ -00034630: 3320 312c 3420 2020 2031 2c33 2032 2c34  3 1,4    1,3 2,4
│ │ │ │ -00034640: 2020 2020 312c 3220 332c 3420 2020 322c      1,2 3,4   2,
│ │ │ │ -00034650: 3320 302c 3420 2020 2030 2c33 2032 2c34  3 0,4    0,3 2,4
│ │ │ │ -00034660: 2020 2020 302c 3220 332c 3420 2020 7c0a      0,2 3,4   |.
│ │ │ │ -00034670: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +000345c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000345d0: 2020 2020 7c0a 7c6f 3820 3d20 6964 6561      |.|o8 = idea
│ │ │ │ +000345e0: 6c20 2870 2020 2070 2020 2020 2d20 7020  l (p   p    - p 
│ │ │ │ +000345f0: 2020 7020 2020 202b 2070 2020 2070 2020    p    + p   p  
│ │ │ │ +00034600: 202c 2070 2020 2070 2020 2020 2d20 7020   , p   p    - p 
│ │ │ │ +00034610: 2020 7020 2020 202b 2070 2020 2070 2020    p    + p   p  
│ │ │ │ +00034620: 202c 2020 7c0a 7c20 2020 2020 2020 2020   ,  |.|         
│ │ │ │ +00034630: 2020 2020 322c 3320 312c 3420 2020 2031      2,3 1,4    1
│ │ │ │ +00034640: 2c33 2032 2c34 2020 2020 312c 3220 332c  ,3 2,4    1,2 3,
│ │ │ │ +00034650: 3420 2020 322c 3320 302c 3420 2020 2030  4   2,3 0,4    0
│ │ │ │ +00034660: 2c33 2032 2c34 2020 2020 302c 3220 332c  ,3 2,4    0,2 3,
│ │ │ │ +00034670: 3420 2020 7c0a 7c20 2020 2020 2d2d 2d2d  4   |.|     ----
│ │ │ │  00034680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000346a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000346b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -000346c0: 7c20 2020 2020 7020 2020 7020 2020 202d  |     p   p    -
│ │ │ │ -000346d0: 2070 2020 2070 2020 2020 2b20 7020 2020   p   p    + p   
│ │ │ │ -000346e0: 7020 2020 2c20 7020 2020 7020 2020 202d  p   , p   p    -
│ │ │ │ -000346f0: 2070 2020 2070 2020 2020 2b20 7020 2020   p   p    + p   
│ │ │ │ -00034700: 7020 2020 2c20 7020 2020 7020 2020 7c0a  p   , p   p   |.
│ │ │ │ -00034710: 7c20 2020 2020 2031 2c33 2030 2c34 2020  |      1,3 0,4  
│ │ │ │ -00034720: 2020 302c 3320 312c 3420 2020 2030 2c31    0,3 1,4    0,1
│ │ │ │ -00034730: 2033 2c34 2020 2031 2c32 2030 2c34 2020   3,4   1,2 0,4  
│ │ │ │ -00034740: 2020 302c 3220 312c 3420 2020 2030 2c31    0,2 1,4    0,1
│ │ │ │ -00034750: 2032 2c34 2020 2031 2c32 2030 2c33 7c0a   2,4   1,2 0,3|.
│ │ │ │ -00034760: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +000346b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000346c0: 2d2d 2d2d 7c0a 7c20 2020 2020 7020 2020  ----|.|     p   
│ │ │ │ +000346d0: 7020 2020 202d 2070 2020 2070 2020 2020  p    - p   p    
│ │ │ │ +000346e0: 2b20 7020 2020 7020 2020 2c20 7020 2020  + p   p   , p   
│ │ │ │ +000346f0: 7020 2020 202d 2070 2020 2070 2020 2020  p    - p   p    
│ │ │ │ +00034700: 2b20 7020 2020 7020 2020 2c20 7020 2020  + p   p   , p   
│ │ │ │ +00034710: 7020 2020 7c0a 7c20 2020 2020 2031 2c33  p   |.|      1,3
│ │ │ │ +00034720: 2030 2c34 2020 2020 302c 3320 312c 3420   0,4    0,3 1,4 
│ │ │ │ +00034730: 2020 2030 2c31 2033 2c34 2020 2031 2c32     0,1 3,4   1,2
│ │ │ │ +00034740: 2030 2c34 2020 2020 302c 3220 312c 3420   0,4    0,2 1,4 
│ │ │ │ +00034750: 2020 2030 2c31 2032 2c34 2020 2031 2c32     0,1 2,4   1,2
│ │ │ │ +00034760: 2030 2c33 7c0a 7c20 2020 2020 2d2d 2d2d   0,3|.|     ----
│ │ │ │  00034770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000347a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -000347b0: 7c20 2020 2020 2d20 7020 2020 7020 2020  |     - p   p   
│ │ │ │ -000347c0: 202b 2070 2020 2070 2020 2029 2020 2020   + p   p   )    
│ │ │ │ -000347d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000347a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000347b0: 2d2d 2d2d 7c0a 7c20 2020 2020 2d20 7020  ----|.|     - p 
│ │ │ │ +000347c0: 2020 7020 2020 202b 2070 2020 2070 2020    p    + p   p  
│ │ │ │ +000347d0: 2029 2020 2020 2020 2020 2020 2020 2020   )              
│ │ │ │  000347e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000347f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034800: 7c20 2020 2020 2020 2030 2c32 2031 2c33  |        0,2 1,3
│ │ │ │ -00034810: 2020 2020 302c 3120 322c 3320 2020 2020      0,1 2,3     
│ │ │ │ -00034820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000347f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034800: 2020 2020 7c0a 7c20 2020 2020 2020 2030      |.|        0
│ │ │ │ +00034810: 2c32 2031 2c33 2020 2020 302c 3120 322c  ,2 1,3    0,1 2,
│ │ │ │ +00034820: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  00034830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034850: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00034860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000348a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000348b0: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +00034890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000348a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000348b0: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │  000348c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000348d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000348e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000348f0: 7c6f 3820 3a20 4964 6561 6c20 6f66 202d  |o8 : Ideal of -
│ │ │ │ -00034900: 2d2d 2d2d 2d5b 7020 2020 2e2e 7020 2020  -----[p   ..p   
│ │ │ │ -00034910: 2c20 7020 2020 2c20 7020 2020 2c20 7020  , p   , p   , p 
│ │ │ │ +000348e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000348f0: 2020 2020 7c0a 7c6f 3820 3a20 4964 6561      |.|o8 : Idea
│ │ │ │ +00034900: 6c20 6f66 202d 2d2d 2d2d 2d5b 7020 2020  l of ------[p   
│ │ │ │ +00034910: 2e2e 7020 2020 2c20 7020 2020 2c20 7020  ..p   , p   , p 
│ │ │ │  00034920: 2020 2c20 7020 2020 2c20 7020 2020 2c20    , p   , p   , 
│ │ │ │ -00034930: 7020 2020 2c20 7020 2020 2c20 7020 7c0a  p   , p   , p |.
│ │ │ │ -00034940: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ -00034950: 3930 3138 3120 2030 2c31 2020 2030 2c32  90181  0,1   0,2
│ │ │ │ -00034960: 2020 2031 2c32 2020 2030 2c33 2020 2031     1,2   0,3   1
│ │ │ │ -00034970: 2c33 2020 2032 2c33 2020 2030 2c34 2020  ,3   2,3   0,4  
│ │ │ │ -00034980: 2031 2c34 2020 2032 2c34 2020 2033 7c0a   1,4   2,4   3|.
│ │ │ │ -00034990: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00034930: 7020 2020 2c20 7020 2020 2c20 7020 2020  p   , p   , p   
│ │ │ │ +00034940: 2c20 7020 7c0a 7c20 2020 2020 2020 2020  , p |.|         
│ │ │ │ +00034950: 2020 2020 2031 3930 3138 3120 2030 2c31       190181  0,1
│ │ │ │ +00034960: 2020 2030 2c32 2020 2031 2c32 2020 2030     0,2   1,2   0
│ │ │ │ +00034970: 2c33 2020 2031 2c33 2020 2032 2c33 2020  ,3   1,3   2,3  
│ │ │ │ +00034980: 2030 2c34 2020 2031 2c34 2020 2032 2c34   0,4   1,4   2,4
│ │ │ │ +00034990: 2020 2033 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d     3|.|---------
│ │ │ │  000349a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000349b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000349c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000349d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -000349e0: 7c20 205d 2020 2020 2020 2020 2020 2020  |  ]            
│ │ │ │ +000349d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000349e0: 2d2d 2d2d 7c0a 7c20 205d 2020 2020 2020  ----|.|  ]      
│ │ │ │  000349f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034a20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034a30: 7c2c 3420 2020 2020 2020 2020 2020 2020  |,4             
│ │ │ │ +00034a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a30: 2020 2020 7c0a 7c2c 3420 2020 2020 2020      |.|,4       
│ │ │ │  00034a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034a70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034a80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00034a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00034a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00034ad0: 7c69 3920 3a20 7469 6d65 2043 6865 726e  |i9 : time Chern
│ │ │ │ -00034ae0: 436c 6173 7320 4720 2020 2020 2020 2020  Class G         
│ │ │ │ +00034ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034ad0: 2d2d 2d2d 2b0a 7c69 3920 3a20 7469 6d65  ----+.|i9 : time
│ │ │ │ +00034ae0: 2043 6865 726e 436c 6173 7320 4720 2020   ChernClass G   
│ │ │ │  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 7c0a                |.
│ │ │ │ -00034b20: 7c20 2d2d 2075 7365 6420 302e 3330 3336  | -- used 0.3036
│ │ │ │ -00034b30: 3735 7320 2863 7075 293b 2030 2e31 3439  75s (cpu); 0.149
│ │ │ │ -00034b40: 3831 3773 2028 7468 7265 6164 293b 2030  817s (thread); 0
│ │ │ │ -00034b50: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ -00034b60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034b70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034b20: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00034b30: 302e 3233 3539 3335 7320 2863 7075 293b  0.235935s (cpu);
│ │ │ │ +00034b40: 2030 2e31 3635 3833 7320 2874 6872 6561   0.16583s (threa
│ │ │ │ +00034b50: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00034b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034b70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00034b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034bb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034bc0: 7c20 2020 2020 2020 2039 2020 2020 2020  |        9      
│ │ │ │ -00034bd0: 3820 2020 2020 2037 2020 2020 2020 3620  8      7      6 
│ │ │ │ -00034be0: 2020 2020 2035 2020 2020 2020 3420 2020       5      4   
│ │ │ │ -00034bf0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00034c00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034c10: 7c6f 3920 3d20 3130 4820 202b 2033 3048  |o9 = 10H  + 30H
│ │ │ │ -00034c20: 2020 2b20 3630 4820 202b 2037 3548 2020    + 60H  + 75H  
│ │ │ │ -00034c30: 2b20 3537 4820 202b 2032 3548 2020 2b20  + 57H  + 25H  + 
│ │ │ │ -00034c40: 3548 2020 2020 2020 2020 2020 2020 2020  5H              
│ │ │ │ -00034c50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034c60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034bc0: 2020 2020 7c0a 7c20 2020 2020 2020 2039      |.|        9
│ │ │ │ +00034bd0: 2020 2020 2020 3820 2020 2020 2037 2020        8      7  
│ │ │ │ +00034be0: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ +00034bf0: 2020 3420 2020 2020 3320 2020 2020 2020    4     3       
│ │ │ │ +00034c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034c10: 2020 2020 7c0a 7c6f 3920 3d20 3130 4820      |.|o9 = 10H 
│ │ │ │ +00034c20: 202b 2033 3048 2020 2b20 3630 4820 202b   + 30H  + 60H  +
│ │ │ │ +00034c30: 2037 3548 2020 2b20 3537 4820 202b 2032   75H  + 57H  + 2
│ │ │ │ +00034c40: 3548 2020 2b20 3548 2020 2020 2020 2020  5H  + 5H        
│ │ │ │ +00034c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034c60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00034c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034cb0: 7c20 2020 2020 5a5a 5b48 5d20 2020 2020  |     ZZ[H]     
│ │ │ │ -00034cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034cb0: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b48      |.|     ZZ[H
│ │ │ │ +00034cc0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  00034cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034cf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034d00: 7c6f 3920 3a20 2d2d 2d2d 2d20 2020 2020  |o9 : -----     
│ │ │ │ -00034d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034d00: 2020 2020 7c0a 7c6f 3920 3a20 2d2d 2d2d      |.|o9 : ----
│ │ │ │ +00034d10: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │  00034d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034d50: 7c20 2020 2020 2020 3130 2020 2020 2020  |       10      
│ │ │ │ +00034d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034d50: 2020 2020 7c0a 7c20 2020 2020 2020 3130      |.|       10
│ │ │ │  00034d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034da0: 7c20 2020 2020 2048 2020 2020 2020 2020  |      H        
│ │ │ │ +00034d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034da0: 2020 2020 7c0a 7c20 2020 2020 2048 2020      |.|      H  
│ │ │ │  00034db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034de0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034df0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00034de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034df0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00034e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00034e40: 7c69 3130 203a 2074 696d 6520 4368 6572  |i10 : time Cher
│ │ │ │ -00034e50: 6e43 6c61 7373 2847 2c43 6572 7469 6679  nClass(G,Certify
│ │ │ │ -00034e60: 3d3e 7472 7565 2920 2020 2020 2020 2020  =>true)         
│ │ │ │ +00034e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034e40: 2d2d 2d2d 2b0a 7c69 3130 203a 2074 696d  ----+.|i10 : tim
│ │ │ │ +00034e50: 6520 4368 6572 6e43 6c61 7373 2847 2c43  e ChernClass(G,C
│ │ │ │ +00034e60: 6572 7469 6679 3d3e 7472 7565 2920 2020  ertify=>true)   
│ │ │ │  00034e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034e90: 7c20 2d2d 2075 7365 6420 302e 3030 3732  | -- used 0.0072
│ │ │ │ -00034ea0: 3734 3937 7320 2863 7075 293b 2030 2e30  7497s (cpu); 0.0
│ │ │ │ -00034eb0: 3130 3039 3635 7320 2874 6872 6561 6429  100965s (thread)
│ │ │ │ -00034ec0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ -00034ed0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034ee0: 7c43 6572 7469 6679 3a20 6f75 7470 7574  |Certify: output
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│ │ │ │ -00034f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034e90: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00034ea0: 302e 3034 3738 3537 3873 2028 6370 7529  0.0478578s (cpu)
│ │ │ │ +00034eb0: 3b20 302e 3031 3737 3237 3973 2028 7468  ; 0.0177279s (th
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│ │ │ │ +00034ee0: 2020 2020 7c0a 7c43 6572 7469 6679 3a20      |.|Certify: 
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│ │ │ │  00034f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034f20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034f30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034f30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00034f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034f80: 7c20 2020 2020 2020 2020 3920 2020 2020  |         9     
│ │ │ │ -00034f90: 2038 2020 2020 2020 3720 2020 2020 2036   8      7      6
│ │ │ │ -00034fa0: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ -00034fb0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00034fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034fd0: 7c6f 3130 203d 2031 3048 2020 2b20 3330  |o10 = 10H  + 30
│ │ │ │ -00034fe0: 4820 202b 2036 3048 2020 2b20 3735 4820  H  + 60H  + 75H 
│ │ │ │ -00034ff0: 202b 2035 3748 2020 2b20 3235 4820 202b   + 57H  + 25H  +
│ │ │ │ -00035000: 2035 4820 2020 2020 2020 2020 2020 2020   5H             
│ │ │ │ -00035010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00035020: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034f80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00034f90: 3920 2020 2020 2038 2020 2020 2020 3720  9      8      7 
│ │ │ │ +00034fa0: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00034fb0: 2020 2034 2020 2020 2033 2020 2020 2020     4     3      
│ │ │ │ +00034fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034fd0: 2020 2020 7c0a 7c6f 3130 203d 2031 3048      |.|o10 = 10H
│ │ │ │ +00034fe0: 2020 2b20 3330 4820 202b 2036 3048 2020    + 30H  + 60H  
│ │ │ │ +00034ff0: 2b20 3735 4820 202b 2035 3748 2020 2b20  + 75H  + 57H  + 
│ │ │ │ +00035000: 3235 4820 202b 2035 4820 2020 2020 2020  25H  + 5H       
│ │ │ │ +00035010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035020: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00035030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035060: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00035070: 7c20 2020 2020 205a 5a5b 485d 2020 2020  |      ZZ[H]    
│ │ │ │ -00035080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035070: 2020 2020 7c0a 7c20 2020 2020 205a 5a5b      |.|      ZZ[
│ │ │ │ +00035080: 485d 2020 2020 2020 2020 2020 2020 2020  H]              
│ │ │ │  00035090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000350b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000350c0: 7c6f 3130 203a 202d 2d2d 2d2d 2020 2020  |o10 : -----    
│ │ │ │ -000350d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000350b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000350c0: 2020 2020 7c0a 7c6f 3130 203a 202d 2d2d      |.|o10 : ---
│ │ │ │ +000350d0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  000350e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00035110: 7c20 2020 2020 2020 2031 3020 2020 2020  |        10     
│ │ │ │ -00035120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035110: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │ +00035120: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  00035130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00035160: 7c20 2020 2020 2020 4820 2020 2020 2020  |       H       
│ │ │ │ +00035150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035160: 2020 2020 7c0a 7c20 2020 2020 2020 4820      |.|       H 
│ │ │ │  00035170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000351a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000351b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000351a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000351b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000351c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000351d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000351e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000351f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00035200: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -00035210: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 5365  ==..  * *note Se
│ │ │ │ -00035220: 6772 6543 6c61 7373 3a20 5365 6772 6543  greClass: SegreC
│ │ │ │ -00035230: 6c61 7373 2c20 2d2d 2053 6567 7265 2063  lass, -- Segre c
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│ │ │ │ -00035250: 2073 7562 7363 6865 6d65 206f 6620 610a   subscheme of a.
│ │ │ │ -00035260: 2020 2020 7072 6f6a 6563 7469 7665 2076      projective v
│ │ │ │ -00035270: 6172 6965 7479 0a20 202a 202a 6e6f 7465  ariety.  * *note
│ │ │ │ -00035280: 2045 756c 6572 4368 6172 6163 7465 7269   EulerCharacteri
│ │ │ │ -00035290: 7374 6963 3a20 4575 6c65 7243 6861 7261  stic: EulerChara
│ │ │ │ -000352a0: 6374 6572 6973 7469 632c 202d 2d20 746f  cteristic, -- to
│ │ │ │ -000352b0: 706f 6c6f 6769 6361 6c20 4575 6c65 720a  pological Euler.
│ │ │ │ -000352c0: 2020 2020 6368 6172 6163 7465 7269 7374      characterist
│ │ │ │ -000352d0: 6963 206f 6620 6120 2873 6d6f 6f74 6829  ic of a (smooth)
│ │ │ │ -000352e0: 2070 726f 6a65 6374 6976 6520 7661 7269   projective vari
│ │ │ │ -000352f0: 6574 790a 2020 2a20 2a6e 6f74 6520 6575  ety.  * *note eu
│ │ │ │ -00035300: 6c65 7228 5072 6f6a 6563 7469 7665 5661  ler(ProjectiveVa
│ │ │ │ -00035310: 7269 6574 7929 3a20 2856 6172 6965 7469  riety): (Varieti
│ │ │ │ -00035320: 6573 2965 756c 6572 5f6c 7050 726f 6a65  es)euler_lpProje
│ │ │ │ -00035330: 6374 6976 6556 6172 6965 7479 5f72 702c  ctiveVariety_rp,
│ │ │ │ -00035340: 202d 2d0a 2020 2020 746f 706f 6c6f 6769   --.    topologi
│ │ │ │ -00035350: 6361 6c20 4575 6c65 7220 6368 6172 6163  cal Euler charac
│ │ │ │ -00035360: 7465 7269 7374 6963 206f 6620 6120 2873  teristic of a (s
│ │ │ │ -00035370: 6d6f 6f74 6829 2070 726f 6a65 6374 6976  mooth) projectiv
│ │ │ │ -00035380: 6520 7661 7269 6574 790a 0a57 6179 7320  e variety..Ways 
│ │ │ │ -00035390: 746f 2075 7365 2043 6865 726e 5363 6877  to use ChernSchw
│ │ │ │ -000353a0: 6172 747a 4d61 6350 6865 7273 6f6e 3a0a  artzMacPherson:.
│ │ │ │ -000353b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000351f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00035200: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ +00035210: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00035220: 6f74 6520 5365 6772 6543 6c61 7373 3a20  ote SegreClass: 
│ │ │ │ +00035230: 5365 6772 6543 6c61 7373 2c20 2d2d 2053  SegreClass, -- S
│ │ │ │ +00035240: 6567 7265 2063 6c61 7373 206f 6620 6120  egre class of a 
│ │ │ │ +00035250: 636c 6f73 6564 2073 7562 7363 6865 6d65  closed subscheme
│ │ │ │ +00035260: 206f 6620 610a 2020 2020 7072 6f6a 6563   of a.    projec
│ │ │ │ +00035270: 7469 7665 2076 6172 6965 7479 0a20 202a  tive variety.  *
│ │ │ │ +00035280: 202a 6e6f 7465 2045 756c 6572 4368 6172   *note EulerChar
│ │ │ │ +00035290: 6163 7465 7269 7374 6963 3a20 4575 6c65  acteristic: Eule
│ │ │ │ +000352a0: 7243 6861 7261 6374 6572 6973 7469 632c  rCharacteristic,
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│ │ │ │ +000352c0: 4575 6c65 720a 2020 2020 6368 6172 6163  Euler.    charac
│ │ │ │ +000352d0: 7465 7269 7374 6963 206f 6620 6120 2873  teristic of a (s
│ │ │ │ +000352e0: 6d6f 6f74 6829 2070 726f 6a65 6374 6976  mooth) projectiv
│ │ │ │ +000352f0: 6520 7661 7269 6574 790a 2020 2a20 2a6e  e variety.  * *n
│ │ │ │ +00035300: 6f74 6520 6575 6c65 7228 5072 6f6a 6563  ote euler(Projec
│ │ │ │ +00035310: 7469 7665 5661 7269 6574 7929 3a20 2856  tiveVariety): (V
│ │ │ │ +00035320: 6172 6965 7469 6573 2965 756c 6572 5f6c  arieties)euler_l
│ │ │ │ +00035330: 7050 726f 6a65 6374 6976 6556 6172 6965  pProjectiveVarie
│ │ │ │ +00035340: 7479 5f72 702c 202d 2d0a 2020 2020 746f  ty_rp, --.    to
│ │ │ │ +00035350: 706f 6c6f 6769 6361 6c20 4575 6c65 7220  pological Euler 
│ │ │ │ +00035360: 6368 6172 6163 7465 7269 7374 6963 206f  characteristic o
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│ │ │ │ +00035380: 6a65 6374 6976 6520 7661 7269 6574 790a  jective variety.
│ │ │ │ +00035390: 0a57 6179 7320 746f 2075 7365 2043 6865  .Ways to use Che
│ │ │ │ +000353a0: 726e 5363 6877 6172 747a 4d61 6350 6865  rnSchwartzMacPhe
│ │ │ │ +000353b0: 7273 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  rson:.==========
│ │ │ │  000353c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00035410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00035420: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
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│ │ │ │ -00035450: 726e 5363 6877 6172 747a 4d61 6350 6865  rnSchwartzMacPhe
│ │ │ │ -00035460: 7273 6f6e 2c20 6973 2061 202a 6e6f 7465  rson, is a *note
│ │ │ │ -00035470: 0a6d 6574 686f 6420 6675 6e63 7469 6f6e  .method function
│ │ │ │ -00035480: 2077 6974 6820 6f70 7469 6f6e 733a 2028   with options: (
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│ │ │ │ -000354a0: 686f 6446 756e 6374 696f 6e57 6974 684f  hodFunctionWithO
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│ │ │ │ -00035580: 2a6e 6f74 6520 6170 7072 6f78 696d 6174  *note approximat
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│ │ │ │ -000355a0: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
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│ │ │ │ -00035620: 702e 2049 6e20 6d6f 7374 2063 6173 6573  p. In most cases
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│ │ │ │ -00035640: 2074 6865 206f 7074 696d 616c 2076 616c   the optimal val
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│ │ │ │ -00035660: 6173 2069 6e20 7468 6520 666f 6c6c 6f77  as in the follow
│ │ │ │ -00035670: 696e 6720 6578 616d 706c 652e 0a0a 0a2b  ing example....+
│ │ │ │ -00035680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000353d0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +000353e0: 2243 6865 726e 5363 6877 6172 747a 4d61  "ChernSchwartzMa
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│ │ │ │ +00035410: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +00035420: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
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│ │ │ │ +00035440: 6368 7761 7274 7a4d 6163 5068 6572 736f  chwartzMacPherso
│ │ │ │ +00035450: 6e3a 2043 6865 726e 5363 6877 6172 747a  n: ChernSchwartz
│ │ │ │ +00035460: 4d61 6350 6865 7273 6f6e 2c20 6973 2061  MacPherson, is a
│ │ │ │ +00035470: 202a 6e6f 7465 0a6d 6574 686f 6420 6675   *note.method fu
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│ │ │ │ +00035590: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
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│ │ │ │ +00035eb0: 6420 436f 6469 6d42 7349 6e76 3a0a 3d3d  d CodimBsInv:.==
│ │ │ │  00035ec0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00035ed0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00035ee0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00035ef0: 2261 7070 726f 7869 6d61 7465 496e 7665  "approximateInve
│ │ │ │ -00035f00: 7273 654d 6170 282e 2e2e 2c43 6f64 696d  rseMap(...,Codim
│ │ │ │ -00035f10: 4273 496e 763d 3e2e 2e2e 2922 0a0a 466f  BsInv=>...)"..Fo
│ │ │ │ -00035f20: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00035f30: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00035f40: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00035f50: 2a6e 6f74 6520 436f 6469 6d42 7349 6e76  *note CodimBsInv
│ │ │ │ -00035f60: 3a20 436f 6469 6d42 7349 6e76 2c20 6973  : CodimBsInv, is
│ │ │ │ -00035f70: 2061 202a 6e6f 7465 2073 796d 626f 6c3a   a *note symbol:
│ │ │ │ -00035f80: 0a28 4d61 6361 756c 6179 3244 6f63 2953  .(Macaulay2Doc)S
│ │ │ │ -00035f90: 796d 626f 6c2c 2e0a 1f0a 4669 6c65 3a20  ymbol,....File: 
│ │ │ │ -00035fa0: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ -00035fb0: 6465 3a20 636f 6566 6669 6369 656e 7452  de: coefficientR
│ │ │ │ -00035fc0: 696e 675f 6c70 5261 7469 6f6e 616c 4d61  ing_lpRationalMa
│ │ │ │ -00035fd0: 705f 7270 2c20 4e65 7874 3a20 636f 6566  p_rp, Next: coef
│ │ │ │ -00035fe0: 6669 6369 656e 7473 5f6c 7052 6174 696f  ficients_lpRatio
│ │ │ │ -00035ff0: 6e61 6c4d 6170 5f72 702c 2050 7265 763a  nalMap_rp, Prev:
│ │ │ │ -00036000: 2043 6f64 696d 4273 496e 762c 2055 703a   CodimBsInv, Up:
│ │ │ │ -00036010: 2054 6f70 0a0a 636f 6566 6669 6369 656e   Top..coefficien
│ │ │ │ -00036020: 7452 696e 6728 5261 7469 6f6e 616c 4d61  tRing(RationalMa
│ │ │ │ -00036030: 7029 202d 2d20 636f 6566 6669 6369 656e  p) -- coefficien
│ │ │ │ -00036040: 7420 7269 6e67 206f 6620 6120 7261 7469  t ring of a rati
│ │ │ │ -00036050: 6f6e 616c 206d 6170 0a2a 2a2a 2a2a 2a2a  onal map.*******
│ │ │ │ +00035ee0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00035ef0: 0a0a 2020 2a20 2261 7070 726f 7869 6d61  ..  * "approxima
│ │ │ │ +00035f00: 7465 496e 7665 7273 654d 6170 282e 2e2e  teInverseMap(...
│ │ │ │ +00035f10: 2c43 6f64 696d 4273 496e 763d 3e2e 2e2e  ,CodimBsInv=>...
│ │ │ │ +00035f20: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00035f30: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00035f40: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00035f50: 626a 6563 7420 2a6e 6f74 6520 436f 6469  bject *note Codi
│ │ │ │ +00035f60: 6d42 7349 6e76 3a20 436f 6469 6d42 7349  mBsInv: CodimBsI
│ │ │ │ +00035f70: 6e76 2c20 6973 2061 202a 6e6f 7465 2073  nv, is a *note s
│ │ │ │ +00035f80: 796d 626f 6c3a 0a28 4d61 6361 756c 6179  ymbol:.(Macaulay
│ │ │ │ +00035f90: 3244 6f63 2953 796d 626f 6c2c 2e0a 1f0a  2Doc)Symbol,....
│ │ │ │ +00035fa0: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ +00035fb0: 666f 2c20 4e6f 6465 3a20 636f 6566 6669  fo, Node: coeffi
│ │ │ │ +00035fc0: 6369 656e 7452 696e 675f 6c70 5261 7469  cientRing_lpRati
│ │ │ │ +00035fd0: 6f6e 616c 4d61 705f 7270 2c20 4e65 7874  onalMap_rp, Next
│ │ │ │ +00035fe0: 3a20 636f 6566 6669 6369 656e 7473 5f6c  : coefficients_l
│ │ │ │ +00035ff0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +00036000: 2050 7265 763a 2043 6f64 696d 4273 496e   Prev: CodimBsIn
│ │ │ │ +00036010: 762c 2055 703a 2054 6f70 0a0a 636f 6566  v, Up: Top..coef
│ │ │ │ +00036020: 6669 6369 656e 7452 696e 6728 5261 7469  ficientRing(Rati
│ │ │ │ +00036030: 6f6e 616c 4d61 7029 202d 2d20 636f 6566  onalMap) -- coef
│ │ │ │ +00036040: 6669 6369 656e 7420 7269 6e67 206f 6620  ficient ring of 
│ │ │ │ +00036050: 6120 7261 7469 6f6e 616c 206d 6170 0a2a  a rational map.*
│ │ │ │  00036060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00036070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00036080: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00036090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ -000360a0: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ -000360b0: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e    * Function: *n
│ │ │ │ -000360c0: 6f74 6520 636f 6566 6669 6369 656e 7452  ote coefficientR
│ │ │ │ -000360d0: 696e 673a 2028 4d61 6361 756c 6179 3244  ing: (Macaulay2D
│ │ │ │ -000360e0: 6f63 2963 6f65 6666 6963 6965 6e74 5269  oc)coefficientRi
│ │ │ │ -000360f0: 6e67 2c0a 2020 2a20 5573 6167 653a 200a  ng,.  * Usage: .
│ │ │ │ -00036100: 2020 2020 2020 2020 636f 6566 6669 6369          coeffici
│ │ │ │ -00036110: 656e 7452 696e 6720 7068 690a 2020 2a20  entRing phi.  * 
│ │ │ │ -00036120: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00036130: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ -00036140: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00036150: 6e61 6c4d 6170 2c0a 2020 2a20 4f75 7470  nalMap,.  * Outp
│ │ │ │ -00036160: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ -00036170: 6e6f 7465 2072 696e 673a 2028 4d61 6361  note ring: (Maca
│ │ │ │ -00036180: 756c 6179 3244 6f63 2952 696e 672c 2c20  ulay2Doc)Ring,, 
│ │ │ │ -00036190: 7468 6520 636f 6566 6669 6369 656e 7420  the coefficient 
│ │ │ │ -000361a0: 7269 6e67 206f 6620 7068 690a 0a53 6565  ring of phi..See
│ │ │ │ -000361b0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -000361c0: 2020 2a20 2a6e 6f74 6520 5261 7469 6f6e    * *note Ration
│ │ │ │ -000361d0: 616c 4d61 7020 2a2a 2052 696e 673a 2052  alMap ** Ring: R
│ │ │ │ -000361e0: 6174 696f 6e61 6c4d 6170 205f 7374 5f73  ationalMap _st_s
│ │ │ │ -000361f0: 7420 5269 6e67 2c20 2d2d 2063 6861 6e67  t Ring, -- chang
│ │ │ │ -00036200: 6520 7468 650a 2020 2020 636f 6566 6669  e the.    coeffi
│ │ │ │ -00036210: 6369 656e 7420 7269 6e67 206f 6620 6120  cient ring of a 
│ │ │ │ -00036220: 7261 7469 6f6e 616c 206d 6170 0a0a 5761  rational map..Wa
│ │ │ │ -00036230: 7973 2074 6f20 7573 6520 7468 6973 206d  ys to use this m
│ │ │ │ -00036240: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d  ethod:.=========
│ │ │ │ -00036250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00036260: 0a20 202a 202a 6e6f 7465 2063 6f65 6666  .  * *note coeff
│ │ │ │ -00036270: 6963 6965 6e74 5269 6e67 2852 6174 696f  icientRing(Ratio
│ │ │ │ -00036280: 6e61 6c4d 6170 293a 2063 6f65 6666 6963  nalMap): coeffic
│ │ │ │ -00036290: 6965 6e74 5269 6e67 5f6c 7052 6174 696f  ientRing_lpRatio
│ │ │ │ -000362a0: 6e61 6c4d 6170 5f72 702c 202d 2d0a 2020  nalMap_rp, --.  
│ │ │ │ -000362b0: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ -000362c0: 6e67 206f 6620 6120 7261 7469 6f6e 616c  ng of a rational
│ │ │ │ -000362d0: 206d 6170 0a1f 0a46 696c 653a 2043 7265   map...File: Cre
│ │ │ │ -000362e0: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
│ │ │ │ -000362f0: 2063 6f65 6666 6963 6965 6e74 735f 6c70   coefficients_lp
│ │ │ │ -00036300: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -00036310: 4e65 7874 3a20 6465 6772 6565 5f6c 7052  Next: degree_lpR
│ │ │ │ -00036320: 6174 696f 6e61 6c4d 6170 5f72 702c 2050  ationalMap_rp, P
│ │ │ │ -00036330: 7265 763a 2063 6f65 6666 6963 6965 6e74  rev: coefficient
│ │ │ │ -00036340: 5269 6e67 5f6c 7052 6174 696f 6e61 6c4d  Ring_lpRationalM
│ │ │ │ -00036350: 6170 5f72 702c 2055 703a 2054 6f70 0a0a  ap_rp, Up: Top..
│ │ │ │ -00036360: 636f 6566 6669 6369 656e 7473 2852 6174  coefficients(Rat
│ │ │ │ -00036370: 696f 6e61 6c4d 6170 2920 2d2d 2063 6f65  ionalMap) -- coe
│ │ │ │ -00036380: 6666 6963 6965 6e74 206d 6174 7269 7820  fficient matrix 
│ │ │ │ -00036390: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ -000363a0: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  p.**************
│ │ │ │ +00036090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000360a0: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ +000360b0: 3d3d 3d3d 0a0a 2020 2a20 4675 6e63 7469  ====..  * Functi
│ │ │ │ +000360c0: 6f6e 3a20 2a6e 6f74 6520 636f 6566 6669  on: *note coeffi
│ │ │ │ +000360d0: 6369 656e 7452 696e 673a 2028 4d61 6361  cientRing: (Maca
│ │ │ │ +000360e0: 756c 6179 3244 6f63 2963 6f65 6666 6963  ulay2Doc)coeffic
│ │ │ │ +000360f0: 6965 6e74 5269 6e67 2c0a 2020 2a20 5573  ientRing,.  * Us
│ │ │ │ +00036100: 6167 653a 200a 2020 2020 2020 2020 636f  age: .        co
│ │ │ │ +00036110: 6566 6669 6369 656e 7452 696e 6720 7068  efficientRing ph
│ │ │ │ +00036120: 690a 2020 2a20 496e 7075 7473 3a0a 2020  i.  * Inputs:.  
│ │ │ │ +00036130: 2020 2020 2a20 7068 692c 2061 202a 6e6f      * phi, a *no
│ │ │ │ +00036140: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ +00036150: 2052 6174 696f 6e61 6c4d 6170 2c0a 2020   RationalMap,.  
│ │ │ │ +00036160: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00036170: 202a 2061 202a 6e6f 7465 2072 696e 673a   * a *note ring:
│ │ │ │ +00036180: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ +00036190: 696e 672c 2c20 7468 6520 636f 6566 6669  ing,, the coeffi
│ │ │ │ +000361a0: 6369 656e 7420 7269 6e67 206f 6620 7068  cient ring of ph
│ │ │ │ +000361b0: 690a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  i..See also.====
│ │ │ │ +000361c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +000361d0: 5261 7469 6f6e 616c 4d61 7020 2a2a 2052  RationalMap ** R
│ │ │ │ +000361e0: 696e 673a 2052 6174 696f 6e61 6c4d 6170  ing: RationalMap
│ │ │ │ +000361f0: 205f 7374 5f73 7420 5269 6e67 2c20 2d2d   _st_st Ring, --
│ │ │ │ +00036200: 2063 6861 6e67 6520 7468 650a 2020 2020   change the.    
│ │ │ │ +00036210: 636f 6566 6669 6369 656e 7420 7269 6e67  coefficient ring
│ │ │ │ +00036220: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ +00036230: 6170 0a0a 5761 7973 2074 6f20 7573 6520  ap..Ways to use 
│ │ │ │ +00036240: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ +00036250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00036260: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +00036270: 2063 6f65 6666 6963 6965 6e74 5269 6e67   coefficientRing
│ │ │ │ +00036280: 2852 6174 696f 6e61 6c4d 6170 293a 2063  (RationalMap): c
│ │ │ │ +00036290: 6f65 6666 6963 6965 6e74 5269 6e67 5f6c  oefficientRing_l
│ │ │ │ +000362a0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +000362b0: 202d 2d0a 2020 2020 636f 6566 6669 6369   --.    coeffici
│ │ │ │ +000362c0: 656e 7420 7269 6e67 206f 6620 6120 7261  ent ring of a ra
│ │ │ │ +000362d0: 7469 6f6e 616c 206d 6170 0a1f 0a46 696c  tional map...Fil
│ │ │ │ +000362e0: 653a 2043 7265 6d6f 6e61 2e69 6e66 6f2c  e: Cremona.info,
│ │ │ │ +000362f0: 204e 6f64 653a 2063 6f65 6666 6963 6965   Node: coefficie
│ │ │ │ +00036300: 6e74 735f 6c70 5261 7469 6f6e 616c 4d61  nts_lpRationalMa
│ │ │ │ +00036310: 705f 7270 2c20 4e65 7874 3a20 6465 6772  p_rp, Next: degr
│ │ │ │ +00036320: 6565 5f6c 7052 6174 696f 6e61 6c4d 6170  ee_lpRationalMap
│ │ │ │ +00036330: 5f72 702c 2050 7265 763a 2063 6f65 6666  _rp, Prev: coeff
│ │ │ │ +00036340: 6963 6965 6e74 5269 6e67 5f6c 7052 6174  icientRing_lpRat
│ │ │ │ +00036350: 696f 6e61 6c4d 6170 5f72 702c 2055 703a  ionalMap_rp, Up:
│ │ │ │ +00036360: 2054 6f70 0a0a 636f 6566 6669 6369 656e   Top..coefficien
│ │ │ │ +00036370: 7473 2852 6174 696f 6e61 6c4d 6170 2920  ts(RationalMap) 
│ │ │ │ +00036380: 2d2d 2063 6f65 6666 6963 6965 6e74 206d  -- coefficient m
│ │ │ │ +00036390: 6174 7269 7820 6f66 2061 2072 6174 696f  atrix of a ratio
│ │ │ │ +000363a0: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a  nal map.********
│ │ │ │  000363b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000363c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000363d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000363e0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ -000363f0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675 6e63  ======..  * Func
│ │ │ │ -00036400: 7469 6f6e 3a20 2a6e 6f74 6520 636f 6566  tion: *note coef
│ │ │ │ -00036410: 6669 6369 656e 7473 3a20 284d 6163 6175  ficients: (Macau
│ │ │ │ -00036420: 6c61 7932 446f 6329 636f 6566 6669 6369  lay2Doc)coeffici
│ │ │ │ -00036430: 656e 7473 2c0a 2020 2a20 5573 6167 653a  ents,.  * Usage:
│ │ │ │ -00036440: 200a 2020 2020 2020 2020 636f 6566 6669   .        coeffi
│ │ │ │ -00036450: 6369 656e 7473 2070 6869 0a20 202a 2049  cients phi.  * I
│ │ │ │ -00036460: 6e70 7574 733a 0a20 2020 2020 202a 2070  nputs:.      * p
│ │ │ │ -00036470: 6869 2c20 6120 2a6e 6f74 6520 7261 7469  hi, a *note rati
│ │ │ │ -00036480: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -00036490: 616c 4d61 702c 0a20 202a 202a 6e6f 7465  alMap,.  * *note
│ │ │ │ -000364a0: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ -000364b0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -000364c0: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ -000364d0: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
│ │ │ │ -000364e0: 7075 7473 2c3a 0a20 2020 2020 202a 202a  puts,:.      * *
│ │ │ │ -000364f0: 6e6f 7465 204d 6f6e 6f6d 6961 6c73 3a20  note Monomials: 
│ │ │ │ -00036500: 284d 6163 6175 6c61 7932 446f 6329 636f  (Macaulay2Doc)co
│ │ │ │ -00036510: 6566 6669 6369 656e 7473 2c20 3d3e 202e  efficients, => .
│ │ │ │ -00036520: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -00036530: 650a 2020 2020 2020 2020 6e75 6c6c 2c0a  e.        null,.
│ │ │ │ -00036540: 2020 2020 2020 2a20 2a6e 6f74 6520 5661        * *note Va
│ │ │ │ -00036550: 7269 6162 6c65 733a 2028 4d61 6361 756c  riables: (Macaul
│ │ │ │ -00036560: 6179 3244 6f63 2963 6f65 6666 6963 6965  ay2Doc)coefficie
│ │ │ │ -00036570: 6e74 732c 203d 3e20 2e2e 2e2c 2064 6566  nts, => ..., def
│ │ │ │ -00036580: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -00036590: 2020 206e 756c 6c2c 0a20 202a 204f 7574     null,.  * Out
│ │ │ │ -000365a0: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ -000365b0: 2a6e 6f74 6520 6d61 7472 6978 3a20 284d  *note matrix: (M
│ │ │ │ -000365c0: 6163 6175 6c61 7932 446f 6329 4d61 7472  acaulay2Doc)Matr
│ │ │ │ -000365d0: 6978 2c2c 2074 6865 2063 6f65 6666 6963  ix,, the coeffic
│ │ │ │ -000365e0: 6965 6e74 206d 6174 7269 7820 6f66 2074  ient matrix of t
│ │ │ │ -000365f0: 6865 0a20 2020 2020 2020 2070 6f6c 796e  he.        polyn
│ │ │ │ -00036600: 6f6d 6961 6c73 2064 6566 696e 696e 6720  omials defining 
│ │ │ │ -00036610: 7068 690a 0a44 6573 6372 6970 7469 6f6e  phi..Description
│ │ │ │ -00036620: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ -00036630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000363e0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ +000363f0: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ +00036400: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74  * Function: *not
│ │ │ │ +00036410: 6520 636f 6566 6669 6369 656e 7473 3a20  e coefficients: 
│ │ │ │ +00036420: 284d 6163 6175 6c61 7932 446f 6329 636f  (Macaulay2Doc)co
│ │ │ │ +00036430: 6566 6669 6369 656e 7473 2c0a 2020 2a20  efficients,.  * 
│ │ │ │ +00036440: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00036450: 636f 6566 6669 6369 656e 7473 2070 6869  coefficients phi
│ │ │ │ +00036460: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +00036470: 2020 202a 2070 6869 2c20 6120 2a6e 6f74     * phi, a *not
│ │ │ │ +00036480: 6520 7261 7469 6f6e 616c 206d 6170 3a20  e rational map: 
│ │ │ │ +00036490: 5261 7469 6f6e 616c 4d61 702c 0a20 202a  RationalMap,.  *
│ │ │ │ +000364a0: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ +000364b0: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ +000364c0: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ +000364d0: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ +000364e0: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ +000364f0: 2020 202a 202a 6e6f 7465 204d 6f6e 6f6d     * *note Monom
│ │ │ │ +00036500: 6961 6c73 3a20 284d 6163 6175 6c61 7932  ials: (Macaulay2
│ │ │ │ +00036510: 446f 6329 636f 6566 6669 6369 656e 7473  Doc)coefficients
│ │ │ │ +00036520: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +00036530: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ +00036540: 6e75 6c6c 2c0a 2020 2020 2020 2a20 2a6e  null,.      * *n
│ │ │ │ +00036550: 6f74 6520 5661 7269 6162 6c65 733a 2028  ote Variables: (
│ │ │ │ +00036560: 4d61 6361 756c 6179 3244 6f63 2963 6f65  Macaulay2Doc)coe
│ │ │ │ +00036570: 6666 6963 6965 6e74 732c 203d 3e20 2e2e  fficients, => ..
│ │ │ │ +00036580: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +00036590: 0a20 2020 2020 2020 206e 756c 6c2c 0a20  .        null,. 
│ │ │ │ +000365a0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +000365b0: 2020 2a20 6120 2a6e 6f74 6520 6d61 7472    * a *note matr
│ │ │ │ +000365c0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ +000365d0: 6329 4d61 7472 6978 2c2c 2074 6865 2063  c)Matrix,, the c
│ │ │ │ +000365e0: 6f65 6666 6963 6965 6e74 206d 6174 7269  oefficient matri
│ │ │ │ +000365f0: 7820 6f66 2074 6865 0a20 2020 2020 2020  x of the.       
│ │ │ │ +00036600: 2070 6f6c 796e 6f6d 6961 6c73 2064 6566   polynomials def
│ │ │ │ +00036610: 696e 696e 6720 7068 690a 0a44 6573 6372  ining phi..Descr
│ │ │ │ +00036620: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00036630: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ==..+-----------
│ │ │ │  00036640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036660: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 204b  -------+.|i1 : K
│ │ │ │ -00036670: 203d 2051 513b 2072 696e 6750 3920 3d20   = QQ; ringP9 = 
│ │ │ │ -00036680: 4b5b 785f 302e 2e78 5f39 5d3b 2020 2020  K[x_0..x_9];    
│ │ │ │ -00036690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000366a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00036660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00036670: 6931 203a 204b 203d 2051 513b 2072 696e  i1 : K = QQ; rin
│ │ │ │ +00036680: 6750 3920 3d20 4b5b 785f 302e 2e78 5f39  gP9 = K[x_0..x_9
│ │ │ │ +00036690: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ +000366a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  000366b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000366c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000366d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000366e0: 6933 203a 204d 203d 2072 616e 646f 6d28  i3 : M = random(
│ │ │ │ -000366f0: 4b5e 3130 2c4b 5e31 3029 2020 2020 2020  K^10,K^10)      
│ │ │ │ +000366d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000366e0: 2d2d 2d2b 0a7c 6933 203a 204d 203d 2072  ---+.|i3 : M = r
│ │ │ │ +000366f0: 616e 646f 6d28 4b5e 3130 2c4b 5e31 3029  andom(K^10,K^10)
│ │ │ │  00036700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036710: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00036720: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00036730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036750: 2020 207c 0a7c 6f33 203d 207c 2039 2f32     |.|o3 = | 9/2
│ │ │ │ -00036760: 2037 2f31 3020 3220 2020 2031 2f32 2035   7/10 2    1/2 5
│ │ │ │ -00036770: 2f33 2020 342f 3320 2032 2f33 2031 2f38  /3  4/3  2/3 1/8
│ │ │ │ -00036780: 2020 3120 2020 352f 3220 7c20 2020 7c0a    1   5/2 |   |.
│ │ │ │ -00036790: 7c20 2020 2020 7c20 312f 3220 312f 3220  |     | 1/2 1/2 
│ │ │ │ -000367a0: 2036 2020 2020 3130 2020 372f 3220 2033   6    10  7/2  3
│ │ │ │ -000367b0: 2f37 2020 3620 2020 3130 2f33 2037 2f35  /7  6   10/3 7/5
│ │ │ │ -000367c0: 2035 2f32 207c 2020 207c 0a7c 2020 2020   5/2 |   |.|    
│ │ │ │ -000367d0: 207c 2039 2f34 2037 2f31 3020 352f 3420   | 9/4 7/10 5/4 
│ │ │ │ -000367e0: 2033 2020 2032 2f35 2020 392f 3130 2035   3   2/5  9/10 5
│ │ │ │ -000367f0: 2f34 2033 2f34 2020 332f 3220 312f 3620  /4 3/4  3/2 1/6 
│ │ │ │ -00036800: 7c20 2020 7c0a 7c20 2020 2020 7c20 312f  |   |.|     | 1/
│ │ │ │ -00036810: 3220 372f 3320 2032 2f39 2020 3320 2020  2 7/3  2/9  3   
│ │ │ │ -00036820: 362f 3520 2034 2f37 2020 322f 3920 3420  6/5  4/7  2/9 4 
│ │ │ │ -00036830: 2020 2031 2f35 2033 2f34 207c 2020 207c     1/5 3/4 |   |
│ │ │ │ -00036840: 0a7c 2020 2020 207c 2031 2020 2037 2020  .|     | 1   7  
│ │ │ │ -00036850: 2020 3520 2020 2033 2f32 2035 2f34 2020    5    3/2 5/4  
│ │ │ │ -00036860: 352f 3220 2038 2f35 2031 2f34 2020 3520  5/2  8/5 1/4  5 
│ │ │ │ -00036870: 2020 3420 2020 7c20 2020 7c0a 7c20 2020    4   |   |.|   
│ │ │ │ -00036880: 2020 7c20 332f 3420 332f 3720 2033 2f31    | 3/4 3/7  3/1
│ │ │ │ -00036890: 3020 342f 3320 352f 3720 2035 2f39 2020  0 4/3 5/7  5/9  
│ │ │ │ -000368a0: 392f 3420 312f 3320 2035 2f37 2038 2f35  9/4 1/3  5/7 8/5
│ │ │ │ -000368b0: 207c 2020 207c 0a7c 2020 2020 207c 2033   |   |.|     | 3
│ │ │ │ -000368c0: 2f32 2035 2f32 2020 3120 2020 2037 2f38  /2 5/2  1    7/8
│ │ │ │ -000368d0: 2035 2f39 2020 352f 3920 2032 2f39 2034   5/9  5/9  2/9 4
│ │ │ │ -000368e0: 2f33 2020 332f 3820 3130 2020 7c20 2020  /3  3/8 10  |   
│ │ │ │ -000368f0: 7c0a 7c20 2020 2020 7c20 332f 3420 362f  |.|     | 3/4 6/
│ │ │ │ -00036900: 3720 2033 2f37 2020 352f 3620 352f 3320  7  3/7  5/6 5/3 
│ │ │ │ -00036910: 2036 2f37 2020 3320 2020 392f 3130 2033   6/7  3   9/10 3
│ │ │ │ -00036920: 2020 2032 2020 207c 2020 207c 0a7c 2020     2   |   |.|  
│ │ │ │ -00036930: 2020 207c 2037 2f34 2032 2f33 2020 3520     | 7/4 2/3  5 
│ │ │ │ -00036940: 2020 2035 2020 2034 2f35 2020 3220 2020     5   4/5  2   
│ │ │ │ -00036950: 2039 2f38 2035 2f34 2020 3120 2020 312f   9/8 5/4  1   1/
│ │ │ │ -00036960: 3320 7c20 2020 7c0a 7c20 2020 2020 7c20  3 |   |.|     | 
│ │ │ │ -00036970: 372f 3920 3120 2020 2031 302f 3920 322f  7/9 1    10/9 2/
│ │ │ │ -00036980: 3520 312f 3130 2031 2020 2020 312f 3220  5 1/10 1    1/2 
│ │ │ │ -00036990: 312f 3720 2031 2f32 2035 2f32 207c 2020  1/7  1/2 5/2 |  
│ │ │ │ -000369a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00036750: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +00036760: 207c 2039 2f32 2037 2f31 3020 3220 2020   | 9/2 7/10 2   
│ │ │ │ +00036770: 2031 2f32 2035 2f33 2020 342f 3320 2032   1/2 5/3  4/3  2
│ │ │ │ +00036780: 2f33 2031 2f38 2020 3120 2020 352f 3220  /3 1/8  1   5/2 
│ │ │ │ +00036790: 7c20 2020 7c0a 7c20 2020 2020 7c20 312f  |   |.|     | 1/
│ │ │ │ +000367a0: 3220 312f 3220 2036 2020 2020 3130 2020  2 1/2  6    10  
│ │ │ │ +000367b0: 372f 3220 2033 2f37 2020 3620 2020 3130  7/2  3/7  6   10
│ │ │ │ +000367c0: 2f33 2037 2f35 2035 2f32 207c 2020 207c  /3 7/5 5/2 |   |
│ │ │ │ +000367d0: 0a7c 2020 2020 207c 2039 2f34 2037 2f31  .|     | 9/4 7/1
│ │ │ │ +000367e0: 3020 352f 3420 2033 2020 2032 2f35 2020  0 5/4  3   2/5  
│ │ │ │ +000367f0: 392f 3130 2035 2f34 2033 2f34 2020 332f  9/10 5/4 3/4  3/
│ │ │ │ +00036800: 3220 312f 3620 7c20 2020 7c0a 7c20 2020  2 1/6 |   |.|   
│ │ │ │ +00036810: 2020 7c20 312f 3220 372f 3320 2032 2f39    | 1/2 7/3  2/9
│ │ │ │ +00036820: 2020 3320 2020 362f 3520 2034 2f37 2020    3   6/5  4/7  
│ │ │ │ +00036830: 322f 3920 3420 2020 2031 2f35 2033 2f34  2/9 4    1/5 3/4
│ │ │ │ +00036840: 207c 2020 207c 0a7c 2020 2020 207c 2031   |   |.|     | 1
│ │ │ │ +00036850: 2020 2037 2020 2020 3520 2020 2033 2f32     7    5    3/2
│ │ │ │ +00036860: 2035 2f34 2020 352f 3220 2038 2f35 2031   5/4  5/2  8/5 1
│ │ │ │ +00036870: 2f34 2020 3520 2020 3420 2020 7c20 2020  /4  5   4   |   
│ │ │ │ +00036880: 7c0a 7c20 2020 2020 7c20 332f 3420 332f  |.|     | 3/4 3/
│ │ │ │ +00036890: 3720 2033 2f31 3020 342f 3320 352f 3720  7  3/10 4/3 5/7 
│ │ │ │ +000368a0: 2035 2f39 2020 392f 3420 312f 3320 2035   5/9  9/4 1/3  5
│ │ │ │ +000368b0: 2f37 2038 2f35 207c 2020 207c 0a7c 2020  /7 8/5 |   |.|  
│ │ │ │ +000368c0: 2020 207c 2033 2f32 2035 2f32 2020 3120     | 3/2 5/2  1 
│ │ │ │ +000368d0: 2020 2037 2f38 2035 2f39 2020 352f 3920     7/8 5/9  5/9 
│ │ │ │ +000368e0: 2032 2f39 2034 2f33 2020 332f 3820 3130   2/9 4/3  3/8 10
│ │ │ │ +000368f0: 2020 7c20 2020 7c0a 7c20 2020 2020 7c20    |   |.|     | 
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│ │ │ │ +00036910: 3620 352f 3320 2036 2f37 2020 3320 2020  6 5/3  6/7  3   
│ │ │ │ +00036920: 392f 3130 2033 2020 2032 2020 207c 2020  9/10 3   2   |  
│ │ │ │ +00036930: 207c 0a7c 2020 2020 207c 2037 2f34 2032   |.|     | 7/4 2
│ │ │ │ +00036940: 2f33 2020 3520 2020 2035 2020 2034 2f35  /3  5    5   4/5
│ │ │ │ +00036950: 2020 3220 2020 2039 2f38 2035 2f34 2020    2    9/8 5/4  
│ │ │ │ +00036960: 3120 2020 312f 3320 7c20 2020 7c0a 7c20  1   1/3 |   |.| 
│ │ │ │ +00036970: 2020 2020 7c20 372f 3920 3120 2020 2031      | 7/9 1    1
│ │ │ │ +00036980: 302f 3920 322f 3520 312f 3130 2031 2020  0/9 2/5 1/10 1  
│ │ │ │ +00036990: 2020 312f 3220 312f 3720 2031 2f32 2035    1/2 1/7  1/2 5
│ │ │ │ +000369a0: 2f32 207c 2020 207c 0a7c 2020 2020 2020  /2 |   |.|      
│ │ │ │  000369b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000369c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000369d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000369e0: 2020 2020 2020 2020 2020 2020 2031 3020               10 
│ │ │ │ -000369f0: 2020 2020 2020 3130 2020 2020 2020 2020        10        
│ │ │ │ +000369d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000369e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000369f0: 2020 2031 3020 2020 2020 2020 3130 2020     10       10  
│ │ │ │  00036a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a10: 2020 2020 2020 207c 0a7c 6f33 203a 204d         |.|o3 : M
│ │ │ │ -00036a20: 6174 7269 7820 5151 2020 203c 2d2d 2051  atrix QQ   <-- Q
│ │ │ │ -00036a30: 5120 2020 2020 2020 2020 2020 2020 2020  Q               
│ │ │ │ +00036a10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036a20: 6f33 203a 204d 6174 7269 7820 5151 2020  o3 : Matrix QQ  
│ │ │ │ +00036a30: 203c 2d2d 2051 5120 2020 2020 2020 2020   <-- QQ         
│ │ │ │  00036a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00036a50: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  00036a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
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│ │ │ │ -00036ab0: 6e67 5039 2920 2a20 2874 7261 6e73 706f  ngP9) * (transpo
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│ │ │ │ -00036ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036a90: 2d2d 2d2b 0a7c 6934 203a 2070 6869 203d  ---+.|i4 : phi =
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│ │ │ │ +00036ab0: 6172 7320 7269 6e67 5039 2920 2a20 2874  ars ringP9) * (t
│ │ │ │ +00036ac0: 7261 6e73 706f 7365 204d 2929 3b20 7c0a  ranspose M)); |.
│ │ │ │ +00036ad0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00036ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036b00: 2020 207c 0a7c 6f34 203a 2052 6174 696f     |.|o4 : Ratio
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│ │ │ │ -00036b40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00036b00: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
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│ │ │ │ +00036b40: 505e 3929 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d  P^9)|.+---------
│ │ │ │  00036b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036b70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -00036b80: 204d 2720 3d20 636f 6566 6669 6369 656e   M' = coefficien
│ │ │ │ -00036b90: 7473 2070 6869 2020 2020 2020 2020 2020  ts phi          
│ │ │ │ +00036b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00036b80: 0a7c 6935 203a 204d 2720 3d20 636f 6566  .|i5 : M' = coef
│ │ │ │ +00036b90: 6669 6369 656e 7473 2070 6869 2020 2020  ficients phi    
│ │ │ │  00036ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036bb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00036bb0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 207c                 |
│ │ │ │ -00036bf0: 0a7c 6f35 203d 207c 2039 2f32 2037 2f31  .|o5 = | 9/2 7/1
│ │ │ │ -00036c00: 3020 3220 2020 2031 2f32 2035 2f33 2020  0 2    1/2 5/3  
│ │ │ │ -00036c10: 342f 3320 2032 2f33 2031 2f38 2020 3120  4/3  2/3 1/8  1 
│ │ │ │ -00036c20: 2020 352f 3220 7c20 2020 7c0a 7c20 2020    5/2 |   |.|   
│ │ │ │ -00036c30: 2020 7c20 312f 3220 312f 3220 2036 2020    | 1/2 1/2  6  
│ │ │ │ -00036c40: 2020 3130 2020 372f 3220 2033 2f37 2020    10  7/2  3/7  
│ │ │ │ -00036c50: 3620 2020 3130 2f33 2037 2f35 2035 2f32  6   10/3 7/5 5/2
│ │ │ │ -00036c60: 207c 2020 207c 0a7c 2020 2020 207c 2039   |   |.|     | 9
│ │ │ │ -00036c70: 2f34 2037 2f31 3020 352f 3420 2033 2020  /4 7/10 5/4  3  
│ │ │ │ -00036c80: 2032 2f35 2020 392f 3130 2035 2f34 2033   2/5  9/10 5/4 3
│ │ │ │ -00036c90: 2f34 2020 332f 3220 312f 3620 7c20 2020  /4  3/2 1/6 |   
│ │ │ │ -00036ca0: 7c0a 7c20 2020 2020 7c20 312f 3220 372f  |.|     | 1/2 7/
│ │ │ │ -00036cb0: 3320 2032 2f39 2020 3320 2020 362f 3520  3  2/9  3   6/5 
│ │ │ │ -00036cc0: 2034 2f37 2020 322f 3920 3420 2020 2031   4/7  2/9 4    1
│ │ │ │ -00036cd0: 2f35 2033 2f34 207c 2020 207c 0a7c 2020  /5 3/4 |   |.|  
│ │ │ │ -00036ce0: 2020 207c 2031 2020 2037 2020 2020 3520     | 1   7    5 
│ │ │ │ -00036cf0: 2020 2033 2f32 2035 2f34 2020 352f 3220     3/2 5/4  5/2 
│ │ │ │ -00036d00: 2038 2f35 2031 2f34 2020 3520 2020 3420   8/5 1/4  5   4 
│ │ │ │ -00036d10: 2020 7c20 2020 7c0a 7c20 2020 2020 7c20    |   |.|     | 
│ │ │ │ -00036d20: 332f 3420 332f 3720 2033 2f31 3020 342f  3/4 3/7  3/10 4/
│ │ │ │ -00036d30: 3320 352f 3720 2035 2f39 2020 392f 3420  3 5/7  5/9  9/4 
│ │ │ │ -00036d40: 312f 3320 2035 2f37 2038 2f35 207c 2020  1/3  5/7 8/5 |  
│ │ │ │ -00036d50: 207c 0a7c 2020 2020 207c 2033 2f32 2035   |.|     | 3/2 5
│ │ │ │ -00036d60: 2f32 2020 3120 2020 2037 2f38 2035 2f39  /2  1    7/8 5/9
│ │ │ │ -00036d70: 2020 352f 3920 2032 2f39 2034 2f33 2020    5/9  2/9 4/3  
│ │ │ │ -00036d80: 332f 3820 3130 2020 7c20 2020 7c0a 7c20  3/8 10  |   |.| 
│ │ │ │ -00036d90: 2020 2020 7c20 332f 3420 362f 3720 2033      | 3/4 6/7  3
│ │ │ │ -00036da0: 2f37 2020 352f 3620 352f 3320 2036 2f37  /7  5/6 5/3  6/7
│ │ │ │ -00036db0: 2020 3320 2020 392f 3130 2033 2020 2032    3   9/10 3   2
│ │ │ │ -00036dc0: 2020 207c 2020 207c 0a7c 2020 2020 207c     |   |.|     |
│ │ │ │ -00036dd0: 2037 2f34 2032 2f33 2020 3520 2020 2035   7/4 2/3  5    5
│ │ │ │ -00036de0: 2020 2034 2f35 2020 3220 2020 2039 2f38     4/5  2    9/8
│ │ │ │ -00036df0: 2035 2f34 2020 3120 2020 312f 3320 7c20   5/4  1   1/3 | 
│ │ │ │ -00036e00: 2020 7c0a 7c20 2020 2020 7c20 372f 3920    |.|     | 7/9 
│ │ │ │ -00036e10: 3120 2020 2031 302f 3920 322f 3520 312f  1    10/9 2/5 1/
│ │ │ │ -00036e20: 3130 2031 2020 2020 312f 3220 312f 3720  10 1    1/2 1/7 
│ │ │ │ -00036e30: 2031 2f32 2035 2f32 207c 2020 207c 0a7c   1/2 5/2 |   |.|
│ │ │ │ -00036e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036bf0: 2020 2020 207c 0a7c 6f35 203d 207c 2039       |.|o5 = | 9
│ │ │ │ +00036c00: 2f32 2037 2f31 3020 3220 2020 2031 2f32  /2 7/10 2    1/2
│ │ │ │ +00036c10: 2035 2f33 2020 342f 3320 2032 2f33 2031   5/3  4/3  2/3 1
│ │ │ │ +00036c20: 2f38 2020 3120 2020 352f 3220 7c20 2020  /8  1   5/2 |   
│ │ │ │ +00036c30: 7c0a 7c20 2020 2020 7c20 312f 3220 312f  |.|     | 1/2 1/
│ │ │ │ +00036c40: 3220 2036 2020 2020 3130 2020 372f 3220  2  6    10  7/2 
│ │ │ │ +00036c50: 2033 2f37 2020 3620 2020 3130 2f33 2037   3/7  6   10/3 7
│ │ │ │ +00036c60: 2f35 2035 2f32 207c 2020 207c 0a7c 2020  /5 5/2 |   |.|  
│ │ │ │ +00036c70: 2020 207c 2039 2f34 2037 2f31 3020 352f     | 9/4 7/10 5/
│ │ │ │ +00036c80: 3420 2033 2020 2032 2f35 2020 392f 3130  4  3   2/5  9/10
│ │ │ │ +00036c90: 2035 2f34 2033 2f34 2020 332f 3220 312f   5/4 3/4  3/2 1/
│ │ │ │ +00036ca0: 3620 7c20 2020 7c0a 7c20 2020 2020 7c20  6 |   |.|     | 
│ │ │ │ +00036cb0: 312f 3220 372f 3320 2032 2f39 2020 3320  1/2 7/3  2/9  3 
│ │ │ │ +00036cc0: 2020 362f 3520 2034 2f37 2020 322f 3920    6/5  4/7  2/9 
│ │ │ │ +00036cd0: 3420 2020 2031 2f35 2033 2f34 207c 2020  4    1/5 3/4 |  
│ │ │ │ +00036ce0: 207c 0a7c 2020 2020 207c 2031 2020 2037   |.|     | 1   7
│ │ │ │ +00036cf0: 2020 2020 3520 2020 2033 2f32 2035 2f34      5    3/2 5/4
│ │ │ │ +00036d00: 2020 352f 3220 2038 2f35 2031 2f34 2020    5/2  8/5 1/4  
│ │ │ │ +00036d10: 3520 2020 3420 2020 7c20 2020 7c0a 7c20  5   4   |   |.| 
│ │ │ │ +00036d20: 2020 2020 7c20 332f 3420 332f 3720 2033      | 3/4 3/7  3
│ │ │ │ +00036d30: 2f31 3020 342f 3320 352f 3720 2035 2f39  /10 4/3 5/7  5/9
│ │ │ │ +00036d40: 2020 392f 3420 312f 3320 2035 2f37 2038    9/4 1/3  5/7 8
│ │ │ │ +00036d50: 2f35 207c 2020 207c 0a7c 2020 2020 207c  /5 |   |.|     |
│ │ │ │ +00036d60: 2033 2f32 2035 2f32 2020 3120 2020 2037   3/2 5/2  1    7
│ │ │ │ +00036d70: 2f38 2035 2f39 2020 352f 3920 2032 2f39  /8 5/9  5/9  2/9
│ │ │ │ +00036d80: 2034 2f33 2020 332f 3820 3130 2020 7c20   4/3  3/8 10  | 
│ │ │ │ +00036d90: 2020 7c0a 7c20 2020 2020 7c20 332f 3420    |.|     | 3/4 
│ │ │ │ +00036da0: 362f 3720 2033 2f37 2020 352f 3620 352f  6/7  3/7  5/6 5/
│ │ │ │ +00036db0: 3320 2036 2f37 2020 3320 2020 392f 3130  3  6/7  3   9/10
│ │ │ │ +00036dc0: 2033 2020 2032 2020 207c 2020 207c 0a7c   3   2   |   |.|
│ │ │ │ +00036dd0: 2020 2020 207c 2037 2f34 2032 2f33 2020       | 7/4 2/3  
│ │ │ │ +00036de0: 3520 2020 2035 2020 2034 2f35 2020 3220  5    5   4/5  2 
│ │ │ │ +00036df0: 2020 2039 2f38 2035 2f34 2020 3120 2020     9/8 5/4  1   
│ │ │ │ +00036e00: 312f 3320 7c20 2020 7c0a 7c20 2020 2020  1/3 |   |.|     
│ │ │ │ +00036e10: 7c20 372f 3920 3120 2020 2031 302f 3920  | 7/9 1    10/9 
│ │ │ │ +00036e20: 322f 3520 312f 3130 2031 2020 2020 312f  2/5 1/10 1    1/
│ │ │ │ +00036e30: 3220 312f 3720 2031 2f32 2035 2f32 207c  2 1/7  1/2 5/2 |
│ │ │ │ +00036e40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00036e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036e80: 2020 2020 2020 2020 2031 3020 2020 2020           10     
│ │ │ │ -00036e90: 2020 3130 2020 2020 2020 2020 2020 2020    10            
│ │ │ │ +00036e70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00036e80: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ +00036e90: 3020 2020 2020 2020 3130 2020 2020 2020  0       10      
│ │ │ │  00036ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036eb0: 2020 207c 0a7c 6f35 203a 204d 6174 7269     |.|o5 : Matri
│ │ │ │ -00036ec0: 7820 5151 2020 203c 2d2d 2051 5120 2020  x QQ   <-- QQ   
│ │ │ │ -00036ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ee0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00036ef0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00036eb0: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +00036ec0: 204d 6174 7269 7820 5151 2020 203c 2d2d   Matrix QQ   <--
│ │ │ │ +00036ed0: 2051 5120 2020 2020 2020 2020 2020 2020   QQ             
│ │ │ │ +00036ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00036f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036f20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ -00036f30: 204d 203d 3d20 4d27 2020 2020 2020 2020   M == M'        
│ │ │ │ +00036f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00036f30: 0a7c 6936 203a 204d 203d 3d20 4d27 2020  .|i6 : M == M'  
│ │ │ │  00036f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00036f60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 207c                 |
│ │ │ │ -00036fa0: 0a7c 6f36 203d 2074 7275 6520 2020 2020  .|o6 = true     
│ │ │ │ -00036fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036fa0: 2020 2020 207c 0a7c 6f36 203d 2074 7275       |.|o6 = tru
│ │ │ │ +00036fb0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  00036fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036fd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00036fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036fe0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00036ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037010: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20  -----+..Ways to 
│ │ │ │ -00037020: 7573 6520 7468 6973 206d 6574 686f 643a  use this method:
│ │ │ │ -00037030: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00037040: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  =========..  * *
│ │ │ │ -00037050: 6e6f 7465 2063 6f65 6666 6963 6965 6e74  note coefficient
│ │ │ │ -00037060: 7328 5261 7469 6f6e 616c 4d61 7029 3a20  s(RationalMap): 
│ │ │ │ -00037070: 636f 6566 6669 6369 656e 7473 5f6c 7052  coefficients_lpR
│ │ │ │ -00037080: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ -00037090: 2d0a 2020 2020 636f 6566 6669 6369 656e  -.    coefficien
│ │ │ │ -000370a0: 7420 6d61 7472 6978 206f 6620 6120 7261  t matrix of a ra
│ │ │ │ -000370b0: 7469 6f6e 616c 206d 6170 0a1f 0a46 696c  tional map...Fil
│ │ │ │ -000370c0: 653a 2043 7265 6d6f 6e61 2e69 6e66 6f2c  e: Cremona.info,
│ │ │ │ -000370d0: 204e 6f64 653a 2064 6567 7265 655f 6c70   Node: degree_lp
│ │ │ │ -000370e0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -000370f0: 4e65 7874 3a20 6465 6772 6565 4d61 702c  Next: degreeMap,
│ │ │ │ -00037100: 2050 7265 763a 2063 6f65 6666 6963 6965   Prev: coefficie
│ │ │ │ -00037110: 6e74 735f 6c70 5261 7469 6f6e 616c 4d61  nts_lpRationalMa
│ │ │ │ -00037120: 705f 7270 2c20 5570 3a20 546f 700a 0a64  p_rp, Up: Top..d
│ │ │ │ -00037130: 6567 7265 6528 5261 7469 6f6e 616c 4d61  egree(RationalMa
│ │ │ │ -00037140: 7029 202d 2d20 6465 6772 6565 206f 6620  p) -- degree of 
│ │ │ │ -00037150: 6120 7261 7469 6f6e 616c 206d 6170 0a2a  a rational map.*
│ │ │ │ -00037160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00037010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761  -----------+..Wa
│ │ │ │ +00037020: 7973 2074 6f20 7573 6520 7468 6973 206d  ys to use this m
│ │ │ │ +00037030: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d  ethod:.=========
│ │ │ │ +00037040: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00037050: 0a20 202a 202a 6e6f 7465 2063 6f65 6666  .  * *note coeff
│ │ │ │ +00037060: 6963 6965 6e74 7328 5261 7469 6f6e 616c  icients(Rational
│ │ │ │ +00037070: 4d61 7029 3a20 636f 6566 6669 6369 656e  Map): coefficien
│ │ │ │ +00037080: 7473 5f6c 7052 6174 696f 6e61 6c4d 6170  ts_lpRationalMap
│ │ │ │ +00037090: 5f72 702c 202d 2d0a 2020 2020 636f 6566  _rp, --.    coef
│ │ │ │ +000370a0: 6669 6369 656e 7420 6d61 7472 6978 206f  ficient matrix o
│ │ │ │ +000370b0: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ +000370c0: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ +000370d0: 2e69 6e66 6f2c 204e 6f64 653a 2064 6567  .info, Node: deg
│ │ │ │ +000370e0: 7265 655f 6c70 5261 7469 6f6e 616c 4d61  ree_lpRationalMa
│ │ │ │ +000370f0: 705f 7270 2c20 4e65 7874 3a20 6465 6772  p_rp, Next: degr
│ │ │ │ +00037100: 6565 4d61 702c 2050 7265 763a 2063 6f65  eeMap, Prev: coe
│ │ │ │ +00037110: 6666 6963 6965 6e74 735f 6c70 5261 7469  fficients_lpRati
│ │ │ │ +00037120: 6f6e 616c 4d61 705f 7270 2c20 5570 3a20  onalMap_rp, Up: 
│ │ │ │ +00037130: 546f 700a 0a64 6567 7265 6528 5261 7469  Top..degree(Rati
│ │ │ │ +00037140: 6f6e 616c 4d61 7029 202d 2d20 6465 6772  onalMap) -- degr
│ │ │ │ +00037150: 6565 206f 6620 6120 7261 7469 6f6e 616c  ee of a rational
│ │ │ │ +00037160: 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a   map.***********
│ │ │ │  00037170: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00037180: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -00037190: 5379 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d  Synopsis.=======
│ │ │ │ -000371a0: 3d0a 0a20 202a 2046 756e 6374 696f 6e3a  =..  * Function:
│ │ │ │ -000371b0: 202a 6e6f 7465 2064 6567 7265 653a 2028   *note degree: (
│ │ │ │ -000371c0: 4d61 6361 756c 6179 3244 6f63 2964 6567  Macaulay2Doc)deg
│ │ │ │ -000371d0: 7265 652c 0a20 202a 2055 7361 6765 3a20  ree,.  * Usage: 
│ │ │ │ -000371e0: 0a20 2020 2020 2020 2064 6567 7265 6520  .        degree 
│ │ │ │ -000371f0: 7068 690a 2020 2a20 496e 7075 7473 3a0a  phi.  * Inputs:.
│ │ │ │ -00037200: 2020 2020 2020 2a20 7068 692c 2061 202a        * phi, a *
│ │ │ │ -00037210: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ -00037220: 703a 2052 6174 696f 6e61 6c4d 6170 2c0a  p: RationalMap,.
│ │ │ │ -00037230: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00037240: 2020 202a 2061 6e20 2a6e 6f74 6520 696e     * an *note in
│ │ │ │ -00037250: 7465 6765 723a 2028 4d61 6361 756c 6179  teger: (Macaulay
│ │ │ │ -00037260: 3244 6f63 295a 5a2c 2c20 7468 6520 6465  2Doc)ZZ,, the de
│ │ │ │ -00037270: 6772 6565 206f 6620 7068 692e 2053 6f20  gree of phi. So 
│ │ │ │ -00037280: 7468 6973 2076 616c 7565 0a20 2020 2020  this value.     
│ │ │ │ -00037290: 2020 2069 7320 3120 6966 2061 6e64 206f     is 1 if and o
│ │ │ │ -000372a0: 6e6c 7920 6966 2074 6865 206d 6170 2069  nly if the map i
│ │ │ │ -000372b0: 7320 6269 7261 7469 6f6e 616c 206f 6e74  s birational ont
│ │ │ │ -000372c0: 6f20 6974 7320 696d 6167 652e 0a0a 4465  o its image...De
│ │ │ │ -000372d0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ -000372e0: 3d3d 3d3d 3d0a 0a54 6869 7320 6973 2061  =====..This is a
│ │ │ │ -000372f0: 2073 686f 7274 6375 7420 666f 7220 6465   shortcut for de
│ │ │ │ -00037300: 6772 6565 4d61 7028 7068 692c 4365 7274  greeMap(phi,Cert
│ │ │ │ -00037310: 6966 793d 3e74 7275 652c 5665 7262 6f73  ify=>true,Verbos
│ │ │ │ -00037320: 653d 3e66 616c 7365 292c 2073 6565 202a  e=>false), see *
│ │ │ │ -00037330: 6e6f 7465 0a64 6567 7265 654d 6170 2852  note.degreeMap(R
│ │ │ │ -00037340: 6174 696f 6e61 6c4d 6170 293a 2064 6567  ationalMap): deg
│ │ │ │ -00037350: 7265 654d 6170 5f6c 7052 6174 696f 6e61  reeMap_lpRationa
│ │ │ │ -00037360: 6c4d 6170 5f72 702c 2e0a 0a57 6179 7320  lMap_rp,...Ways 
│ │ │ │ -00037370: 746f 2075 7365 2074 6869 7320 6d65 7468  to use this meth
│ │ │ │ -00037380: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  od:.============
│ │ │ │ -00037390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -000373a0: 2a20 2a6e 6f74 6520 6465 6772 6565 2852  * *note degree(R
│ │ │ │ -000373b0: 6174 696f 6e61 6c4d 6170 293a 2064 6567  ationalMap): deg
│ │ │ │ -000373c0: 7265 655f 6c70 5261 7469 6f6e 616c 4d61  ree_lpRationalMa
│ │ │ │ -000373d0: 705f 7270 2c20 2d2d 2064 6567 7265 6520  p_rp, -- degree 
│ │ │ │ -000373e0: 6f66 2061 2072 6174 696f 6e61 6c0a 2020  of a rational.  
│ │ │ │ -000373f0: 2020 6d61 700a 1f0a 4669 6c65 3a20 4372    map...File: Cr
│ │ │ │ -00037400: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ -00037410: 3a20 6465 6772 6565 4d61 702c 204e 6578  : degreeMap, Nex
│ │ │ │ -00037420: 743a 2064 6567 7265 654d 6170 5f6c 7052  t: degreeMap_lpR
│ │ │ │ -00037430: 6174 696f 6e61 6c4d 6170 5f72 702c 2050  ationalMap_rp, P
│ │ │ │ -00037440: 7265 763a 2064 6567 7265 655f 6c70 5261  rev: degree_lpRa
│ │ │ │ -00037450: 7469 6f6e 616c 4d61 705f 7270 2c20 5570  tionalMap_rp, Up
│ │ │ │ -00037460: 3a20 546f 700a 0a64 6567 7265 654d 6170  : Top..degreeMap
│ │ │ │ -00037470: 202d 2d20 6465 6772 6565 206f 6620 6120   -- degree of a 
│ │ │ │ -00037480: 7261 7469 6f6e 616c 206d 6170 2062 6574  rational map bet
│ │ │ │ -00037490: 7765 656e 2070 726f 6a65 6374 6976 6520  ween projective 
│ │ │ │ -000374a0: 7661 7269 6574 6965 730a 2a2a 2a2a 2a2a  varieties.******
│ │ │ │ +00037180: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00037190: 2a2a 2a2a 0a0a 5379 6e6f 7073 6973 0a3d  ****..Synopsis.=
│ │ │ │ +000371a0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2046 756e  =======..  * Fun
│ │ │ │ +000371b0: 6374 696f 6e3a 202a 6e6f 7465 2064 6567  ction: *note deg
│ │ │ │ +000371c0: 7265 653a 2028 4d61 6361 756c 6179 3244  ree: (Macaulay2D
│ │ │ │ +000371d0: 6f63 2964 6567 7265 652c 0a20 202a 2055  oc)degree,.  * U
│ │ │ │ +000371e0: 7361 6765 3a20 0a20 2020 2020 2020 2064  sage: .        d
│ │ │ │ +000371f0: 6567 7265 6520 7068 690a 2020 2a20 496e  egree phi.  * In
│ │ │ │ +00037200: 7075 7473 3a0a 2020 2020 2020 2a20 7068  puts:.      * ph
│ │ │ │ +00037210: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ +00037220: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ +00037230: 6c4d 6170 2c0a 2020 2a20 4f75 7470 7574  lMap,.  * Output
│ │ │ │ +00037240: 733a 0a20 2020 2020 202a 2061 6e20 2a6e  s:.      * an *n
│ │ │ │ +00037250: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ +00037260: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ +00037270: 7468 6520 6465 6772 6565 206f 6620 7068  the degree of ph
│ │ │ │ +00037280: 692e 2053 6f20 7468 6973 2076 616c 7565  i. So this value
│ │ │ │ +00037290: 0a20 2020 2020 2020 2069 7320 3120 6966  .        is 1 if
│ │ │ │ +000372a0: 2061 6e64 206f 6e6c 7920 6966 2074 6865   and only if the
│ │ │ │ +000372b0: 206d 6170 2069 7320 6269 7261 7469 6f6e   map is biration
│ │ │ │ +000372c0: 616c 206f 6e74 6f20 6974 7320 696d 6167  al onto its imag
│ │ │ │ +000372d0: 652e 0a0a 4465 7363 7269 7074 696f 6e0a  e...Description.
│ │ │ │ +000372e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ +000372f0: 7320 6973 2061 2073 686f 7274 6375 7420  s is a shortcut 
│ │ │ │ +00037300: 666f 7220 6465 6772 6565 4d61 7028 7068  for degreeMap(ph
│ │ │ │ +00037310: 692c 4365 7274 6966 793d 3e74 7275 652c  i,Certify=>true,
│ │ │ │ +00037320: 5665 7262 6f73 653d 3e66 616c 7365 292c  Verbose=>false),
│ │ │ │ +00037330: 2073 6565 202a 6e6f 7465 0a64 6567 7265   see *note.degre
│ │ │ │ +00037340: 654d 6170 2852 6174 696f 6e61 6c4d 6170  eMap(RationalMap
│ │ │ │ +00037350: 293a 2064 6567 7265 654d 6170 5f6c 7052  ): degreeMap_lpR
│ │ │ │ +00037360: 6174 696f 6e61 6c4d 6170 5f72 702c 2e0a  ationalMap_rp,..
│ │ │ │ +00037370: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ +00037380: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ +00037390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000373a0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6465  ==..  * *note de
│ │ │ │ +000373b0: 6772 6565 2852 6174 696f 6e61 6c4d 6170  gree(RationalMap
│ │ │ │ +000373c0: 293a 2064 6567 7265 655f 6c70 5261 7469  ): degree_lpRati
│ │ │ │ +000373d0: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2064  onalMap_rp, -- d
│ │ │ │ +000373e0: 6567 7265 6520 6f66 2061 2072 6174 696f  egree of a ratio
│ │ │ │ +000373f0: 6e61 6c0a 2020 2020 6d61 700a 1f0a 4669  nal.    map...Fi
│ │ │ │ +00037400: 6c65 3a20 4372 656d 6f6e 612e 696e 666f  le: Cremona.info
│ │ │ │ +00037410: 2c20 4e6f 6465 3a20 6465 6772 6565 4d61  , Node: degreeMa
│ │ │ │ +00037420: 702c 204e 6578 743a 2064 6567 7265 654d  p, Next: degreeM
│ │ │ │ +00037430: 6170 5f6c 7052 6174 696f 6e61 6c4d 6170  ap_lpRationalMap
│ │ │ │ +00037440: 5f72 702c 2050 7265 763a 2064 6567 7265  _rp, Prev: degre
│ │ │ │ +00037450: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +00037460: 7270 2c20 5570 3a20 546f 700a 0a64 6567  rp, Up: Top..deg
│ │ │ │ +00037470: 7265 654d 6170 202d 2d20 6465 6772 6565  reeMap -- degree
│ │ │ │ +00037480: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ +00037490: 6170 2062 6574 7765 656e 2070 726f 6a65  ap between proje
│ │ │ │ +000374a0: 6374 6976 6520 7661 7269 6574 6965 730a  ctive varieties.
│ │ │ │  000374b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000374c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000374d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000374e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -000374f0: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -00037500: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -00037510: 2020 2020 2064 6567 7265 654d 6170 2070       degreeMap p
│ │ │ │ -00037520: 6869 0a20 202a 2049 6e70 7574 733a 0a20  hi.  * Inputs:. 
│ │ │ │ -00037530: 2020 2020 202a 2070 6869 2c20 6120 2a6e       * phi, a *n
│ │ │ │ -00037540: 6f74 6520 7269 6e67 206d 6170 3a20 284d  ote ring map: (M
│ │ │ │ -00037550: 6163 6175 6c61 7932 446f 6329 5269 6e67  acaulay2Doc)Ring
│ │ │ │ -00037560: 4d61 702c 2c20 7768 6963 6820 7265 7072  Map,, which repr
│ │ │ │ -00037570: 6573 656e 7473 2061 0a20 2020 2020 2020  esents a.       
│ │ │ │ -00037580: 2072 6174 696f 6e61 6c20 6d61 7020 245c   rational map $\
│ │ │ │ -00037590: 5068 6924 2062 6574 7765 656e 2070 726f  Phi$ between pro
│ │ │ │ -000375a0: 6a65 6374 6976 6520 7661 7269 6574 6965  jective varietie
│ │ │ │ -000375b0: 730a 2020 2a20 2a6e 6f74 6520 4f70 7469  s.  * *note Opti
│ │ │ │ -000375c0: 6f6e 616c 2069 6e70 7574 733a 2028 4d61  onal inputs: (Ma
│ │ │ │ -000375d0: 6361 756c 6179 3244 6f63 2975 7369 6e67  caulay2Doc)using
│ │ │ │ -000375e0: 2066 756e 6374 696f 6e73 2077 6974 6820   functions with 
│ │ │ │ -000375f0: 6f70 7469 6f6e 616c 2069 6e70 7574 732c  optional inputs,
│ │ │ │ -00037600: 3a0a 2020 2020 2020 2a20 2a6e 6f74 6520  :.      * *note 
│ │ │ │ -00037610: 426c 6f77 5570 5374 7261 7465 6779 3a20  BlowUpStrategy: 
│ │ │ │ -00037620: 426c 6f77 5570 5374 7261 7465 6779 2c20  BlowUpStrategy, 
│ │ │ │ -00037630: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -00037640: 7661 6c75 650a 2020 2020 2020 2020 2245  value.        "E
│ │ │ │ -00037650: 6c69 6d69 6e61 7465 222c 0a20 2020 2020  liminate",.     
│ │ │ │ -00037660: 202a 202a 6e6f 7465 2043 6572 7469 6679   * *note Certify
│ │ │ │ -00037670: 3a20 4365 7274 6966 792c 203d 3e20 2e2e  : Certify, => ..
│ │ │ │ -00037680: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -00037690: 2066 616c 7365 2c20 7768 6574 6865 7220   false, whether 
│ │ │ │ -000376a0: 746f 2065 6e73 7572 650a 2020 2020 2020  to ensure.      
│ │ │ │ -000376b0: 2020 636f 7272 6563 746e 6573 7320 6f66    correctness of
│ │ │ │ -000376c0: 206f 7574 7075 740a 2020 2020 2020 2a20   output.      * 
│ │ │ │ -000376d0: 2a6e 6f74 6520 5665 7262 6f73 653a 2069  *note Verbose: i
│ │ │ │ -000376e0: 6e76 6572 7365 4d61 705f 6c70 5f70 645f  nverseMap_lp_pd_
│ │ │ │ -000376f0: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ -00037700: 3e5f 7064 5f70 645f 7064 5f72 702c 203d  >_pd_pd_pd_rp, =
│ │ │ │ -00037710: 3e20 2e2e 2e2c 0a20 2020 2020 2020 2064  > ...,.        d
│ │ │ │ -00037720: 6566 6175 6c74 2076 616c 7565 2074 7275  efault value tru
│ │ │ │ -00037730: 652c 0a20 202a 204f 7574 7075 7473 3a0a  e,.  * Outputs:.
│ │ │ │ -00037740: 2020 2020 2020 2a20 616e 202a 6e6f 7465        * an *note
│ │ │ │ -00037750: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ -00037760: 6c61 7932 446f 6329 5a5a 2c2c 2074 6865  lay2Doc)ZZ,, the
│ │ │ │ -00037770: 2064 6567 7265 6520 6f66 2024 5c50 6869   degree of $\Phi
│ │ │ │ -00037780: 242e 2053 6f20 7468 6973 0a20 2020 2020  $. So this.     
│ │ │ │ -00037790: 2020 2076 616c 7565 2069 7320 3120 6966     value is 1 if
│ │ │ │ -000377a0: 2061 6e64 206f 6e6c 7920 6966 2074 6865   and only if the
│ │ │ │ -000377b0: 206d 6170 2069 7320 6269 7261 7469 6f6e   map is biration
│ │ │ │ -000377c0: 616c 206f 6e74 6f20 6974 7320 696d 6167  al onto its imag
│ │ │ │ -000377d0: 652e 0a0a 4465 7363 7269 7074 696f 6e0a  e...Description.
│ │ │ │ -000377e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4f 6e65  ===========..One
│ │ │ │ -000377f0: 2069 6d70 6f72 7461 6e74 2063 6173 6520   important case 
│ │ │ │ -00037800: 6973 2077 6865 6e20 245c 5068 693a 5c6d  is when $\Phi:\m
│ │ │ │ -00037810: 6174 6862 627b 507d 5e6e 3d50 726f 6a28  athbb{P}^n=Proj(
│ │ │ │ -00037820: 4b5b 785f 302c 5c6c 646f 7473 2c78 5f6e  K[x_0,\ldots,x_n
│ │ │ │ -00037830: 5d29 0a5c 6461 7368 7269 6768 7461 7272  ]).\dashrightarr
│ │ │ │ -00037840: 6f77 205c 6d61 7468 6262 7b50 7d5e 6d3d  ow \mathbb{P}^m=
│ │ │ │ -00037850: 5072 6f6a 284b 5b79 5f30 2c5c 6c64 6f74  Proj(K[y_0,\ldot
│ │ │ │ -00037860: 732c 795f 6d5d 2924 2069 7320 6120 7261  s,y_m])$ is a ra
│ │ │ │ -00037870: 7469 6f6e 616c 206d 6170 2062 6574 7765  tional map betwe
│ │ │ │ -00037880: 656e 0a70 726f 6a65 6374 6976 6520 7370  en.projective sp
│ │ │ │ -00037890: 6163 6573 2c20 636f 7272 6573 706f 6e64  aces, correspond
│ │ │ │ -000378a0: 696e 6720 746f 2061 2072 696e 6720 6d61  ing to a ring ma
│ │ │ │ -000378b0: 7020 245c 7068 6924 2e20 4966 2024 7024  p $\phi$. If $p$
│ │ │ │ -000378c0: 2069 7320 6120 6765 6e65 7261 6c0a 706f   is a general.po
│ │ │ │ -000378d0: 696e 7420 6f66 2024 5c6d 6174 6862 627b  int of $\mathbb{
│ │ │ │ -000378e0: 507d 5e6e 242c 2064 656e 6f74 6520 6279  P}^n$, denote by
│ │ │ │ -000378f0: 2024 465f 7028 5c50 6869 2924 2074 6865   $F_p(\Phi)$ the
│ │ │ │ -00037900: 2063 6c6f 7375 7265 206f 660a 245c 5068   closure of.$\Ph
│ │ │ │ -00037910: 695e 7b2d 317d 285c 5068 6928 7029 295c  i^{-1}(\Phi(p))\
│ │ │ │ -00037920: 7375 6273 6574 6571 205c 6d61 7468 6262  subseteq \mathbb
│ │ │ │ -00037930: 7b50 7d5e 6e24 2e20 5468 6520 6465 6772  {P}^n$. The degr
│ │ │ │ -00037940: 6565 206f 6620 245c 5068 6924 2069 7320  ee of $\Phi$ is 
│ │ │ │ -00037950: 6465 6669 6e65 6420 6173 0a74 6865 2064  defined as.the d
│ │ │ │ -00037960: 6567 7265 6520 6f66 2024 465f 7028 5c50  egree of $F_p(\P
│ │ │ │ -00037970: 6869 2924 2069 6620 2464 696d 2046 5f70  hi)$ if $dim F_p
│ │ │ │ -00037980: 285c 5068 6929 203d 2030 2420 616e 6420  (\Phi) = 0$ and 
│ │ │ │ -00037990: 2430 2420 6f74 6865 7277 6973 652e 2049  $0$ otherwise. I
│ │ │ │ -000379a0: 6620 245c 5068 6924 0a69 7320 6465 6669  f $\Phi$.is defi
│ │ │ │ -000379b0: 6e65 6420 6279 2066 6f72 6d73 2024 465f  ned by forms $F_
│ │ │ │ -000379c0: 3028 785f 302c 5c6c 646f 7473 2c78 5f6e  0(x_0,\ldots,x_n
│ │ │ │ -000379d0: 292c 5c6c 646f 7473 2c46 5f6d 2878 5f30  ),\ldots,F_m(x_0
│ │ │ │ -000379e0: 2c5c 6c64 6f74 732c 785f 6e29 2420 616e  ,\ldots,x_n)$ an
│ │ │ │ -000379f0: 6420 2449 5f70 240a 6973 2074 6865 2069  d $I_p$.is the i
│ │ │ │ -00037a00: 6465 616c 206f 6620 7468 6520 706f 696e  deal of the poin
│ │ │ │ -00037a10: 7420 2470 242c 2074 6865 6e20 7468 6520  t $p$, then the 
│ │ │ │ -00037a20: 6964 6561 6c20 6f66 2024 465f 7028 5c50  ideal of $F_p(\P
│ │ │ │ -00037a30: 6869 2924 2069 7320 6e6f 7468 696e 6720  hi)$ is nothing 
│ │ │ │ -00037a40: 6275 7420 7468 650a 7361 7475 7261 7469  but the.saturati
│ │ │ │ -00037a50: 6f6e 2024 7b28 5c70 6869 285c 7068 695e  on ${(\phi(\phi^
│ │ │ │ -00037a60: 7b2d 317d 2849 5f70 2929 293a 2846 5f30  {-1}(I_p))):(F_0
│ │ │ │ -00037a70: 2c2e 2e2e 2e2c 465f 6d29 7d5e 7b5c 696e  ,....,F_m)}^{\in
│ │ │ │ -00037a80: 6674 797d 242e 0a0a 2b2d 2d2d 2d2d 2d2d  fty}$...+-------
│ │ │ │ +000374e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000374f0: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +00037500: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ +00037510: 3a20 0a20 2020 2020 2020 2064 6567 7265  : .        degre
│ │ │ │ +00037520: 654d 6170 2070 6869 0a20 202a 2049 6e70  eMap phi.  * Inp
│ │ │ │ +00037530: 7574 733a 0a20 2020 2020 202a 2070 6869  uts:.      * phi
│ │ │ │ +00037540: 2c20 6120 2a6e 6f74 6520 7269 6e67 206d  , a *note ring m
│ │ │ │ +00037550: 6170 3a20 284d 6163 6175 6c61 7932 446f  ap: (Macaulay2Do
│ │ │ │ +00037560: 6329 5269 6e67 4d61 702c 2c20 7768 6963  c)RingMap,, whic
│ │ │ │ +00037570: 6820 7265 7072 6573 656e 7473 2061 0a20  h represents a. 
│ │ │ │ +00037580: 2020 2020 2020 2072 6174 696f 6e61 6c20         rational 
│ │ │ │ +00037590: 6d61 7020 245c 5068 6924 2062 6574 7765  map $\Phi$ betwe
│ │ │ │ +000375a0: 656e 2070 726f 6a65 6374 6976 6520 7661  en projective va
│ │ │ │ +000375b0: 7269 6574 6965 730a 2020 2a20 2a6e 6f74  rieties.  * *not
│ │ │ │ +000375c0: 6520 4f70 7469 6f6e 616c 2069 6e70 7574  e Optional input
│ │ │ │ +000375d0: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ +000375e0: 2975 7369 6e67 2066 756e 6374 696f 6e73  )using functions
│ │ │ │ +000375f0: 2077 6974 6820 6f70 7469 6f6e 616c 2069   with optional i
│ │ │ │ +00037600: 6e70 7574 732c 3a0a 2020 2020 2020 2a20  nputs,:.      * 
│ │ │ │ +00037610: 2a6e 6f74 6520 426c 6f77 5570 5374 7261  *note BlowUpStra
│ │ │ │ +00037620: 7465 6779 3a20 426c 6f77 5570 5374 7261  tegy: BlowUpStra
│ │ │ │ +00037630: 7465 6779 2c20 3d3e 202e 2e2e 2c20 6465  tegy, => ..., de
│ │ │ │ +00037640: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ +00037650: 2020 2020 2245 6c69 6d69 6e61 7465 222c      "Eliminate",
│ │ │ │ +00037660: 0a20 2020 2020 202a 202a 6e6f 7465 2043  .      * *note C
│ │ │ │ +00037670: 6572 7469 6679 3a20 4365 7274 6966 792c  ertify: Certify,
│ │ │ │ +00037680: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +00037690: 2076 616c 7565 2066 616c 7365 2c20 7768   value false, wh
│ │ │ │ +000376a0: 6574 6865 7220 746f 2065 6e73 7572 650a  ether to ensure.
│ │ │ │ +000376b0: 2020 2020 2020 2020 636f 7272 6563 746e          correctn
│ │ │ │ +000376c0: 6573 7320 6f66 206f 7574 7075 740a 2020  ess of output.  
│ │ │ │ +000376d0: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
│ │ │ │ +000376e0: 6f73 653a 2069 6e76 6572 7365 4d61 705f  ose: inverseMap_
│ │ │ │ +000376f0: 6c70 5f70 645f 7064 5f70 645f 636d 5665  lp_pd_pd_pd_cmVe
│ │ │ │ +00037700: 7262 6f73 653d 3e5f 7064 5f70 645f 7064  rbose=>_pd_pd_pd
│ │ │ │ +00037710: 5f72 702c 203d 3e20 2e2e 2e2c 0a20 2020  _rp, => ...,.   
│ │ │ │ +00037720: 2020 2020 2064 6566 6175 6c74 2076 616c       default val
│ │ │ │ +00037730: 7565 2074 7275 652c 0a20 202a 204f 7574  ue true,.  * Out
│ │ │ │ +00037740: 7075 7473 3a0a 2020 2020 2020 2a20 616e  puts:.      * an
│ │ │ │ +00037750: 202a 6e6f 7465 2069 6e74 6567 6572 3a20   *note integer: 
│ │ │ │ +00037760: 284d 6163 6175 6c61 7932 446f 6329 5a5a  (Macaulay2Doc)ZZ
│ │ │ │ +00037770: 2c2c 2074 6865 2064 6567 7265 6520 6f66  ,, the degree of
│ │ │ │ +00037780: 2024 5c50 6869 242e 2053 6f20 7468 6973   $\Phi$. So this
│ │ │ │ +00037790: 0a20 2020 2020 2020 2076 616c 7565 2069  .        value i
│ │ │ │ +000377a0: 7320 3120 6966 2061 6e64 206f 6e6c 7920  s 1 if and only 
│ │ │ │ +000377b0: 6966 2074 6865 206d 6170 2069 7320 6269  if the map is bi
│ │ │ │ +000377c0: 7261 7469 6f6e 616c 206f 6e74 6f20 6974  rational onto it
│ │ │ │ +000377d0: 7320 696d 6167 652e 0a0a 4465 7363 7269  s image...Descri
│ │ │ │ +000377e0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +000377f0: 3d0a 0a4f 6e65 2069 6d70 6f72 7461 6e74  =..One important
│ │ │ │ +00037800: 2063 6173 6520 6973 2077 6865 6e20 245c   case is when $\
│ │ │ │ +00037810: 5068 693a 5c6d 6174 6862 627b 507d 5e6e  Phi:\mathbb{P}^n
│ │ │ │ +00037820: 3d50 726f 6a28 4b5b 785f 302c 5c6c 646f  =Proj(K[x_0,\ldo
│ │ │ │ +00037830: 7473 2c78 5f6e 5d29 0a5c 6461 7368 7269  ts,x_n]).\dashri
│ │ │ │ +00037840: 6768 7461 7272 6f77 205c 6d61 7468 6262  ghtarrow \mathbb
│ │ │ │ +00037850: 7b50 7d5e 6d3d 5072 6f6a 284b 5b79 5f30  {P}^m=Proj(K[y_0
│ │ │ │ +00037860: 2c5c 6c64 6f74 732c 795f 6d5d 2924 2069  ,\ldots,y_m])$ i
│ │ │ │ +00037870: 7320 6120 7261 7469 6f6e 616c 206d 6170  s a rational map
│ │ │ │ +00037880: 2062 6574 7765 656e 0a70 726f 6a65 6374   between.project
│ │ │ │ +00037890: 6976 6520 7370 6163 6573 2c20 636f 7272  ive spaces, corr
│ │ │ │ +000378a0: 6573 706f 6e64 696e 6720 746f 2061 2072  esponding to a r
│ │ │ │ +000378b0: 696e 6720 6d61 7020 245c 7068 6924 2e20  ing map $\phi$. 
│ │ │ │ +000378c0: 4966 2024 7024 2069 7320 6120 6765 6e65  If $p$ is a gene
│ │ │ │ +000378d0: 7261 6c0a 706f 696e 7420 6f66 2024 5c6d  ral.point of $\m
│ │ │ │ +000378e0: 6174 6862 627b 507d 5e6e 242c 2064 656e  athbb{P}^n$, den
│ │ │ │ +000378f0: 6f74 6520 6279 2024 465f 7028 5c50 6869  ote by $F_p(\Phi
│ │ │ │ +00037900: 2924 2074 6865 2063 6c6f 7375 7265 206f  )$ the closure o
│ │ │ │ +00037910: 660a 245c 5068 695e 7b2d 317d 285c 5068  f.$\Phi^{-1}(\Ph
│ │ │ │ +00037920: 6928 7029 295c 7375 6273 6574 6571 205c  i(p))\subseteq \
│ │ │ │ +00037930: 6d61 7468 6262 7b50 7d5e 6e24 2e20 5468  mathbb{P}^n$. Th
│ │ │ │ +00037940: 6520 6465 6772 6565 206f 6620 245c 5068  e degree of $\Ph
│ │ │ │ +00037950: 6924 2069 7320 6465 6669 6e65 6420 6173  i$ is defined as
│ │ │ │ +00037960: 0a74 6865 2064 6567 7265 6520 6f66 2024  .the degree of $
│ │ │ │ +00037970: 465f 7028 5c50 6869 2924 2069 6620 2464  F_p(\Phi)$ if $d
│ │ │ │ +00037980: 696d 2046 5f70 285c 5068 6929 203d 2030  im F_p(\Phi) = 0
│ │ │ │ +00037990: 2420 616e 6420 2430 2420 6f74 6865 7277  $ and $0$ otherw
│ │ │ │ +000379a0: 6973 652e 2049 6620 245c 5068 6924 0a69  ise. If $\Phi$.i
│ │ │ │ +000379b0: 7320 6465 6669 6e65 6420 6279 2066 6f72  s defined by for
│ │ │ │ +000379c0: 6d73 2024 465f 3028 785f 302c 5c6c 646f  ms $F_0(x_0,\ldo
│ │ │ │ +000379d0: 7473 2c78 5f6e 292c 5c6c 646f 7473 2c46  ts,x_n),\ldots,F
│ │ │ │ +000379e0: 5f6d 2878 5f30 2c5c 6c64 6f74 732c 785f  _m(x_0,\ldots,x_
│ │ │ │ +000379f0: 6e29 2420 616e 6420 2449 5f70 240a 6973  n)$ and $I_p$.is
│ │ │ │ +00037a00: 2074 6865 2069 6465 616c 206f 6620 7468   the ideal of th
│ │ │ │ +00037a10: 6520 706f 696e 7420 2470 242c 2074 6865  e point $p$, the
│ │ │ │ +00037a20: 6e20 7468 6520 6964 6561 6c20 6f66 2024  n the ideal of $
│ │ │ │ +00037a30: 465f 7028 5c50 6869 2924 2069 7320 6e6f  F_p(\Phi)$ is no
│ │ │ │ +00037a40: 7468 696e 6720 6275 7420 7468 650a 7361  thing but the.sa
│ │ │ │ +00037a50: 7475 7261 7469 6f6e 2024 7b28 5c70 6869  turation ${(\phi
│ │ │ │ +00037a60: 285c 7068 695e 7b2d 317d 2849 5f70 2929  (\phi^{-1}(I_p))
│ │ │ │ +00037a70: 293a 2846 5f30 2c2e 2e2e 2e2c 465f 6d29  ):(F_0,....,F_m)
│ │ │ │ +00037a80: 7d5e 7b5c 696e 6674 797d 242e 0a0a 2b2d  }^{\infty}$...+-
│ │ │ │  00037a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037ad0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 2d2d  ------+.|i1 : --
│ │ │ │ -00037ae0: 2054 616b 6520 6120 7261 7469 6f6e 616c   Take a rational
│ │ │ │ -00037af0: 206d 6170 2070 6869 3a50 5e38 2d2d 2d3e   map phi:P^8--->
│ │ │ │ -00037b00: 4728 312c 3529 2073 7562 7365 7420 505e  G(1,5) subset P^
│ │ │ │ -00037b10: 3134 2064 6566 696e 6564 2062 7920 2020  14 defined by   
│ │ │ │ -00037b20: 2020 2020 2020 7c0a 7c20 2020 2020 2d2d        |.|     --
│ │ │ │ -00037b30: 206f 6620 6120 6765 6e65 7269 6320 3220   of a generic 2 
│ │ │ │ -00037b40: 7820 3620 6d61 7472 6978 206f 6620 6c69  x 6 matrix of li
│ │ │ │ -00037b50: 6e65 6172 2066 6f72 6d73 206f 6e20 505e  near forms on P^
│ │ │ │ -00037b60: 3820 2874 6875 7320 7068 6920 6973 2020  8 (thus phi is  
│ │ │ │ -00037b70: 2020 2020 2020 7c0a 7c20 2020 2020 4b3d        |.|     K=
│ │ │ │ -00037b80: 5a5a 2f33 3333 313b 2072 696e 6750 383d  ZZ/3331; ringP8=
│ │ │ │ -00037b90: 4b5b 785f 302e 2e78 5f38 5d3b 2072 696e  K[x_0..x_8]; rin
│ │ │ │ -00037ba0: 6750 3134 3d4b 5b74 5f30 2e2e 745f 3134  gP14=K[t_0..t_14
│ │ │ │ -00037bb0: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ -00037bc0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00037ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00037ae0: 3120 3a20 2d2d 2054 616b 6520 6120 7261  1 : -- Take a ra
│ │ │ │ +00037af0: 7469 6f6e 616c 206d 6170 2070 6869 3a50  tional map phi:P
│ │ │ │ +00037b00: 5e38 2d2d 2d3e 4728 312c 3529 2073 7562  ^8--->G(1,5) sub
│ │ │ │ +00037b10: 7365 7420 505e 3134 2064 6566 696e 6564  set P^14 defined
│ │ │ │ +00037b20: 2062 7920 2020 2020 2020 2020 7c0a 7c20   by         |.| 
│ │ │ │ +00037b30: 2020 2020 2d2d 206f 6620 6120 6765 6e65      -- of a gene
│ │ │ │ +00037b40: 7269 6320 3220 7820 3620 6d61 7472 6978  ric 2 x 6 matrix
│ │ │ │ +00037b50: 206f 6620 6c69 6e65 6172 2066 6f72 6d73   of linear forms
│ │ │ │ +00037b60: 206f 6e20 505e 3820 2874 6875 7320 7068   on P^8 (thus ph
│ │ │ │ +00037b70: 6920 6973 2020 2020 2020 2020 7c0a 7c20  i is        |.| 
│ │ │ │ +00037b80: 2020 2020 4b3d 5a5a 2f33 3333 313b 2072      K=ZZ/3331; r
│ │ │ │ +00037b90: 696e 6750 383d 4b5b 785f 302e 2e78 5f38  ingP8=K[x_0..x_8
│ │ │ │ +00037ba0: 5d3b 2072 696e 6750 3134 3d4b 5b74 5f30  ]; ringP14=K[t_0
│ │ │ │ +00037bb0: 2e2e 745f 3134 5d3b 2020 2020 2020 2020  ..t_14];        
│ │ │ │ +00037bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  00037bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037c10: 2d2d 2d2d 2d2d 7c0a 7c74 6865 206d 6178  ------|.|the max
│ │ │ │ -00037c20: 696d 616c 206d 696e 6f72 7320 2020 2020  imal minors     
│ │ │ │ -00037c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c74  ------------|.|t
│ │ │ │ +00037c20: 6865 206d 6178 696d 616c 206d 696e 6f72  he maximal minor
│ │ │ │ +00037c30: 7320 2020 2020 2020 2020 2020 2020 2020  s               
│ │ │ │  00037c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c60: 2020 2020 2020 7c0a 7c20 6269 7261 7469        |.| birati
│ │ │ │ -00037c70: 6f6e 616c 206f 6e74 6f20 6974 7320 696d  onal onto its im
│ │ │ │ -00037c80: 6167 6529 2020 2020 2020 2020 2020 2020  age)            
│ │ │ │ +00037c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00037c70: 6269 7261 7469 6f6e 616c 206f 6e74 6f20  birational onto 
│ │ │ │ +00037c80: 6974 7320 696d 6167 6529 2020 2020 2020  its image)      
│ │ │ │  00037c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037cb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00037cb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00037cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037d00: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7068  ------+.|i4 : ph
│ │ │ │ -00037d10: 693d 6d61 7028 7269 6e67 5038 2c72 696e  i=map(ringP8,rin
│ │ │ │ -00037d20: 6750 3134 2c67 656e 7320 6d69 6e6f 7273  gP14,gens minors
│ │ │ │ -00037d30: 2832 2c6d 6174 7269 7820 7061 636b 2836  (2,matrix pack(6
│ │ │ │ -00037d40: 2c66 6f72 2069 2074 6f20 2020 2020 2020  ,for i to       
│ │ │ │ -00037d50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00037d10: 3420 3a20 7068 693d 6d61 7028 7269 6e67  4 : phi=map(ring
│ │ │ │ +00037d20: 5038 2c72 696e 6750 3134 2c67 656e 7320  P8,ringP14,gens 
│ │ │ │ +00037d30: 6d69 6e6f 7273 2832 2c6d 6174 7269 7820  minors(2,matrix 
│ │ │ │ +00037d40: 7061 636b 2836 2c66 6f72 2069 2074 6f20  pack(6,for i to 
│ │ │ │ +00037d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00037da0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00037db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037dc0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00037dd0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00037de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037df0: 2020 2020 2020 7c0a 7c6f 3420 3d20 6d61        |.|o4 = ma
│ │ │ │ -00037e00: 7020 2872 696e 6750 382c 2072 696e 6750  p (ringP8, ringP
│ │ │ │ -00037e10: 3134 2c20 7b2d 2039 3578 2020 2b20 3138  14, {- 95x  + 18
│ │ │ │ -00037e20: 3178 2078 2020 2b20 3130 3238 7820 202d  1x x  + 1028x  -
│ │ │ │ -00037e30: 2031 3338 3478 2078 2020 2020 2020 2020   1384x x        
│ │ │ │ -00037e40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037dd0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00037de0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00037df0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00037e00: 3420 3d20 6d61 7020 2872 696e 6750 382c  4 = map (ringP8,
│ │ │ │ +00037e10: 2072 696e 6750 3134 2c20 7b2d 2039 3578   ringP14, {- 95x
│ │ │ │ +00037e20: 2020 2b20 3138 3178 2078 2020 2b20 3130    + 181x x  + 10
│ │ │ │ +00037e30: 3238 7820 202d 2031 3338 3478 2078 2020  28x  - 1384x x  
│ │ │ │ +00037e40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00037e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e60: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ -00037e70: 2020 3020 3120 2020 2020 2020 2031 2020    0 1        1  
│ │ │ │ -00037e80: 2020 2020 2020 3020 3220 2020 2020 2020        0 2       
│ │ │ │ -00037e90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037e70: 3020 2020 2020 2020 3020 3120 2020 2020  0       0 1     
│ │ │ │ +00037e80: 2020 2031 2020 2020 2020 2020 3020 3220     1        0 2 
│ │ │ │ +00037e90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ee0: 2020 2020 2020 7c0a 7c6f 3420 3a20 5269        |.|o4 : Ri
│ │ │ │ -00037ef0: 6e67 4d61 7020 7269 6e67 5038 203c 2d2d  ngMap ringP8 <--
│ │ │ │ -00037f00: 2072 696e 6750 3134 2020 2020 2020 2020   ringP14        
│ │ │ │ +00037ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00037ef0: 3420 3a20 5269 6e67 4d61 7020 7269 6e67  4 : RingMap ring
│ │ │ │ +00037f00: 5038 203c 2d2d 2072 696e 6750 3134 2020  P8 <-- ringP14  
│ │ │ │  00037f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f30: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00037f30: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  00037f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037f80: 2d2d 2d2d 2d2d 7c0a 7c31 3120 6c69 7374  ------|.|11 list
│ │ │ │ -00037f90: 2072 616e 646f 6d28 312c 7269 6e67 5038   random(1,ringP8
│ │ │ │ -00037fa0: 2929 2929 2020 2020 2020 2020 2020 2020  ))))            
│ │ │ │ +00037f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c31  ------------|.|1
│ │ │ │ +00037f90: 3120 6c69 7374 2072 616e 646f 6d28 312c  1 list random(1,
│ │ │ │ +00037fa0: 7269 6e67 5038 2929 2929 2020 2020 2020  ringP8))))      
│ │ │ │  00037fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037fd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00038020: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00038030: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00038040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038020: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00038030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038040: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00038050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038060: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00038070: 2020 2020 2020 7c0a 7c2d 2031 3435 3578        |.|- 1455x
│ │ │ │ -00038080: 2078 2020 2b20 3535 3978 2020 2d20 3530   x  + 559x  - 50
│ │ │ │ -00038090: 3278 2078 2020 2b20 3132 3634 7820 7820  2x x  + 1264x x 
│ │ │ │ -000380a0: 202d 2031 3632 7820 7820 202b 2031 3230   - 162x x  + 120
│ │ │ │ -000380b0: 3978 2020 2d20 3138 3078 2078 2020 2d20  9x  - 180x x  - 
│ │ │ │ -000380c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000380d0: 3120 3220 2020 2020 2020 3220 2020 2020  1 2       2     
│ │ │ │ -000380e0: 2020 3020 3320 2020 2020 2020 2031 2033    0 3        1 3
│ │ │ │ -000380f0: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ -00038100: 2020 3320 2020 2020 2020 3020 3420 2020    3       0 4   
│ │ │ │ -00038110: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00038060: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00038070: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00038080: 2031 3435 3578 2078 2020 2b20 3535 3978   1455x x  + 559x
│ │ │ │ +00038090: 2020 2d20 3530 3278 2078 2020 2b20 3132    - 502x x  + 12
│ │ │ │ +000380a0: 3634 7820 7820 202d 2031 3632 7820 7820  64x x  - 162x x 
│ │ │ │ +000380b0: 202b 2031 3230 3978 2020 2d20 3138 3078   + 1209x  - 180x
│ │ │ │ +000380c0: 2078 2020 2d20 2020 2020 2020 7c0a 7c20   x  -       |.| 
│ │ │ │ +000380d0: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │ +000380e0: 3220 2020 2020 2020 3020 3320 2020 2020  2       0 3     
│ │ │ │ +000380f0: 2020 2031 2033 2020 2020 2020 2032 2033     1 3       2 3
│ │ │ │ +00038100: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +00038110: 3020 3420 2020 2020 2020 2020 7c0a 7c2d  0 4         |.|-
│ │ │ │  00038120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038160: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00038160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  00038170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038180: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00038190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038190: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  000381a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000381b0: 2020 2020 2020 7c0a 7c35 3034 7820 7820        |.|504x x 
│ │ │ │ -000381c0: 202d 2031 3136 3878 2078 2020 2d20 3637   - 1168x x  - 67
│ │ │ │ -000381d0: 3678 2078 2020 2b20 3530 3178 2020 2b20  6x x  + 501x  + 
│ │ │ │ -000381e0: 3733 7820 7820 202b 2031 3236 3378 2078  73x x  + 1263x x
│ │ │ │ -000381f0: 2020 2b20 3130 3335 7820 7820 202b 2038    + 1035x x  + 8
│ │ │ │ -00038200: 3434 7820 7820 7c0a 7c20 2020 2031 2034  44x x |.|    1 4
│ │ │ │ -00038210: 2020 2020 2020 2020 3220 3420 2020 2020          2 4     
│ │ │ │ -00038220: 2020 3320 3420 2020 2020 2020 3420 2020    3 4       4   
│ │ │ │ -00038230: 2020 2030 2035 2020 2020 2020 2020 3120     0 5        1 
│ │ │ │ -00038240: 3520 2020 2020 2020 2032 2035 2020 2020  5        2 5    
│ │ │ │ -00038250: 2020 2033 2035 7c0a 7c2d 2d2d 2d2d 2d2d     3 5|.|-------
│ │ │ │ +000381b0: 2020 2020 2020 2020 2020 2020 7c0a 7c35              |.|5
│ │ │ │ +000381c0: 3034 7820 7820 202d 2031 3136 3878 2078  04x x  - 1168x x
│ │ │ │ +000381d0: 2020 2d20 3637 3678 2078 2020 2b20 3530    - 676x x  + 50
│ │ │ │ +000381e0: 3178 2020 2b20 3733 7820 7820 202b 2031  1x  + 73x x  + 1
│ │ │ │ +000381f0: 3236 3378 2078 2020 2b20 3130 3335 7820  263x x  + 1035x 
│ │ │ │ +00038200: 7820 202b 2038 3434 7820 7820 7c0a 7c20  x  + 844x x |.| 
│ │ │ │ +00038210: 2020 2031 2034 2020 2020 2020 2020 3220     1 4        2 
│ │ │ │ +00038220: 3420 2020 2020 2020 3320 3420 2020 2020  4       3 4     
│ │ │ │ +00038230: 2020 3420 2020 2020 2030 2035 2020 2020    4      0 5    
│ │ │ │ +00038240: 2020 2020 3120 3520 2020 2020 2020 2032      1 5        2
│ │ │ │ +00038250: 2035 2020 2020 2020 2033 2035 7c0a 7c2d   5       3 5|.|-
│ │ │ │  00038260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000382a0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -000382b0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000382c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000382a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +000382b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000382c0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  000382d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000382e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000382f0: 2020 2020 2020 7c0a 7c2b 2031 3539 3378        |.|+ 1593x
│ │ │ │ -00038300: 2078 2020 2b20 3738 3578 2020 2b20 3938   x  + 785x  + 98
│ │ │ │ -00038310: 3278 2078 2020 2d20 3431 3278 2078 2020  2x x  - 412x x  
│ │ │ │ -00038320: 2b20 3133 3335 7820 7820 202b 2031 3133  + 1335x x  + 113
│ │ │ │ -00038330: 3678 2078 2020 2b20 3832 3678 2078 2020  6x x  + 826x x  
│ │ │ │ -00038340: 2b20 2020 2020 7c0a 7c20 2020 2020 2020  +     |.|       
│ │ │ │ -00038350: 3420 3520 2020 2020 2020 3520 2020 2020  4 5       5     
│ │ │ │ -00038360: 2020 3020 3620 2020 2020 2020 3120 3620    0 6       1 6 
│ │ │ │ -00038370: 2020 2020 2020 2032 2036 2020 2020 2020         2 6      
│ │ │ │ -00038380: 2020 3320 3620 2020 2020 2020 3420 3620    3 6       4 6 
│ │ │ │ -00038390: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +000382f0: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ +00038300: 2031 3539 3378 2078 2020 2b20 3738 3578   1593x x  + 785x
│ │ │ │ +00038310: 2020 2b20 3938 3278 2078 2020 2d20 3431    + 982x x  - 41
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│ │ │ │ -00038c00: 2020 2020 2020 7c0a 7c20 2020 2020 2035        |.|      5
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│ │ │ │ -00039ec0: 7820 202b 2020 7c0a 7c20 2020 2030 2032  x  +  |.|    0 2
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│ │ │ │ -0003a140: 2020 2020 2020 7c0a 7c20 2020 2033 2035        |.|    3 5
│ │ │ │ -0003a150: 2020 2020 2020 2020 3420 3520 2020 2020          4 5     
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│ │ │ │ -0003a180: 3620 2020 2020 2020 2033 2036 2020 2020  6        3 6    
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│ │ │ │ +0003a110: 2020 2b20 3332 7820 202b 2036 3935 7820    + 32x  + 695x 
│ │ │ │ +0003a120: 7820 202b 2033 3035 7820 7820 202b 2031  x  + 305x x  + 1
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│ │ │ │ +0003a140: 7820 202b 2020 2020 2020 2020 7c0a 7c20  x  +        |.| 
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│ │ │ │ +0003a160: 3520 2020 2020 2035 2020 2020 2020 2030  5      5       0
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│ │ │ │ +0003a180: 2020 2020 3220 3620 2020 2020 2020 2033      2 6        3
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│ │ │ │ -0003a280: 3478 2078 2020 7c0a 7c20 2020 2020 3420  4x x  |.|     4 
│ │ │ │ -0003a290: 3620 2020 2020 2020 2035 2036 2020 2020  6        5 6    
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│ │ │ │ -0003a3c0: 2020 2020 2020 7c0a 7c20 2020 2020 2035        |.|      5
│ │ │ │ -0003a3d0: 2037 2020 2020 2020 2036 2037 2020 2020   7       6 7    
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│ │ │ │ -0003a4e0: 202b 2036 3939 7820 2c20 2d20 3232 3878   + 699x , - 228x
│ │ │ │ -0003a4f0: 2020 2d20 3135 3130 7820 7820 202b 2032    - 1510x x  + 2
│ │ │ │ -0003a500: 3737 7820 202d 7c0a 7c20 2020 2020 3420  77x  -|.|     4 
│ │ │ │ -0003a510: 3820 2020 2020 2020 3520 3820 2020 2020  8       5 8     
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│ │ │ │ -0003a530: 2020 2020 2020 2038 2020 2020 2020 2020         8        
│ │ │ │ -0003a540: 3020 2020 2020 2020 2030 2031 2020 2020  0        0 1    
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│ │ │ │ +0003a4d0: 2020 2b20 3130 3678 2078 2020 2d20 3133    + 106x x  - 13
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│ │ │ │ +0003a4f0: 2d20 3232 3878 2020 2d20 3135 3130 7820  - 228x  - 1510x 
│ │ │ │ +0003a500: 7820 202b 2032 3737 7820 202d 7c0a 7c20  x  + 277x  -|.| 
│ │ │ │ +0003a510: 2020 2020 3420 3820 2020 2020 2020 3520      4 8       5 
│ │ │ │ +0003a520: 3820 2020 2020 2020 3620 3820 2020 2020  8       6 8     
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│ │ │ │  0003a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003a5f0: 2020 2020 2020 7c0a 7c34 7820 7820 202d        |.|4x x  -
│ │ │ │ -0003a600: 2032 3278 2078 2020 2d20 3135 3236 7820   22x x  - 1526x 
│ │ │ │ -0003a610: 202b 2032 3334 7820 7820 202b 2039 3639   + 234x x  + 969
│ │ │ │ -0003a620: 7820 7820 202b 2031 3235 3378 2078 2020  x x  + 1253x x  
│ │ │ │ -0003a630: 2d20 3134 3236 7820 202d 2031 3437 3478  - 1426x  - 1474x
│ │ │ │ -0003a640: 2078 2020 2b20 7c0a 7c20 2030 2032 2020   x  + |.|  0 2  
│ │ │ │ -0003a650: 2020 2020 3120 3220 2020 2020 2020 2032      1 2        2
│ │ │ │ -0003a660: 2020 2020 2020 2030 2033 2020 2020 2020         0 3      
│ │ │ │ -0003a670: 2031 2033 2020 2020 2020 2020 3220 3320   1 3        2 3 
│ │ │ │ -0003a680: 2020 2020 2020 2033 2020 2020 2020 2020         3        
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│ │ │ │ +0003a600: 7820 7820 202d 2032 3278 2078 2020 2d20  x x  - 22x x  - 
│ │ │ │ +0003a610: 3135 3236 7820 202b 2032 3334 7820 7820  1526x  + 234x x 
│ │ │ │ +0003a620: 202b 2039 3639 7820 7820 202b 2031 3235   + 969x x  + 125
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│ │ │ │ +0003af20: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +0003af30: 3020 3520 2020 2020 2020 3120 3520 2020  0 5       1 5   
│ │ │ │ +0003af40: 2020 2020 3220 3520 2020 2020 2020 3320      2 5       3 
│ │ │ │ +0003af50: 3520 2020 2020 2020 2034 2035 7c0a 7c2d  5        4 5|.|-
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│ │ │ │  0003af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003afa0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2032  ------|.|      2
│ │ │ │ -0003afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0003afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aff0: 2020 2020 3220 7c0a 7c2b 2038 3130 7820      2 |.|+ 810x 
│ │ │ │ -0003b000: 202d 2034 3438 7820 7820 202d 2033 3537   - 448x x  - 357
│ │ │ │ -0003b010: 7820 7820 202d 2031 3037 7820 7820 202b  x x  - 107x x  +
│ │ │ │ -0003b020: 2034 3078 2078 2020 2b20 3738 3478 2078   40x x  + 784x x
│ │ │ │ -0003b030: 2020 2d20 3134 3233 7820 7820 202b 2031    - 1423x x  + 1
│ │ │ │ -0003b040: 3237 3678 2020 7c0a 7c20 2020 2020 2035  276x  |.|      5
│ │ │ │ -0003b050: 2020 2020 2020 2030 2036 2020 2020 2020         0 6      
│ │ │ │ -0003b060: 2031 2036 2020 2020 2020 2032 2036 2020   1 6       2 6  
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│ │ │ │ -0003b080: 3620 2020 2020 2020 2035 2036 2020 2020  6        5 6    
│ │ │ │ -0003b090: 2020 2020 3620 7c0a 7c2d 2d2d 2d2d 2d2d      6 |.|-------
│ │ │ │ +0003aff0: 2020 2020 2020 2020 2020 3220 7c0a 7c2b            2 |.|+
│ │ │ │ +0003b000: 2038 3130 7820 202d 2034 3438 7820 7820   810x  - 448x x 
│ │ │ │ +0003b010: 202d 2033 3537 7820 7820 202d 2031 3037   - 357x x  - 107
│ │ │ │ +0003b020: 7820 7820 202b 2034 3078 2078 2020 2b20  x x  + 40x x  + 
│ │ │ │ +0003b030: 3738 3478 2078 2020 2d20 3134 3233 7820  784x x  - 1423x 
│ │ │ │ +0003b040: 7820 202b 2031 3237 3678 2020 7c0a 7c20  x  + 1276x  |.| 
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│ │ │ │ +0003b060: 2020 2020 2020 2031 2036 2020 2020 2020         1 6      
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│ │ │ │ +0003b090: 2036 2020 2020 2020 2020 3620 7c0a 7c2d   6        6 |.|-
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│ │ │ │ -0003b1d0: 3720 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d  7     |.|-------
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│ │ │ │ +0003b160: 3037 3778 2078 2020 2d20 3132 3134 7820  077x x  - 1214x 
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│ │ │ │ +0003b1c0: 2037 2020 2020 2020 2035 2037 2020 2020   7       5 7    
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│ │ │ │ -0003b310: 2038 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   8    |.|-------
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│ │ │ │ +0003b2c0: 3131 3030 7820 7820 202b 2020 7c0a 7c20  1100x x  +  |.| 
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│ │ │ │ +0003b2e0: 2020 2020 3120 3820 2020 2020 2020 3220      1 8       2 
│ │ │ │ +0003b2f0: 3820 2020 2020 2033 2038 2020 2020 2020  8      3 8      
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│ │ │ │ -0003b3e0: 3478 2020 2b20 3531 7820 7820 202b 2032  4x  + 51x x  + 2
│ │ │ │ -0003b3f0: 3735 7820 7820 202b 2031 3134 3978 2020  75x x  + 1149x  
│ │ │ │ -0003b400: 2b20 2020 2020 7c0a 7c20 2020 2020 3720  +     |.|     7 
│ │ │ │ -0003b410: 3820 2020 2020 2020 3820 2020 2020 2020  8       8       
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│ │ │ │ +0003b3e0: 2020 2d20 3730 3478 2020 2b20 3531 7820    - 704x  + 51x 
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│ │ │ │ -0003cfb0: 3720 2020 2020 2020 2035 2037 2020 2020  7        5 7    
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│ │ │ │ -0003d1b0: 2020 2020 2020 7c0a 7c39 3838 7820 7820        |.|988x x 
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│ │ │ │ -0003d200: 202d 2020 2020 7c0a 7c20 2020 2037 2038   -    |.|    7 8
│ │ │ │ -0003d210: 2020 2020 2020 2038 2020 2020 2020 2020         8        
│ │ │ │ -0003d220: 2030 2020 2020 2020 3020 3120 2020 2020   0      0 1     
│ │ │ │ -0003d230: 2020 3120 2020 2020 2030 2032 2020 2020    1      0 2    
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│ │ │ │ +0003d1b0: 2020 2020 2032 2020 2020 2020 7c0a 7c39       2      |.|9
│ │ │ │ +0003d1c0: 3838 7820 7820 202b 2031 3133 7820 2c20  88x x  + 113x , 
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│ │ │ │ +0003d1e0: 2020 2b20 3830 3578 2020 2d20 3231 7820    + 805x  - 21x 
│ │ │ │ +0003d1f0: 7820 202d 2031 3635 3578 2078 2020 2b20  x  - 1655x x  + 
│ │ │ │ +0003d200: 3134 3739 7820 202d 2020 2020 7c0a 7c20  1479x  -    |.| 
│ │ │ │ +0003d210: 2020 2037 2038 2020 2020 2020 2038 2020     7 8       8  
│ │ │ │ +0003d220: 2020 2020 2020 2030 2020 2020 2020 3020         0      0 
│ │ │ │ +0003d230: 3120 2020 2020 2020 3120 2020 2020 2030  1       1      0
│ │ │ │ +0003d240: 2032 2020 2020 2020 2020 3120 3220 2020   2        1 2   
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│ │ │ │ -0003d4c0: 2020 2020 2020 2034 2035 2020 2020 2020         4 5      
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│ │ │ │ -0003d730: 3720 2020 2020 2020 3420 3720 2020 2020  7       4 7     
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│ │ │ │ +0003d6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
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│ │ │ │ -0003d990: 2020 2020 2020 2020 3720 3820 2020 2020          7 8     
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│ │ │ │  0003da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003da80: 2d20 3432 3878 2078 2020 2b20 3135 3478  - 428x x  + 154x
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│ │ │ │ -0003dad0: 2020 2020 2020 3020 3320 2020 2020 2020        0 3       
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│ │ │ │ +0003ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  0003ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed30: 2020 2020 2032 7c0a 7c31 3437 7820 7820       2|.|147x x 
│ │ │ │ -0003ed40: 202b 2039 3239 7820 7820 202d 2031 3232   + 929x x  - 122
│ │ │ │ -0003ed50: 3078 2078 2020 2d20 3835 3378 2078 2020  0x x  - 853x x  
│ │ │ │ -0003ed60: 2b20 3439 3378 2078 2020 2b20 3232 3678  + 493x x  + 226x
│ │ │ │ -0003ed70: 2078 2020 2b20 3134 3136 7820 7820 202b   x  + 1416x x  +
│ │ │ │ -0003ed80: 2032 3830 7820 7c0a 7c20 2020 2030 2037   280x |.|    0 7
│ │ │ │ -0003ed90: 2020 2020 2020 2031 2037 2020 2020 2020         1 7      
│ │ │ │ -0003eda0: 2020 3220 3720 2020 2020 2020 3320 3720    2 7       3 7 
│ │ │ │ -0003edb0: 2020 2020 2020 3420 3720 2020 2020 2020        4 7       
│ │ │ │ -0003edc0: 3520 3720 2020 2020 2020 2036 2037 2020  5 7        6 7  
│ │ │ │ -0003edd0: 2020 2020 2037 7c0a 7c2d 2d2d 2d2d 2d2d       7|.|-------
│ │ │ │ +0003ed30: 2020 2020 2020 2020 2020 2032 7c0a 7c31             2|.|1
│ │ │ │ +0003ed40: 3437 7820 7820 202b 2039 3239 7820 7820  47x x  + 929x x 
│ │ │ │ +0003ed50: 202d 2031 3232 3078 2078 2020 2d20 3835   - 1220x x  - 85
│ │ │ │ +0003ed60: 3378 2078 2020 2b20 3439 3378 2078 2020  3x x  + 493x x  
│ │ │ │ +0003ed70: 2b20 3232 3678 2078 2020 2b20 3134 3136  + 226x x  + 1416
│ │ │ │ +0003ed80: 7820 7820 202b 2032 3830 7820 7c0a 7c20  x x  + 280x |.| 
│ │ │ │ +0003ed90: 2020 2030 2037 2020 2020 2020 2031 2037     0 7       1 7
│ │ │ │ +0003eda0: 2020 2020 2020 2020 3220 3720 2020 2020          2 7     
│ │ │ │ +0003edb0: 2020 3320 3720 2020 2020 2020 3420 3720    3 7       4 7 
│ │ │ │ +0003edc0: 2020 2020 2020 3520 3720 2020 2020 2020        5 7       
│ │ │ │ +0003edd0: 2036 2037 2020 2020 2020 2037 7c0a 7c2d   6 7       7|.|-
│ │ │ │  0003ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ee00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ee10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ee20: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0003ee20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  0003ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee70: 2020 2020 2020 7c0a 7c2d 2037 7820 7820        |.|- 7x x 
│ │ │ │ -0003ee80: 202b 2031 3633 3278 2078 2020 2b20 3532   + 1632x x  + 52
│ │ │ │ -0003ee90: 3078 2078 2020 2b20 3132 3539 7820 7820  0x x  + 1259x x 
│ │ │ │ -0003eea0: 202b 2031 3537 7820 7820 202b 2031 3539   + 157x x  + 159
│ │ │ │ -0003eeb0: 3678 2078 2020 2b20 3635 3578 2078 2020  6x x  + 655x x  
│ │ │ │ -0003eec0: 2d20 2020 2020 7c0a 7c20 2020 2030 2038  -     |.|    0 8
│ │ │ │ -0003eed0: 2020 2020 2020 2020 3120 3820 2020 2020          1 8     
│ │ │ │ -0003eee0: 2020 3220 3820 2020 2020 2020 2033 2038    2 8        3 8
│ │ │ │ -0003eef0: 2020 2020 2020 2034 2038 2020 2020 2020         4 8      
│ │ │ │ -0003ef00: 2020 3520 3820 2020 2020 2020 3620 3820    5 8       6 8 
│ │ │ │ -0003ef10: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0003ee70: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0003ee80: 2037 7820 7820 202b 2031 3633 3278 2078   7x x  + 1632x x
│ │ │ │ +0003ee90: 2020 2b20 3532 3078 2078 2020 2b20 3132    + 520x x  + 12
│ │ │ │ +0003eea0: 3539 7820 7820 202b 2031 3537 7820 7820  59x x  + 157x x 
│ │ │ │ +0003eeb0: 202b 2031 3539 3678 2078 2020 2b20 3635   + 1596x x  + 65
│ │ │ │ +0003eec0: 3578 2078 2020 2d20 2020 2020 7c0a 7c20  5x x  -     |.| 
│ │ │ │ +0003eed0: 2020 2030 2038 2020 2020 2020 2020 3120     0 8        1 
│ │ │ │ +0003eee0: 3820 2020 2020 2020 3220 3820 2020 2020  8       2 8     
│ │ │ │ +0003eef0: 2020 2033 2038 2020 2020 2020 2034 2038     3 8       4 8
│ │ │ │ +0003ef00: 2020 2020 2020 2020 3520 3820 2020 2020          5 8     
│ │ │ │ +0003ef10: 2020 3620 3820 2020 2020 2020 7c0a 7c2d    6 8       |.|-
│ │ │ │  0003ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ef50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ef60: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -0003ef70: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0003ef60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0003ef70: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  0003ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003efb0: 2020 2020 2020 7c0a 7c34 3278 2078 2020        |.|42x x  
│ │ │ │ -0003efc0: 2d20 3538 3678 207d 2920 2020 2020 2020  - 586x })       
│ │ │ │ +0003efb0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +0003efc0: 3278 2078 2020 2d20 3538 3678 207d 2920  2x x  - 586x }) 
│ │ │ │  0003efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f000: 2020 2020 2020 7c0a 7c20 2020 3720 3820        |.|   7 8 
│ │ │ │ -0003f010: 2020 2020 2020 3820 2020 2020 2020 2020        8         
│ │ │ │ +0003f000: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003f010: 2020 3720 3820 2020 2020 2020 3820 2020    7 8       8   
│ │ │ │  0003f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f050: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0003f050: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0003f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f0a0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7469  ------+.|i5 : ti
│ │ │ │ -0003f0b0: 6d65 2064 6567 7265 654d 6170 2070 6869  me degreeMap phi
│ │ │ │ -0003f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0003f0b0: 3520 3a20 7469 6d65 2064 6567 7265 654d  5 : time degreeM
│ │ │ │ +0003f0c0: 6170 2070 6869 2020 2020 2020 2020 2020  ap phi          
│ │ │ │  0003f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0f0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -0003f100: 6420 302e 3133 3134 3735 7320 2863 7075  d 0.131475s (cpu
│ │ │ │ -0003f110: 293b 2030 2e30 3537 3130 3636 7320 2874  ); 0.0571066s (t
│ │ │ │ -0003f120: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -0003f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003f0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003f100: 2d2d 2075 7365 6420 302e 3134 3337 3432  -- used 0.143742
│ │ │ │ +0003f110: 7320 2863 7075 293b 2030 2e30 3730 3434  s (cpu); 0.07044
│ │ │ │ +0003f120: 3832 7320 2874 6872 6561 6429 3b20 3073  82s (thread); 0s
│ │ │ │ +0003f130: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0003f140: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0003f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f190: 2020 2020 2020 7c0a 7c6f 3520 3d20 3120        |.|o5 = 1 
│ │ │ │ -0003f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f190: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0003f1a0: 3520 3d20 3120 2020 2020 2020 2020 2020  5 = 1           
│ │ │ │  0003f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f1e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0003f1e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0003f1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f230: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 2d2d  ------+.|i6 : --
│ │ │ │ -0003f240: 2043 6f6d 706f 7365 2070 6869 3a50 5e38   Compose phi:P^8
│ │ │ │ -0003f250: 2d2d 2d3e 505e 3134 2077 6974 6820 6120  --->P^14 with a 
│ │ │ │ -0003f260: 6c69 6e65 6172 2070 726f 6a65 6374 696f  linear projectio
│ │ │ │ -0003f270: 6e20 505e 3134 2d2d 2d3e 505e 3820 6672  n P^14--->P^8 fr
│ │ │ │ -0003f280: 6f6d 2020 2020 7c0a 7c20 2020 2020 2d2d  om    |.|     --
│ │ │ │ -0003f290: 206f 6620 6469 6d65 6e73 696f 6e20 3520   of dimension 5 
│ │ │ │ -0003f2a0: 2873 6f20 7468 6174 2074 6865 2063 6f6d  (so that the com
│ │ │ │ -0003f2b0: 706f 7369 7469 6f6e 2070 6869 273a 505e  position phi':P^
│ │ │ │ -0003f2c0: 382d 2d2d 3e50 5e38 206d 7573 7420 6861  8--->P^8 must ha
│ │ │ │ -0003f2d0: 7665 2020 2020 7c0a 7c20 2020 2020 7068  ve    |.|     ph
│ │ │ │ -0003f2e0: 6927 3d70 6869 2a6d 6170 2872 696e 6750  i'=phi*map(ringP
│ │ │ │ -0003f2f0: 3134 2c72 696e 6750 382c 666f 7220 6920  14,ringP8,for i 
│ │ │ │ -0003f300: 746f 2038 206c 6973 7420 7261 6e64 6f6d  to 8 list random
│ │ │ │ -0003f310: 2831 2c72 696e 6750 3134 2929 2020 2020  (1,ringP14))    
│ │ │ │ -0003f320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003f230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0003f240: 3620 3a20 2d2d 2043 6f6d 706f 7365 2070  6 : -- Compose p
│ │ │ │ +0003f250: 6869 3a50 5e38 2d2d 2d3e 505e 3134 2077  hi:P^8--->P^14 w
│ │ │ │ +0003f260: 6974 6820 6120 6c69 6e65 6172 2070 726f  ith a linear pro
│ │ │ │ +0003f270: 6a65 6374 696f 6e20 505e 3134 2d2d 2d3e  jection P^14--->
│ │ │ │ +0003f280: 505e 3820 6672 6f6d 2020 2020 7c0a 7c20  P^8 from    |.| 
│ │ │ │ +0003f290: 2020 2020 2d2d 206f 6620 6469 6d65 6e73      -- of dimens
│ │ │ │ +0003f2a0: 696f 6e20 3520 2873 6f20 7468 6174 2074  ion 5 (so that t
│ │ │ │ +0003f2b0: 6865 2063 6f6d 706f 7369 7469 6f6e 2070  he composition p
│ │ │ │ +0003f2c0: 6869 273a 505e 382d 2d2d 3e50 5e38 206d  hi':P^8--->P^8 m
│ │ │ │ +0003f2d0: 7573 7420 6861 7665 2020 2020 7c0a 7c20  ust have    |.| 
│ │ │ │ +0003f2e0: 2020 2020 7068 6927 3d70 6869 2a6d 6170      phi'=phi*map
│ │ │ │ +0003f2f0: 2872 696e 6750 3134 2c72 696e 6750 382c  (ringP14,ringP8,
│ │ │ │ +0003f300: 666f 7220 6920 746f 2038 206c 6973 7420  for i to 8 list 
│ │ │ │ +0003f310: 7261 6e64 6f6d 2831 2c72 696e 6750 3134  random(1,ringP14
│ │ │ │ +0003f320: 2929 2020 2020 2020 2020 2020 7c0a 7c20  ))          |.| 
│ │ │ │  0003f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f370: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003f370: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0003f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f390: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0003f3a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0003f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f3c0: 2020 2020 2020 7c0a 7c6f 3620 3d20 6d61        |.|o6 = ma
│ │ │ │ -0003f3d0: 7020 2872 696e 6750 382c 2072 696e 6750  p (ringP8, ringP
│ │ │ │ -0003f3e0: 382c 207b 2d20 3738 3078 2020 2d20 3530  8, {- 780x  - 50
│ │ │ │ -0003f3f0: 3678 2078 2020 2b20 3135 3337 7820 202d  6x x  + 1537x  -
│ │ │ │ -0003f400: 2031 3332 7820 7820 202d 2039 3238 7820   132x x  - 928x 
│ │ │ │ -0003f410: 7820 2020 2020 7c0a 7c20 2020 2020 2020  x     |.|       
│ │ │ │ +0003f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f3a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003f3b0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0003f3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0003f3d0: 3620 3d20 6d61 7020 2872 696e 6750 382c  6 = map (ringP8,
│ │ │ │ +0003f3e0: 2072 696e 6750 382c 207b 2d20 3738 3078   ringP8, {- 780x
│ │ │ │ +0003f3f0: 2020 2d20 3530 3678 2078 2020 2b20 3135    - 506x x  + 15
│ │ │ │ +0003f400: 3337 7820 202d 2031 3332 7820 7820 202d  37x  - 132x x  -
│ │ │ │ +0003f410: 2039 3238 7820 7820 2020 2020 7c0a 7c20   928x x     |.| 
│ │ │ │  0003f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f430: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ -0003f440: 2020 3020 3120 2020 2020 2020 2031 2020    0 1        1  
│ │ │ │ -0003f450: 2020 2020 2030 2032 2020 2020 2020 2031       0 2       1
│ │ │ │ -0003f460: 2032 2020 2020 7c0a 7c20 2020 2020 2020   2    |.|       
│ │ │ │ +0003f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f440: 3020 2020 2020 2020 3020 3120 2020 2020  0       0 1     
│ │ │ │ +0003f450: 2020 2031 2020 2020 2020 2030 2032 2020     1       0 2  
│ │ │ │ +0003f460: 2020 2020 2031 2032 2020 2020 7c0a 7c20       1 2    |.| 
│ │ │ │  0003f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f4b0: 2020 2020 2020 7c0a 7c6f 3620 3a20 5269        |.|o6 : Ri
│ │ │ │ -0003f4c0: 6e67 4d61 7020 7269 6e67 5038 203c 2d2d  ngMap ringP8 <--
│ │ │ │ -0003f4d0: 2072 696e 6750 3820 2020 2020 2020 2020   ringP8         
│ │ │ │ +0003f4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0003f4c0: 3620 3a20 5269 6e67 4d61 7020 7269 6e67  6 : RingMap ring
│ │ │ │ +0003f4d0: 5038 203c 2d2d 2072 696e 6750 3820 2020  P8 <-- ringP8   
│ │ │ │  0003f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f500: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0003f500: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  0003f510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f550: 2d2d 2d2d 2d2d 7c0a 7c61 2067 656e 6572  ------|.|a gener
│ │ │ │ -0003f560: 616c 2073 7562 7370 6163 6520 6f66 2050  al subspace of P
│ │ │ │ -0003f570: 5e31 3420 2020 2020 2020 2020 2020 2020  ^14             
│ │ │ │ +0003f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c61  ------------|.|a
│ │ │ │ +0003f560: 2067 656e 6572 616c 2073 7562 7370 6163   general subspac
│ │ │ │ +0003f570: 6520 6f66 2050 5e31 3420 2020 2020 2020  e of P^14       
│ │ │ │  0003f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f5a0: 2020 2020 2020 7c0a 7c64 6567 7265 6520        |.|degree 
│ │ │ │ -0003f5b0: 6571 7561 6c20 746f 2064 6567 2847 2831  equal to deg(G(1
│ │ │ │ -0003f5c0: 2c35 2929 3d31 3429 2020 2020 2020 2020  ,5))=14)        
│ │ │ │ +0003f5a0: 2020 2020 2020 2020 2020 2020 7c0a 7c64              |.|d
│ │ │ │ +0003f5b0: 6567 7265 6520 6571 7561 6c20 746f 2064  egree equal to d
│ │ │ │ +0003f5c0: 6567 2847 2831 2c35 2929 3d31 3429 2020  eg(G(1,5))=14)  
│ │ │ │  0003f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f5f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003f5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0003f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f640: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003f640: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0003f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f690: 2020 2020 2020 7c0a 7c20 2020 2020 2032        |.|      2
│ │ │ │ -0003f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003f6a0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  0003f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f6c0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0003f6c0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
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│ │ │ │ +00043c20: 6f74 6520 7072 6f6a 6563 7469 7665 4465  ote projectiveDe
│ │ │ │ +00043c30: 6772 6565 733a 2070 726f 6a65 6374 6976  grees: projectiv
│ │ │ │ +00043c40: 6544 6567 7265 6573 2c20 2d2d 2070 726f  eDegrees, -- pro
│ │ │ │ +00043c50: 6a65 6374 6976 6520 6465 6772 6565 7320  jective degrees 
│ │ │ │ +00043c60: 6f66 2061 0a20 2020 2072 6174 696f 6e61  of a.    rationa
│ │ │ │ +00043c70: 6c20 6d61 7020 6265 7477 6565 6e20 7072  l map between pr
│ │ │ │ +00043c80: 6f6a 6563 7469 7665 2076 6172 6965 7469  ojective varieti
│ │ │ │ +00043c90: 6573 0a0a 5761 7973 2074 6f20 7573 6520  es..Ways to use 
│ │ │ │ +00043ca0: 6465 6772 6565 4d61 703a 0a3d 3d3d 3d3d  degreeMap:.=====
│ │ │ │ +00043cb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00043cc0: 3d0a 0a20 202a 2022 6465 6772 6565 4d61  =..  * "degreeMa
│ │ │ │ +00043cd0: 7028 5269 6e67 4d61 7029 220a 2020 2a20  p(RingMap)".  * 
│ │ │ │ +00043ce0: 2a6e 6f74 6520 6465 6772 6565 4d61 7028  *note degreeMap(
│ │ │ │ +00043cf0: 5261 7469 6f6e 616c 4d61 7029 3a20 6465  RationalMap): de
│ │ │ │ +00043d00: 6772 6565 4d61 705f 6c70 5261 7469 6f6e  greeMap_lpRation
│ │ │ │ +00043d10: 616c 4d61 705f 7270 2c20 2d2d 2064 6567  alMap_rp, -- deg
│ │ │ │ +00043d20: 7265 6520 6f66 2061 0a20 2020 2072 6174  ree of a.    rat
│ │ │ │ +00043d30: 696f 6e61 6c20 6d61 700a 0a46 6f72 2074  ional map..For t
│ │ │ │ +00043d40: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +00043d50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00043d60: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +00043d70: 7465 2064 6567 7265 654d 6170 3a20 6465  te degreeMap: de
│ │ │ │ +00043d80: 6772 6565 4d61 702c 2069 7320 6120 2a6e  greeMap, is a *n
│ │ │ │ +00043d90: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ +00043da0: 696f 6e20 7769 7468 206f 7074 696f 6e73  ion with options
│ │ │ │ +00043db0: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ +00043dc0: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769  MethodFunctionWi
│ │ │ │ +00043dd0: 7468 4f70 7469 6f6e 732c 2e0a 1f0a 4669  thOptions,....Fi
│ │ │ │ +00043de0: 6c65 3a20 4372 656d 6f6e 612e 696e 666f  le: Cremona.info
│ │ │ │ +00043df0: 2c20 4e6f 6465 3a20 6465 6772 6565 4d61  , Node: degreeMa
│ │ │ │ +00043e00: 705f 6c70 5261 7469 6f6e 616c 4d61 705f  p_lpRationalMap_
│ │ │ │ +00043e10: 7270 2c20 4e65 7874 3a20 6465 6772 6565  rp, Next: degree
│ │ │ │ +00043e20: 735f 6c70 5261 7469 6f6e 616c 4d61 705f  s_lpRationalMap_
│ │ │ │ +00043e30: 7270 2c20 5072 6576 3a20 6465 6772 6565  rp, Prev: degree
│ │ │ │ +00043e40: 4d61 702c 2055 703a 2054 6f70 0a0a 6465  Map, Up: Top..de
│ │ │ │ +00043e50: 6772 6565 4d61 7028 5261 7469 6f6e 616c  greeMap(Rational
│ │ │ │ +00043e60: 4d61 7029 202d 2d20 6465 6772 6565 206f  Map) -- degree o
│ │ │ │ +00043e70: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ +00043e80: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  00043e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00043ea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ -00043eb0: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ -00043ec0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -00043ed0: 2a6e 6f74 6520 6465 6772 6565 4d61 703a  *note degreeMap:
│ │ │ │ -00043ee0: 2064 6567 7265 654d 6170 2c0a 2020 2a20   degreeMap,.  * 
│ │ │ │ -00043ef0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00043f00: 6465 6772 6565 4d61 7020 5068 690a 2020  degreeMap Phi.  
│ │ │ │ -00043f10: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00043f20: 2a20 5068 692c 2061 202a 6e6f 7465 2072  * Phi, a *note r
│ │ │ │ -00043f30: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ -00043f40: 696f 6e61 6c4d 6170 2c0a 2020 2a20 2a6e  ionalMap,.  * *n
│ │ │ │ -00043f50: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
│ │ │ │ -00043f60: 7574 733a 2028 4d61 6361 756c 6179 3244  uts: (Macaulay2D
│ │ │ │ -00043f70: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
│ │ │ │ -00043f80: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ -00043f90: 2069 6e70 7574 732c 3a0a 2020 2020 2020   inputs,:.      
│ │ │ │ -00043fa0: 2a20 2a6e 6f74 6520 426c 6f77 5570 5374  * *note BlowUpSt
│ │ │ │ -00043fb0: 7261 7465 6779 3a20 426c 6f77 5570 5374  rategy: BlowUpSt
│ │ │ │ -00043fc0: 7261 7465 6779 2c20 3d3e 202e 2e2e 2c20  rategy, => ..., 
│ │ │ │ -00043fd0: 6465 6661 756c 7420 7661 6c75 650a 2020  default value.  
│ │ │ │ -00043fe0: 2020 2020 2020 2245 6c69 6d69 6e61 7465        "Eliminate
│ │ │ │ -00043ff0: 222c 0a20 2020 2020 202a 202a 6e6f 7465  ",.      * *note
│ │ │ │ -00044000: 2043 6572 7469 6679 3a20 4365 7274 6966   Certify: Certif
│ │ │ │ -00044010: 792c 203d 3e20 2e2e 2e2c 2064 6566 6175  y, => ..., defau
│ │ │ │ -00044020: 6c74 2076 616c 7565 2066 616c 7365 2c20  lt value false, 
│ │ │ │ -00044030: 7768 6574 6865 7220 746f 2065 6e73 7572  whether to ensur
│ │ │ │ -00044040: 650a 2020 2020 2020 2020 636f 7272 6563  e.        correc
│ │ │ │ -00044050: 746e 6573 7320 6f66 206f 7574 7075 740a  tness of output.
│ │ │ │ -00044060: 2020 2020 2020 2a20 2a6e 6f74 6520 5665        * *note Ve
│ │ │ │ -00044070: 7262 6f73 653a 2069 6e76 6572 7365 4d61  rbose: inverseMa
│ │ │ │ -00044080: 705f 6c70 5f70 645f 7064 5f70 645f 636d  p_lp_pd_pd_pd_cm
│ │ │ │ -00044090: 5665 7262 6f73 653d 3e5f 7064 5f70 645f  Verbose=>_pd_pd_
│ │ │ │ -000440a0: 7064 5f72 702c 203d 3e20 2e2e 2e2c 0a20  pd_rp, => ...,. 
│ │ │ │ -000440b0: 2020 2020 2020 2064 6566 6175 6c74 2076         default v
│ │ │ │ -000440c0: 616c 7565 2074 7275 652c 0a20 202a 204f  alue true,.  * O
│ │ │ │ -000440d0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -000440e0: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ -000440f0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00044100: 5a5a 2c2c 2074 6865 2064 6567 7265 6520  ZZ,, the degree 
│ │ │ │ -00044110: 6f66 2050 6869 2e20 536f 2074 6869 7320  of Phi. So this 
│ │ │ │ -00044120: 7661 6c75 650a 2020 2020 2020 2020 6973  value.        is
│ │ │ │ -00044130: 2031 2069 6620 616e 6420 6f6e 6c79 2069   1 if and only i
│ │ │ │ -00044140: 6620 7468 6520 6d61 7020 6973 2062 6972  f the map is bir
│ │ │ │ -00044150: 6174 696f 6e61 6c20 6f6e 746f 2069 7473  ational onto its
│ │ │ │ -00044160: 2069 6d61 6765 2e0a 0a44 6573 6372 6970   image...Descrip
│ │ │ │ -00044170: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00044180: 0a0a 5468 6973 2063 6f6d 7075 7461 7469  ..This computati
│ │ │ │ -00044190: 6f6e 2069 7320 646f 6e65 2074 6872 6f75  on is done throu
│ │ │ │ -000441a0: 6768 2074 6865 2063 6f72 7265 7370 6f6e  gh the correspon
│ │ │ │ -000441b0: 6469 6e67 206d 6574 686f 6420 666f 7220  ding method for 
│ │ │ │ -000441c0: 7269 6e67 206d 6170 732e 2053 6565 0a2a  ring maps. See.*
│ │ │ │ -000441d0: 6e6f 7465 2064 6567 7265 654d 6170 2852  note degreeMap(R
│ │ │ │ -000441e0: 696e 674d 6170 293a 2064 6567 7265 654d  ingMap): degreeM
│ │ │ │ -000441f0: 6170 2c20 666f 7220 6d6f 7265 2064 6574  ap, for more det
│ │ │ │ -00044200: 6169 6c73 2061 6e64 2065 7861 6d70 6c65  ails and example
│ │ │ │ -00044210: 732e 0a0a 5365 6520 616c 736f 0a3d 3d3d  s...See also.===
│ │ │ │ -00044220: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00044230: 2064 6567 7265 654d 6170 2852 696e 674d   degreeMap(RingM
│ │ │ │ -00044240: 6170 293a 2064 6567 7265 654d 6170 2c20  ap): degreeMap, 
│ │ │ │ -00044250: 2d2d 2064 6567 7265 6520 6f66 2061 2072  -- degree of a r
│ │ │ │ -00044260: 6174 696f 6e61 6c20 6d61 7020 6265 7477  ational map betw
│ │ │ │ -00044270: 6565 6e0a 2020 2020 7072 6f6a 6563 7469  een.    projecti
│ │ │ │ -00044280: 7665 2076 6172 6965 7469 6573 0a20 202a  ve varieties.  *
│ │ │ │ -00044290: 202a 6e6f 7465 2070 726f 6a65 6374 6976   *note projectiv
│ │ │ │ -000442a0: 6544 6567 7265 6573 3a20 7072 6f6a 6563  eDegrees: projec
│ │ │ │ -000442b0: 7469 7665 4465 6772 6565 732c 202d 2d20  tiveDegrees, -- 
│ │ │ │ -000442c0: 7072 6f6a 6563 7469 7665 2064 6567 7265  projective degre
│ │ │ │ -000442d0: 6573 206f 6620 610a 2020 2020 7261 7469  es of a.    rati
│ │ │ │ -000442e0: 6f6e 616c 206d 6170 2062 6574 7765 656e  onal map between
│ │ │ │ -000442f0: 2070 726f 6a65 6374 6976 6520 7661 7269   projective vari
│ │ │ │ -00044300: 6574 6965 730a 2020 2a20 2a6e 6f74 6520  eties.  * *note 
│ │ │ │ -00044310: 6465 6772 6565 2852 6174 696f 6e61 6c4d  degree(RationalM
│ │ │ │ -00044320: 6170 293a 2064 6567 7265 655f 6c70 5261  ap): degree_lpRa
│ │ │ │ -00044330: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -00044340: 2064 6567 7265 6520 6f66 2061 2072 6174   degree of a rat
│ │ │ │ -00044350: 696f 6e61 6c0a 2020 2020 6d61 700a 0a57  ional.    map..W
│ │ │ │ -00044360: 6179 7320 746f 2075 7365 2074 6869 7320  ays to use this 
│ │ │ │ -00044370: 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d  method:.========
│ │ │ │ +00043ea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00043eb0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ +00043ec0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675 6e63  ======..  * Func
│ │ │ │ +00043ed0: 7469 6f6e 3a20 2a6e 6f74 6520 6465 6772  tion: *note degr
│ │ │ │ +00043ee0: 6565 4d61 703a 2064 6567 7265 654d 6170  eeMap: degreeMap
│ │ │ │ +00043ef0: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ +00043f00: 2020 2020 2020 6465 6772 6565 4d61 7020        degreeMap 
│ │ │ │ +00043f10: 5068 690a 2020 2a20 496e 7075 7473 3a0a  Phi.  * Inputs:.
│ │ │ │ +00043f20: 2020 2020 2020 2a20 5068 692c 2061 202a        * Phi, a *
│ │ │ │ +00043f30: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ +00043f40: 703a 2052 6174 696f 6e61 6c4d 6170 2c0a  p: RationalMap,.
│ │ │ │ +00043f50: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
│ │ │ │ +00043f60: 616c 2069 6e70 7574 733a 2028 4d61 6361  al inputs: (Maca
│ │ │ │ +00043f70: 756c 6179 3244 6f63 2975 7369 6e67 2066  ulay2Doc)using f
│ │ │ │ +00043f80: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ +00043f90: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a  tional inputs,:.
│ │ │ │ +00043fa0: 2020 2020 2020 2a20 2a6e 6f74 6520 426c        * *note Bl
│ │ │ │ +00043fb0: 6f77 5570 5374 7261 7465 6779 3a20 426c  owUpStrategy: Bl
│ │ │ │ +00043fc0: 6f77 5570 5374 7261 7465 6779 2c20 3d3e  owUpStrategy, =>
│ │ │ │ +00043fd0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +00043fe0: 6c75 650a 2020 2020 2020 2020 2245 6c69  lue.        "Eli
│ │ │ │ +00043ff0: 6d69 6e61 7465 222c 0a20 2020 2020 202a  minate",.      *
│ │ │ │ +00044000: 202a 6e6f 7465 2043 6572 7469 6679 3a20   *note Certify: 
│ │ │ │ +00044010: 4365 7274 6966 792c 203d 3e20 2e2e 2e2c  Certify, => ...,
│ │ │ │ +00044020: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
│ │ │ │ +00044030: 616c 7365 2c20 7768 6574 6865 7220 746f  alse, whether to
│ │ │ │ +00044040: 2065 6e73 7572 650a 2020 2020 2020 2020   ensure.        
│ │ │ │ +00044050: 636f 7272 6563 746e 6573 7320 6f66 206f  correctness of o
│ │ │ │ +00044060: 7574 7075 740a 2020 2020 2020 2a20 2a6e  utput.      * *n
│ │ │ │ +00044070: 6f74 6520 5665 7262 6f73 653a 2069 6e76  ote Verbose: inv
│ │ │ │ +00044080: 6572 7365 4d61 705f 6c70 5f70 645f 7064  erseMap_lp_pd_pd
│ │ │ │ +00044090: 5f70 645f 636d 5665 7262 6f73 653d 3e5f  _pd_cmVerbose=>_
│ │ │ │ +000440a0: 7064 5f70 645f 7064 5f72 702c 203d 3e20  pd_pd_pd_rp, => 
│ │ │ │ +000440b0: 2e2e 2e2c 0a20 2020 2020 2020 2064 6566  ...,.        def
│ │ │ │ +000440c0: 6175 6c74 2076 616c 7565 2074 7275 652c  ault value true,
│ │ │ │ +000440d0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +000440e0: 2020 2020 2a20 616e 202a 6e6f 7465 2069      * an *note i
│ │ │ │ +000440f0: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ +00044100: 7932 446f 6329 5a5a 2c2c 2074 6865 2064  y2Doc)ZZ,, the d
│ │ │ │ +00044110: 6567 7265 6520 6f66 2050 6869 2e20 536f  egree of Phi. So
│ │ │ │ +00044120: 2074 6869 7320 7661 6c75 650a 2020 2020   this value.    
│ │ │ │ +00044130: 2020 2020 6973 2031 2069 6620 616e 6420      is 1 if and 
│ │ │ │ +00044140: 6f6e 6c79 2069 6620 7468 6520 6d61 7020  only if the map 
│ │ │ │ +00044150: 6973 2062 6972 6174 696f 6e61 6c20 6f6e  is birational on
│ │ │ │ +00044160: 746f 2069 7473 2069 6d61 6765 2e0a 0a44  to its image...D
│ │ │ │ +00044170: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00044180: 3d3d 3d3d 3d3d 0a0a 5468 6973 2063 6f6d  ======..This com
│ │ │ │ +00044190: 7075 7461 7469 6f6e 2069 7320 646f 6e65  putation is done
│ │ │ │ +000441a0: 2074 6872 6f75 6768 2074 6865 2063 6f72   through the cor
│ │ │ │ +000441b0: 7265 7370 6f6e 6469 6e67 206d 6574 686f  responding metho
│ │ │ │ +000441c0: 6420 666f 7220 7269 6e67 206d 6170 732e  d for ring maps.
│ │ │ │ +000441d0: 2053 6565 0a2a 6e6f 7465 2064 6567 7265   See.*note degre
│ │ │ │ +000441e0: 654d 6170 2852 696e 674d 6170 293a 2064  eMap(RingMap): d
│ │ │ │ +000441f0: 6567 7265 654d 6170 2c20 666f 7220 6d6f  egreeMap, for mo
│ │ │ │ +00044200: 7265 2064 6574 6169 6c73 2061 6e64 2065  re details and e
│ │ │ │ +00044210: 7861 6d70 6c65 732e 0a0a 5365 6520 616c  xamples...See al
│ │ │ │ +00044220: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +00044230: 202a 6e6f 7465 2064 6567 7265 654d 6170   *note degreeMap
│ │ │ │ +00044240: 2852 696e 674d 6170 293a 2064 6567 7265  (RingMap): degre
│ │ │ │ +00044250: 654d 6170 2c20 2d2d 2064 6567 7265 6520  eMap, -- degree 
│ │ │ │ +00044260: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ +00044270: 7020 6265 7477 6565 6e0a 2020 2020 7072  p between.    pr
│ │ │ │ +00044280: 6f6a 6563 7469 7665 2076 6172 6965 7469  ojective varieti
│ │ │ │ +00044290: 6573 0a20 202a 202a 6e6f 7465 2070 726f  es.  * *note pro
│ │ │ │ +000442a0: 6a65 6374 6976 6544 6567 7265 6573 3a20  jectiveDegrees: 
│ │ │ │ +000442b0: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ +000442c0: 732c 202d 2d20 7072 6f6a 6563 7469 7665  s, -- projective
│ │ │ │ +000442d0: 2064 6567 7265 6573 206f 6620 610a 2020   degrees of a.  
│ │ │ │ +000442e0: 2020 7261 7469 6f6e 616c 206d 6170 2062    rational map b
│ │ │ │ +000442f0: 6574 7765 656e 2070 726f 6a65 6374 6976  etween projectiv
│ │ │ │ +00044300: 6520 7661 7269 6574 6965 730a 2020 2a20  e varieties.  * 
│ │ │ │ +00044310: 2a6e 6f74 6520 6465 6772 6565 2852 6174  *note degree(Rat
│ │ │ │ +00044320: 696f 6e61 6c4d 6170 293a 2064 6567 7265  ionalMap): degre
│ │ │ │ +00044330: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +00044340: 7270 2c20 2d2d 2064 6567 7265 6520 6f66  rp, -- degree of
│ │ │ │ +00044350: 2061 2072 6174 696f 6e61 6c0a 2020 2020   a rational.    
│ │ │ │ +00044360: 6d61 700a 0a57 6179 7320 746f 2075 7365  map..Ways to use
│ │ │ │ +00044370: 2074 6869 7320 6d65 7468 6f64 3a0a 3d3d   this method:.==
│ │ │ │  00044380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00044390: 0a0a 2020 2a20 2a6e 6f74 6520 6465 6772  ..  * *note degr
│ │ │ │ -000443a0: 6565 4d61 7028 5261 7469 6f6e 616c 4d61  eeMap(RationalMa
│ │ │ │ -000443b0: 7029 3a20 6465 6772 6565 4d61 705f 6c70  p): degreeMap_lp
│ │ │ │ -000443c0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -000443d0: 2d2d 2064 6567 7265 6520 6f66 2061 0a20  -- degree of a. 
│ │ │ │ -000443e0: 2020 2072 6174 696f 6e61 6c20 6d61 700a     rational map.
│ │ │ │ -000443f0: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -00044400: 696e 666f 2c20 4e6f 6465 3a20 6465 6772  info, Node: degr
│ │ │ │ -00044410: 6565 735f 6c70 5261 7469 6f6e 616c 4d61  ees_lpRationalMa
│ │ │ │ -00044420: 705f 7270 2c20 4e65 7874 3a20 6465 7363  p_rp, Next: desc
│ │ │ │ -00044430: 7269 6265 5f6c 7052 6174 696f 6e61 6c4d  ribe_lpRationalM
│ │ │ │ -00044440: 6170 5f72 702c 2050 7265 763a 2064 6567  ap_rp, Prev: deg
│ │ │ │ -00044450: 7265 654d 6170 5f6c 7052 6174 696f 6e61  reeMap_lpRationa
│ │ │ │ -00044460: 6c4d 6170 5f72 702c 2055 703a 2054 6f70  lMap_rp, Up: Top
│ │ │ │ -00044470: 0a0a 6465 6772 6565 7328 5261 7469 6f6e  ..degrees(Ration
│ │ │ │ -00044480: 616c 4d61 7029 202d 2d20 7072 6f6a 6563  alMap) -- projec
│ │ │ │ -00044490: 7469 7665 2064 6567 7265 6573 206f 6620  tive degrees of 
│ │ │ │ -000444a0: 6120 7261 7469 6f6e 616c 206d 6170 0a2a  a rational map.*
│ │ │ │ -000444b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00044390: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +000443a0: 6520 6465 6772 6565 4d61 7028 5261 7469  e degreeMap(Rati
│ │ │ │ +000443b0: 6f6e 616c 4d61 7029 3a20 6465 6772 6565  onalMap): degree
│ │ │ │ +000443c0: 4d61 705f 6c70 5261 7469 6f6e 616c 4d61  Map_lpRationalMa
│ │ │ │ +000443d0: 705f 7270 2c20 2d2d 2064 6567 7265 6520  p_rp, -- degree 
│ │ │ │ +000443e0: 6f66 2061 0a20 2020 2072 6174 696f 6e61  of a.    rationa
│ │ │ │ +000443f0: 6c20 6d61 700a 1f0a 4669 6c65 3a20 4372  l map...File: Cr
│ │ │ │ +00044400: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ +00044410: 3a20 6465 6772 6565 735f 6c70 5261 7469  : degrees_lpRati
│ │ │ │ +00044420: 6f6e 616c 4d61 705f 7270 2c20 4e65 7874  onalMap_rp, Next
│ │ │ │ +00044430: 3a20 6465 7363 7269 6265 5f6c 7052 6174  : describe_lpRat
│ │ │ │ +00044440: 696f 6e61 6c4d 6170 5f72 702c 2050 7265  ionalMap_rp, Pre
│ │ │ │ +00044450: 763a 2064 6567 7265 654d 6170 5f6c 7052  v: degreeMap_lpR
│ │ │ │ +00044460: 6174 696f 6e61 6c4d 6170 5f72 702c 2055  ationalMap_rp, U
│ │ │ │ +00044470: 703a 2054 6f70 0a0a 6465 6772 6565 7328  p: Top..degrees(
│ │ │ │ +00044480: 5261 7469 6f6e 616c 4d61 7029 202d 2d20  RationalMap) -- 
│ │ │ │ +00044490: 7072 6f6a 6563 7469 7665 2064 6567 7265  projective degre
│ │ │ │ +000444a0: 6573 206f 6620 6120 7261 7469 6f6e 616c  es of a rational
│ │ │ │ +000444b0: 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a   map.***********
│ │ │ │  000444c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000444d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000444e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ -000444f0: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ -00044500: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e    * Function: *n
│ │ │ │ -00044510: 6f74 6520 6465 6772 6565 733a 2028 4d61  ote degrees: (Ma
│ │ │ │ -00044520: 6361 756c 6179 3244 6f63 2964 6567 7265  caulay2Doc)degre
│ │ │ │ -00044530: 6573 2c0a 2020 2a20 5573 6167 653a 200a  es,.  * Usage: .
│ │ │ │ -00044540: 2020 2020 2020 2020 6465 6772 6565 7320          degrees 
│ │ │ │ -00044550: 7068 6920 0a20 2020 2020 2020 206d 756c  phi .        mul
│ │ │ │ -00044560: 7469 6465 6772 6565 2070 6869 0a20 202a  tidegree phi.  *
│ │ │ │ -00044570: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00044580: 2070 6869 2c20 6120 2a6e 6f74 6520 7261   phi, a *note ra
│ │ │ │ -00044590: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ -000445a0: 6f6e 616c 4d61 702c 0a20 202a 204f 7574  onalMap,.  * Out
│ │ │ │ -000445b0: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ -000445c0: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ -000445d0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ -000445e0: 2074 6865 206c 6973 7420 6f66 2070 726f   the list of pro
│ │ │ │ -000445f0: 6a65 6374 6976 6520 6465 6772 6565 7320  jective degrees 
│ │ │ │ -00044600: 6f66 0a20 2020 2020 2020 2070 6869 0a0a  of.        phi..
│ │ │ │ -00044610: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00044620: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6973  =======..This is
│ │ │ │ -00044630: 2061 2073 686f 7274 6375 7420 666f 7220   a shortcut for 
│ │ │ │ -00044640: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -00044650: 7328 7068 692c 4365 7274 6966 793d 3e74  s(phi,Certify=>t
│ │ │ │ -00044660: 7275 652c 5665 7262 6f73 653d 3e66 616c  rue,Verbose=>fal
│ │ │ │ -00044670: 7365 292c 2073 6565 0a2a 6e6f 7465 2070  se), see.*note p
│ │ │ │ -00044680: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ -00044690: 2852 6174 696f 6e61 6c4d 6170 293a 2070  (RationalMap): p
│ │ │ │ -000446a0: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ -000446b0: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ -000446c0: 702c 2e0a 0a57 6179 7320 746f 2075 7365  p,...Ways to use
│ │ │ │ -000446d0: 2074 6869 7320 6d65 7468 6f64 3a0a 3d3d   this method:.==
│ │ │ │ -000446e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000446f0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -00044700: 6520 6465 6772 6565 7328 5261 7469 6f6e  e degrees(Ration
│ │ │ │ -00044710: 616c 4d61 7029 3a20 6465 6772 6565 735f  alMap): degrees_
│ │ │ │ -00044720: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00044730: 2c20 2d2d 2070 726f 6a65 6374 6976 6520  , -- projective 
│ │ │ │ -00044740: 6465 6772 6565 730a 2020 2020 6f66 2061  degrees.    of a
│ │ │ │ -00044750: 2072 6174 696f 6e61 6c20 6d61 700a 1f0a   rational map...
│ │ │ │ -00044760: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ -00044770: 666f 2c20 4e6f 6465 3a20 6465 7363 7269  fo, Node: descri
│ │ │ │ -00044780: 6265 5f6c 7052 6174 696f 6e61 6c4d 6170  be_lpRationalMap
│ │ │ │ -00044790: 5f72 702c 204e 6578 743a 2044 6f6d 696e  _rp, Next: Domin
│ │ │ │ -000447a0: 616e 742c 2050 7265 763a 2064 6567 7265  ant, Prev: degre
│ │ │ │ -000447b0: 6573 5f6c 7052 6174 696f 6e61 6c4d 6170  es_lpRationalMap
│ │ │ │ -000447c0: 5f72 702c 2055 703a 2054 6f70 0a0a 6465  _rp, Up: Top..de
│ │ │ │ -000447d0: 7363 7269 6265 2852 6174 696f 6e61 6c4d  scribe(RationalM
│ │ │ │ -000447e0: 6170 2920 2d2d 2064 6573 6372 6962 6520  ap) -- describe 
│ │ │ │ -000447f0: 6120 7261 7469 6f6e 616c 206d 6170 0a2a  a rational map.*
│ │ │ │ -00044800: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000444e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000444f0: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ +00044500: 3d3d 3d3d 0a0a 2020 2a20 4675 6e63 7469  ====..  * Functi
│ │ │ │ +00044510: 6f6e 3a20 2a6e 6f74 6520 6465 6772 6565  on: *note degree
│ │ │ │ +00044520: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ +00044530: 2964 6567 7265 6573 2c0a 2020 2a20 5573  )degrees,.  * Us
│ │ │ │ +00044540: 6167 653a 200a 2020 2020 2020 2020 6465  age: .        de
│ │ │ │ +00044550: 6772 6565 7320 7068 6920 0a20 2020 2020  grees phi .     
│ │ │ │ +00044560: 2020 206d 756c 7469 6465 6772 6565 2070     multidegree p
│ │ │ │ +00044570: 6869 0a20 202a 2049 6e70 7574 733a 0a20  hi.  * Inputs:. 
│ │ │ │ +00044580: 2020 2020 202a 2070 6869 2c20 6120 2a6e       * phi, a *n
│ │ │ │ +00044590: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ +000445a0: 3a20 5261 7469 6f6e 616c 4d61 702c 0a20  : RationalMap,. 
│ │ │ │ +000445b0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +000445c0: 2020 2a20 6120 2a6e 6f74 6520 6c69 7374    * a *note list
│ │ │ │ +000445d0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000445e0: 4c69 7374 2c2c 2074 6865 206c 6973 7420  List,, the list 
│ │ │ │ +000445f0: 6f66 2070 726f 6a65 6374 6976 6520 6465  of projective de
│ │ │ │ +00044600: 6772 6565 7320 6f66 0a20 2020 2020 2020  grees of.       
│ │ │ │ +00044610: 2070 6869 0a0a 4465 7363 7269 7074 696f   phi..Descriptio
│ │ │ │ +00044620: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
│ │ │ │ +00044630: 6869 7320 6973 2061 2073 686f 7274 6375  his is a shortcu
│ │ │ │ +00044640: 7420 666f 7220 7072 6f6a 6563 7469 7665  t for projective
│ │ │ │ +00044650: 4465 6772 6565 7328 7068 692c 4365 7274  Degrees(phi,Cert
│ │ │ │ +00044660: 6966 793d 3e74 7275 652c 5665 7262 6f73  ify=>true,Verbos
│ │ │ │ +00044670: 653d 3e66 616c 7365 292c 2073 6565 0a2a  e=>false), see.*
│ │ │ │ +00044680: 6e6f 7465 2070 726f 6a65 6374 6976 6544  note projectiveD
│ │ │ │ +00044690: 6567 7265 6573 2852 6174 696f 6e61 6c4d  egrees(RationalM
│ │ │ │ +000446a0: 6170 293a 2070 726f 6a65 6374 6976 6544  ap): projectiveD
│ │ │ │ +000446b0: 6567 7265 6573 5f6c 7052 6174 696f 6e61  egrees_lpRationa
│ │ │ │ +000446c0: 6c4d 6170 5f72 702c 2e0a 0a57 6179 7320  lMap_rp,...Ways 
│ │ │ │ +000446d0: 746f 2075 7365 2074 6869 7320 6d65 7468  to use this meth
│ │ │ │ +000446e0: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  od:.============
│ │ │ │ +000446f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00044700: 2a20 2a6e 6f74 6520 6465 6772 6565 7328  * *note degrees(
│ │ │ │ +00044710: 5261 7469 6f6e 616c 4d61 7029 3a20 6465  RationalMap): de
│ │ │ │ +00044720: 6772 6565 735f 6c70 5261 7469 6f6e 616c  grees_lpRational
│ │ │ │ +00044730: 4d61 705f 7270 2c20 2d2d 2070 726f 6a65  Map_rp, -- proje
│ │ │ │ +00044740: 6374 6976 6520 6465 6772 6565 730a 2020  ctive degrees.  
│ │ │ │ +00044750: 2020 6f66 2061 2072 6174 696f 6e61 6c20    of a rational 
│ │ │ │ +00044760: 6d61 700a 1f0a 4669 6c65 3a20 4372 656d  map...File: Crem
│ │ │ │ +00044770: 6f6e 612e 696e 666f 2c20 4e6f 6465 3a20  ona.info, Node: 
│ │ │ │ +00044780: 6465 7363 7269 6265 5f6c 7052 6174 696f  describe_lpRatio
│ │ │ │ +00044790: 6e61 6c4d 6170 5f72 702c 204e 6578 743a  nalMap_rp, Next:
│ │ │ │ +000447a0: 2044 6f6d 696e 616e 742c 2050 7265 763a   Dominant, Prev:
│ │ │ │ +000447b0: 2064 6567 7265 6573 5f6c 7052 6174 696f   degrees_lpRatio
│ │ │ │ +000447c0: 6e61 6c4d 6170 5f72 702c 2055 703a 2054  nalMap_rp, Up: T
│ │ │ │ +000447d0: 6f70 0a0a 6465 7363 7269 6265 2852 6174  op..describe(Rat
│ │ │ │ +000447e0: 696f 6e61 6c4d 6170 2920 2d2d 2064 6573  ionalMap) -- des
│ │ │ │ +000447f0: 6372 6962 6520 6120 7261 7469 6f6e 616c  cribe a rational
│ │ │ │ +00044800: 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a   map.***********
│ │ │ │  00044810: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00044820: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00044830: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ -00044840: 3d3d 0a0a 2020 2a20 4675 6e63 7469 6f6e  ==..  * Function
│ │ │ │ -00044850: 3a20 2a6e 6f74 6520 6465 7363 7269 6265  : *note describe
│ │ │ │ -00044860: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00044870: 6465 7363 7269 6265 2c0a 2020 2a20 5573  describe,.  * Us
│ │ │ │ -00044880: 6167 653a 200a 2020 2020 2020 2020 6465  age: .        de
│ │ │ │ -00044890: 7363 7269 6265 2070 6869 0a20 202a 2049  scribe phi.  * I
│ │ │ │ -000448a0: 6e70 7574 733a 0a20 2020 2020 202a 2070  nputs:.      * p
│ │ │ │ -000448b0: 6869 2c20 6120 2a6e 6f74 6520 7261 7469  hi, a *note rati
│ │ │ │ -000448c0: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -000448d0: 616c 4d61 702c 0a20 202a 204f 7574 7075  alMap,.  * Outpu
│ │ │ │ -000448e0: 7473 3a0a 2020 2020 2020 2a20 6120 6465  ts:.      * a de
│ │ │ │ -000448f0: 7363 7269 7074 696f 6e20 6f66 2070 6869  scription of phi
│ │ │ │ -00044900: 2c20 6769 7669 6e67 2073 6f6d 6520 696e  , giving some in
│ │ │ │ -00044910: 6469 6361 7469 6f6e 206f 6620 7768 6174  dication of what
│ │ │ │ -00044920: 2068 6173 2061 6c72 6561 6479 2062 6565   has already bee
│ │ │ │ -00044930: 6e0a 2020 2020 2020 2020 6361 6c63 756c  n.        calcul
│ │ │ │ -00044940: 6174 6564 2e0a 0a44 6573 6372 6970 7469  ated...Descripti
│ │ │ │ -00044950: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00044960: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00044820: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00044830: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +00044840: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675  ========..  * Fu
│ │ │ │ +00044850: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 6465  nction: *note de
│ │ │ │ +00044860: 7363 7269 6265 3a20 284d 6163 6175 6c61  scribe: (Macaula
│ │ │ │ +00044870: 7932 446f 6329 6465 7363 7269 6265 2c0a  y2Doc)describe,.
│ │ │ │ +00044880: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +00044890: 2020 2020 6465 7363 7269 6265 2070 6869      describe phi
│ │ │ │ +000448a0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +000448b0: 2020 202a 2070 6869 2c20 6120 2a6e 6f74     * phi, a *not
│ │ │ │ +000448c0: 6520 7261 7469 6f6e 616c 206d 6170 3a20  e rational map: 
│ │ │ │ +000448d0: 5261 7469 6f6e 616c 4d61 702c 0a20 202a  RationalMap,.  *
│ │ │ │ +000448e0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +000448f0: 2a20 6120 6465 7363 7269 7074 696f 6e20  * a description 
│ │ │ │ +00044900: 6f66 2070 6869 2c20 6769 7669 6e67 2073  of phi, giving s
│ │ │ │ +00044910: 6f6d 6520 696e 6469 6361 7469 6f6e 206f  ome indication o
│ │ │ │ +00044920: 6620 7768 6174 2068 6173 2061 6c72 6561  f what has alrea
│ │ │ │ +00044930: 6479 2062 6565 6e0a 2020 2020 2020 2020  dy been.        
│ │ │ │ +00044940: 6361 6c63 756c 6174 6564 2e0a 0a44 6573  calculated...Des
│ │ │ │ +00044950: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00044960: 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ====..+---------
│ │ │ │  00044970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000449a0: 2b0a 7c69 3120 3a20 5a5a 2f33 3333 3331  +.|i1 : ZZ/33331
│ │ │ │ -000449b0: 5b74 5f30 2e2e 745f 345d 3b20 2020 2020  [t_0..t_4];     
│ │ │ │ -000449c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000449a0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5a5a  ------+.|i1 : ZZ
│ │ │ │ +000449b0: 2f33 3333 3331 5b74 5f30 2e2e 745f 345d  /33331[t_0..t_4]
│ │ │ │ +000449c0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  000449d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000449e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000449e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  000449f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00044a20: 2d2d 2d2d 2b0a 7c69 3220 3a20 7068 6920  ----+.|i2 : phi 
│ │ │ │ -00044a30: 3d20 7261 7469 6f6e 616c 4d61 7020 6d69  = rationalMap mi
│ │ │ │ -00044a40: 6e6f 7273 2832 2c6d 6174 7269 787b 7b74  nors(2,matrix{{t
│ │ │ │ -00044a50: 5f30 2e2e 745f 337d 2c7b 745f 312e 2e74  _0..t_3},{t_1..t
│ │ │ │ -00044a60: 5f34 7d7d 293b 7c0a 7c20 2020 2020 2020  _4}});|.|       
│ │ │ │ +00044a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00044a30: 3a20 7068 6920 3d20 7261 7469 6f6e 616c  : phi = rational
│ │ │ │ +00044a40: 4d61 7020 6d69 6e6f 7273 2832 2c6d 6174  Map minors(2,mat
│ │ │ │ +00044a50: 7269 787b 7b74 5f30 2e2e 745f 337d 2c7b  rix{{t_0..t_3},{
│ │ │ │ +00044a60: 745f 312e 2e74 5f34 7d7d 293b 7c0a 7c20  t_1..t_4}});|.| 
│ │ │ │  00044a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044aa0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -00044ab0: 5261 7469 6f6e 616c 4d61 7020 2871 7561  RationalMap (qua
│ │ │ │ -00044ac0: 6472 6174 6963 2072 6174 696f 6e61 6c20  dratic rational 
│ │ │ │ -00044ad0: 6d61 7020 6672 6f6d 2050 505e 3420 746f  map from PP^4 to
│ │ │ │ -00044ae0: 2050 505e 3529 2020 2020 7c0a 2b2d 2d2d   PP^5)    |.+---
│ │ │ │ -00044af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00044aa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00044ab0: 7c6f 3220 3a20 5261 7469 6f6e 616c 4d61  |o2 : RationalMa
│ │ │ │ +00044ac0: 7020 2871 7561 6472 6174 6963 2072 6174  p (quadratic rat
│ │ │ │ +00044ad0: 696f 6e61 6c20 6d61 7020 6672 6f6d 2050  ional map from P
│ │ │ │ +00044ae0: 505e 3420 746f 2050 505e 3529 2020 2020  P^4 to PP^5)    
│ │ │ │ +00044af0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00044b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00044b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00044b30: 3320 3a20 6465 7363 7269 6265 2070 6869  3 : describe phi
│ │ │ │ -00044b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00044b30: 2d2d 2b0a 7c69 3320 3a20 6465 7363 7269  --+.|i3 : descri
│ │ │ │ +00044b40: 6265 2070 6869 2020 2020 2020 2020 2020  be phi          
│ │ │ │  00044b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044b60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00044b70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00044b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044b70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00044b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044bb0: 7c0a 7c6f 3320 3d20 7261 7469 6f6e 616c  |.|o3 = rational
│ │ │ │ -00044bc0: 206d 6170 2064 6566 696e 6564 2062 7920   map defined by 
│ │ │ │ -00044bd0: 666f 726d 7320 6f66 2064 6567 7265 6520  forms of degree 
│ │ │ │ -00044be0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00044bf0: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ -00044c00: 2076 6172 6965 7479 3a20 5050 5e34 2020   variety: PP^4  
│ │ │ │ -00044c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044bb0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7261        |.|o3 = ra
│ │ │ │ +00044bc0: 7469 6f6e 616c 206d 6170 2064 6566 696e  tional map defin
│ │ │ │ +00044bd0: 6564 2062 7920 666f 726d 7320 6f66 2064  ed by forms of d
│ │ │ │ +00044be0: 6567 7265 6520 3220 2020 2020 2020 2020  egree 2         
│ │ │ │ +00044bf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044c00: 736f 7572 6365 2076 6172 6965 7479 3a20  source variety: 
│ │ │ │ +00044c10: 5050 5e34 2020 2020 2020 2020 2020 2020  PP^4            
│ │ │ │  00044c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044c30: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ -00044c40: 6574 2076 6172 6965 7479 3a20 5050 5e35  et variety: PP^5
│ │ │ │ -00044c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00044c40: 2020 7461 7267 6574 2076 6172 6965 7479    target variety
│ │ │ │ +00044c50: 3a20 5050 5e35 2020 2020 2020 2020 2020  : PP^5          
│ │ │ │  00044c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044c70: 2020 2020 2020 7c0a 7c20 2020 2020 636f        |.|     co
│ │ │ │ -00044c80: 6566 6669 6369 656e 7420 7269 6e67 3a20  efficient ring: 
│ │ │ │ -00044c90: 5a5a 2f33 3333 3331 2020 2020 2020 2020  ZZ/33331        
│ │ │ │ +00044c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00044c80: 2020 2020 636f 6566 6669 6369 656e 7420      coefficient 
│ │ │ │ +00044c90: 7269 6e67 3a20 5a5a 2f33 3333 3331 2020  ring: ZZ/33331  
│ │ │ │  00044ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044cb0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00044cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00044cb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00044cc0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00044cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00044cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -00044d00: 3a20 4920 3d20 696d 6167 6520 7068 693b  : I = image phi;
│ │ │ │ -00044d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00044d00: 2b0a 7c69 3420 3a20 4920 3d20 696d 6167  +.|i4 : I = imag
│ │ │ │ +00044d10: 6520 7068 693b 2020 2020 2020 2020 2020  e phi;          
│ │ │ │  00044d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044d30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00044d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044d40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00044d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044d70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00044d80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00044d90: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +00044d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044d80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00044d90: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │  00044da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044dc0: 7c0a 7c6f 3420 3a20 4964 6561 6c20 6f66  |.|o4 : Ideal of
│ │ │ │ -00044dd0: 202d 2d2d 2d2d 5b78 202e 2e78 205d 2020   -----[x ..x ]  
│ │ │ │ -00044de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044dc0: 2020 2020 2020 7c0a 7c6f 3420 3a20 4964        |.|o4 : Id
│ │ │ │ +00044dd0: 6561 6c20 6f66 202d 2d2d 2d2d 5b78 202e  eal of -----[x .
│ │ │ │ +00044de0: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │  00044df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044e00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00044e10: 2020 2033 3333 3331 2020 3020 2020 3520     33331  0   5 
│ │ │ │ -00044e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044e00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00044e10: 2020 2020 2020 2020 2033 3333 3331 2020           33331  
│ │ │ │ +00044e20: 3020 2020 3520 2020 2020 2020 2020 2020  0   5           
│ │ │ │  00044e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044e40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00044e40: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00044e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00044e80: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6465  ------+.|i5 : de
│ │ │ │ -00044e90: 7363 7269 6265 2070 6869 2020 2020 2020  scribe phi      
│ │ │ │ +00044e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00044e90: 3520 3a20 6465 7363 7269 6265 2070 6869  5 : describe phi
│ │ │ │  00044ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044ec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00044ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044ec0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00044ed0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00044ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044f00: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00044f10: 3d20 7261 7469 6f6e 616c 206d 6170 2064  = rational map d
│ │ │ │ -00044f20: 6566 696e 6564 2062 7920 666f 726d 7320  efined by forms 
│ │ │ │ -00044f30: 6f66 2064 6567 7265 6520 3220 2020 2020  of degree 2     
│ │ │ │ -00044f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00044f50: 2020 2020 736f 7572 6365 2076 6172 6965      source varie
│ │ │ │ -00044f60: 7479 3a20 5050 5e34 2020 2020 2020 2020  ty: PP^4        
│ │ │ │ +00044f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044f10: 7c0a 7c6f 3520 3d20 7261 7469 6f6e 616c  |.|o5 = rational
│ │ │ │ +00044f20: 206d 6170 2064 6566 696e 6564 2062 7920   map defined by 
│ │ │ │ +00044f30: 666f 726d 7320 6f66 2064 6567 7265 6520  forms of degree 
│ │ │ │ +00044f40: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00044f50: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +00044f60: 2076 6172 6965 7479 3a20 5050 5e34 2020   variety: PP^4  
│ │ │ │  00044f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044f80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00044f90: 7c20 2020 2020 7461 7267 6574 2076 6172  |     target var
│ │ │ │ -00044fa0: 6965 7479 3a20 5050 5e35 2020 2020 2020  iety: PP^5      
│ │ │ │ +00044f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044f90: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ +00044fa0: 6574 2076 6172 6965 7479 3a20 5050 5e35  et variety: PP^5
│ │ │ │  00044fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044fd0: 7c0a 7c20 2020 2020 696d 6167 653a 2073  |.|     image: s
│ │ │ │ -00044fe0: 6d6f 6f74 6820 7175 6164 7269 6320 6879  mooth quadric hy
│ │ │ │ -00044ff0: 7065 7273 7572 6661 6365 2069 6e20 5050  persurface in PP
│ │ │ │ -00045000: 5e35 2020 2020 2020 2020 2020 2020 2020  ^5              
│ │ │ │ -00045010: 2020 7c0a 7c20 2020 2020 646f 6d69 6e61    |.|     domina
│ │ │ │ -00045020: 6e63 653a 2066 616c 7365 2020 2020 2020  nce: false      
│ │ │ │ +00044fd0: 2020 2020 2020 7c0a 7c20 2020 2020 696d        |.|     im
│ │ │ │ +00044fe0: 6167 653a 2073 6d6f 6f74 6820 7175 6164  age: smooth quad
│ │ │ │ +00044ff0: 7269 6320 6879 7065 7273 7572 6661 6365  ric hypersurface
│ │ │ │ +00045000: 2069 6e20 5050 5e35 2020 2020 2020 2020   in PP^5        
│ │ │ │ +00045010: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00045020: 646f 6d69 6e61 6e63 653a 2066 616c 7365  dominance: false
│ │ │ │  00045030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045050: 2020 2020 7c0a 7c20 2020 2020 6269 7261      |.|     bira
│ │ │ │ -00045060: 7469 6f6e 616c 6974 793a 2066 616c 7365  tionality: false
│ │ │ │ -00045070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045050: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00045060: 2020 6269 7261 7469 6f6e 616c 6974 793a    birationality:
│ │ │ │ +00045070: 2066 616c 7365 2020 2020 2020 2020 2020   false          
│ │ │ │  00045080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045090: 2020 2020 2020 7c0a 7c20 2020 2020 636f        |.|     co
│ │ │ │ -000450a0: 6566 6669 6369 656e 7420 7269 6e67 3a20  efficient ring: 
│ │ │ │ -000450b0: 5a5a 2f33 3333 3331 2020 2020 2020 2020  ZZ/33331        
│ │ │ │ +00045090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000450a0: 2020 2020 636f 6566 6669 6369 656e 7420      coefficient 
│ │ │ │ +000450b0: 7269 6e67 3a20 5a5a 2f33 3333 3331 2020  ring: ZZ/33331  
│ │ │ │  000450c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000450d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -000450e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000450d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000450e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000450f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045110: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -00045120: 3a20 3f20 4920 2020 2020 2020 2020 2020  : ? I           
│ │ │ │ +00045110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00045120: 2b0a 7c69 3620 3a20 3f20 4920 2020 2020  +.|i6 : ? I     
│ │ │ │  00045130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00045160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045160: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00045170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000451a0: 7c6f 3620 3d20 736d 6f6f 7468 2071 7561  |o6 = smooth qua
│ │ │ │ -000451b0: 6472 6963 2068 7970 6572 7375 7266 6163  dric hypersurfac
│ │ │ │ -000451c0: 6520 696e 2050 505e 3520 2020 2020 2020  e in PP^5       
│ │ │ │ +00045190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000451a0: 2020 2020 7c0a 7c6f 3620 3d20 736d 6f6f      |.|o6 = smoo
│ │ │ │ +000451b0: 7468 2071 7561 6472 6963 2068 7970 6572  th quadric hyper
│ │ │ │ +000451c0: 7375 7266 6163 6520 696e 2050 505e 3520  surface in PP^5 
│ │ │ │  000451d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000451e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000451e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  000451f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045220: 2d2d 2b0a 7c69 3720 3a20 7068 6921 3b20  --+.|i7 : phi!; 
│ │ │ │ -00045230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045220: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ +00045230: 7068 6921 3b20 2020 2020 2020 2020 2020  phi!;           
│ │ │ │  00045240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045260: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00045260: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00045270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000452a0: 2020 2020 2020 7c0a 7c6f 3720 3a20 5261        |.|o7 : Ra
│ │ │ │ -000452b0: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ -000452c0: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ -000452d0: 7020 6672 6f6d 2050 505e 3420 746f 2050  p from PP^4 to P
│ │ │ │ -000452e0: 505e 3529 2020 2020 7c0a 2b2d 2d2d 2d2d  P^5)    |.+-----
│ │ │ │ -000452f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000452a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000452b0: 3720 3a20 5261 7469 6f6e 616c 4d61 7020  7 : RationalMap 
│ │ │ │ +000452c0: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ +000452d0: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ +000452e0: 3420 746f 2050 505e 3529 2020 2020 7c0a  4 to PP^5)    |.
│ │ │ │ +000452f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00045300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045320: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ -00045330: 3a20 6465 7363 7269 6265 2070 6869 2020  : describe phi  
│ │ │ │ -00045340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00045330: 2b0a 7c69 3820 3a20 6465 7363 7269 6265  +.|i8 : describe
│ │ │ │ +00045340: 2070 6869 2020 2020 2020 2020 2020 2020   phi            
│ │ │ │  00045350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045360: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00045370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045370: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00045380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000453a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000453b0: 7c6f 3820 3d20 7261 7469 6f6e 616c 206d  |o8 = rational m
│ │ │ │ -000453c0: 6170 2064 6566 696e 6564 2062 7920 666f  ap defined by fo
│ │ │ │ -000453d0: 726d 7320 6f66 2064 6567 7265 6520 3220  rms of degree 2 
│ │ │ │ -000453e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000453f0: 7c0a 7c20 2020 2020 736f 7572 6365 2076  |.|     source v
│ │ │ │ -00045400: 6172 6965 7479 3a20 5050 5e34 2020 2020  ariety: PP^4    
│ │ │ │ -00045410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000453a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000453b0: 2020 2020 7c0a 7c6f 3820 3d20 7261 7469      |.|o8 = rati
│ │ │ │ +000453c0: 6f6e 616c 206d 6170 2064 6566 696e 6564  onal map defined
│ │ │ │ +000453d0: 2062 7920 666f 726d 7320 6f66 2064 6567   by forms of deg
│ │ │ │ +000453e0: 7265 6520 3220 2020 2020 2020 2020 2020  ree 2           
│ │ │ │ +000453f0: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +00045400: 7572 6365 2076 6172 6965 7479 3a20 5050  urce variety: PP
│ │ │ │ +00045410: 5e34 2020 2020 2020 2020 2020 2020 2020  ^4              
│ │ │ │  00045420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045430: 2020 7c0a 7c20 2020 2020 7461 7267 6574    |.|     target
│ │ │ │ -00045440: 2076 6172 6965 7479 3a20 5050 5e35 2020   variety: PP^5  
│ │ │ │ -00045450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ +00045600: 2020 2020 2020 7c0a 7c20 2020 2020 6469        |.|     di
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│ │ │ │ +00045640: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ -000456a0: 2f33 3333 3331 2020 2020 2020 2020 2020  /33331          
│ │ │ │ +00045680: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00045690: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
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│ │ │ │  000456b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000456c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000456c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  000456d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000456e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000456f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045700: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -00045710: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -00045720: 2a20 2a6e 6f74 6520 5261 7469 6f6e 616c  * *note Rational
│ │ │ │ -00045730: 4d61 7020 213a 2052 6174 696f 6e61 6c4d  Map !: RationalM
│ │ │ │ -00045740: 6170 2021 2c20 2d2d 2063 616c 6375 6c61  ap !, -- calcula
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│ │ │ │ -00045770: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ -00045780: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ -00045790: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -000457a0: 202a 6e6f 7465 2064 6573 6372 6962 6528   *note describe(
│ │ │ │ -000457b0: 5261 7469 6f6e 616c 4d61 7029 3a20 6465  RationalMap): de
│ │ │ │ -000457c0: 7363 7269 6265 5f6c 7052 6174 696f 6e61  scribe_lpRationa
│ │ │ │ -000457d0: 6c4d 6170 5f72 702c 202d 2d20 6465 7363  lMap_rp, -- desc
│ │ │ │ -000457e0: 7269 6265 2061 0a20 2020 2072 6174 696f  ribe a.    ratio
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│ │ │ │ -00045810: 6465 3a20 446f 6d69 6e61 6e74 2c20 4e65  de: Dominant, Ne
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│ │ │ │ -00045830: 7469 6f6e 616c 4d61 705f 7270 2c20 5072  tionalMap_rp, Pr
│ │ │ │ -00045840: 6576 3a20 6465 7363 7269 6265 5f6c 7052  ev: describe_lpR
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│ │ │ │ -00045860: 703a 2054 6f70 0a0a 446f 6d69 6e61 6e74  p: Top..Dominant
│ │ │ │ -00045870: 0a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372  .********..Descr
│ │ │ │ -00045880: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -00045890: 3d3d 0a0a 5468 6973 2069 7320 616e 206f  ==..This is an o
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│ │ │ │ -000458b0: 2066 6f72 202a 6e6f 7465 2074 6f4d 6170   for *note toMap
│ │ │ │ -000458c0: 3a20 746f 4d61 702c 2e20 5768 656e 202a  : toMap,. When *
│ │ │ │ -000458d0: 6e6f 7465 2074 7275 653a 0a28 4d61 6361  note true:.(Maca
│ │ │ │ -000458e0: 756c 6179 3244 6f63 2974 7275 652c 206f  ulay2Doc)true, o
│ │ │ │ -000458f0: 7220 2a6e 6f74 6520 696e 6669 6e69 7479  r *note infinity
│ │ │ │ -00045900: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00045910: 696e 6669 6e69 7479 2c20 6973 2070 6173  infinity, is pas
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│ │ │ │ -00045930: 6f6e 2c20 7468 6520 6b65 726e 656c 206f  on, the kernel o
│ │ │ │ -00045940: 6620 7468 6520 7265 7475 726e 6564 2072  f the returned r
│ │ │ │ -00045950: 696e 6720 6d61 7020 7769 6c6c 2062 6520  ing map will be 
│ │ │ │ -00045960: 7a65 726f 2e0a 0a46 756e 6374 696f 6e73  zero...Functions
│ │ │ │ -00045970: 2077 6974 6820 6f70 7469 6f6e 616c 2061   with optional a
│ │ │ │ -00045980: 7267 756d 656e 7420 6e61 6d65 6420 446f  rgument named Do
│ │ │ │ -00045990: 6d69 6e61 6e74 3a0a 3d3d 3d3d 3d3d 3d3d  minant:.========
│ │ │ │ +00045700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00045710: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00045720: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 5261  ==..  * *note Ra
│ │ │ │ +00045730: 7469 6f6e 616c 4d61 7020 213a 2052 6174  tionalMap !: Rat
│ │ │ │ +00045740: 696f 6e61 6c4d 6170 2021 2c20 2d2d 2063  ionalMap !, -- c
│ │ │ │ +00045750: 616c 6375 6c61 7465 7320 6576 6572 7920  alculates every 
│ │ │ │ +00045760: 706f 7373 6962 6c65 2074 6869 6e67 0a0a  possible thing..
│ │ │ │ +00045770: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
│ │ │ │ +00045780: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   method:.=======
│ │ │ │ +00045790: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000457a0: 3d0a 0a20 202a 202a 6e6f 7465 2064 6573  =..  * *note des
│ │ │ │ +000457b0: 6372 6962 6528 5261 7469 6f6e 616c 4d61  cribe(RationalMa
│ │ │ │ +000457c0: 7029 3a20 6465 7363 7269 6265 5f6c 7052  p): describe_lpR
│ │ │ │ +000457d0: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +000457e0: 2d20 6465 7363 7269 6265 2061 0a20 2020  - describe a.   
│ │ │ │ +000457f0: 2072 6174 696f 6e61 6c20 6d61 700a 1f0a   rational map...
│ │ │ │ +00045800: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ +00045810: 666f 2c20 4e6f 6465 3a20 446f 6d69 6e61  fo, Node: Domina
│ │ │ │ +00045820: 6e74 2c20 4e65 7874 3a20 656e 7472 6965  nt, Next: entrie
│ │ │ │ +00045830: 735f 6c70 5261 7469 6f6e 616c 4d61 705f  s_lpRationalMap_
│ │ │ │ +00045840: 7270 2c20 5072 6576 3a20 6465 7363 7269  rp, Prev: descri
│ │ │ │ +00045850: 6265 5f6c 7052 6174 696f 6e61 6c4d 6170  be_lpRationalMap
│ │ │ │ +00045860: 5f72 702c 2055 703a 2054 6f70 0a0a 446f  _rp, Up: Top..Do
│ │ │ │ +00045870: 6d69 6e61 6e74 0a2a 2a2a 2a2a 2a2a 2a0a  minant.********.
│ │ │ │ +00045880: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +00045890: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2069  ========..This i
│ │ │ │ +000458a0: 7320 616e 206f 7074 696f 6e61 6c20 6172  s an optional ar
│ │ │ │ +000458b0: 6775 6d65 6e74 2066 6f72 202a 6e6f 7465  gument for *note
│ │ │ │ +000458c0: 2074 6f4d 6170 3a20 746f 4d61 702c 2e20   toMap: toMap,. 
│ │ │ │ +000458d0: 5768 656e 202a 6e6f 7465 2074 7275 653a  When *note true:
│ │ │ │ +000458e0: 0a28 4d61 6361 756c 6179 3244 6f63 2974  .(Macaulay2Doc)t
│ │ │ │ +000458f0: 7275 652c 206f 7220 2a6e 6f74 6520 696e  rue, or *note in
│ │ │ │ +00045900: 6669 6e69 7479 3a20 284d 6163 6175 6c61  finity: (Macaula
│ │ │ │ +00045910: 7932 446f 6329 696e 6669 6e69 7479 2c20  y2Doc)infinity, 
│ │ │ │ +00045920: 6973 2070 6173 7365 6420 746f 0a74 6869  is passed to.thi
│ │ │ │ +00045930: 7320 6f70 7469 6f6e 2c20 7468 6520 6b65  s option, the ke
│ │ │ │ +00045940: 726e 656c 206f 6620 7468 6520 7265 7475  rnel of the retu
│ │ │ │ +00045950: 726e 6564 2072 696e 6720 6d61 7020 7769  rned ring map wi
│ │ │ │ +00045960: 6c6c 2062 6520 7a65 726f 2e0a 0a46 756e  ll be zero...Fun
│ │ │ │ +00045970: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ +00045980: 6f6e 616c 2061 7267 756d 656e 7420 6e61  onal argument na
│ │ │ │ +00045990: 6d65 6420 446f 6d69 6e61 6e74 3a0a 3d3d  med Dominant:.==
│ │ │ │  000459a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  000459b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000459c0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2272  ========..  * "r
│ │ │ │ -000459d0: 6174 696f 6e61 6c4d 6170 282e 2e2e 2c44  ationalMap(...,D
│ │ │ │ -000459e0: 6f6d 696e 616e 743d 3e2e 2e2e 2922 0a20  ominant=>...)". 
│ │ │ │ -000459f0: 202a 2022 746f 4d61 7028 2e2e 2e2c 446f   * "toMap(...,Do
│ │ │ │ -00045a00: 6d69 6e61 6e74 3d3e 2e2e 2e29 220a 0a46  minant=>...)"..F
│ │ │ │ -00045a10: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00045a20: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00045a30: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00045a40: 202a 6e6f 7465 2044 6f6d 696e 616e 743a   *note Dominant:
│ │ │ │ -00045a50: 2044 6f6d 696e 616e 742c 2069 7320 6120   Dominant, is a 
│ │ │ │ -00045a60: 2a6e 6f74 6520 7379 6d62 6f6c 3a20 284d  *note symbol: (M
│ │ │ │ -00045a70: 6163 6175 6c61 7932 446f 6329 5379 6d62  acaulay2Doc)Symb
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│ │ │ │ -00045a90: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
│ │ │ │ -00045aa0: 2065 6e74 7269 6573 5f6c 7052 6174 696f   entries_lpRatio
│ │ │ │ -00045ab0: 6e61 6c4d 6170 5f72 702c 204e 6578 743a  nalMap_rp, Next:
│ │ │ │ -00045ac0: 2045 756c 6572 4368 6172 6163 7465 7269   EulerCharacteri
│ │ │ │ -00045ad0: 7374 6963 2c20 5072 6576 3a20 446f 6d69  stic, Prev: Domi
│ │ │ │ -00045ae0: 6e61 6e74 2c20 5570 3a20 546f 700a 0a65  nant, Up: Top..e
│ │ │ │ -00045af0: 6e74 7269 6573 2852 6174 696f 6e61 6c4d  ntries(RationalM
│ │ │ │ -00045b00: 6170 2920 2d2d 2074 6865 2065 6e74 7269  ap) -- the entri
│ │ │ │ -00045b10: 6573 206f 6620 7468 6520 6d61 7472 6978  es of the matrix
│ │ │ │ -00045b20: 2061 7373 6f63 6961 7465 6420 746f 2061   associated to a
│ │ │ │ -00045b30: 2072 6174 696f 6e61 6c20 6d61 700a 2a2a   rational map.**
│ │ │ │ -00045b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000459c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +000459d0: 2020 2a20 2272 6174 696f 6e61 6c4d 6170    * "rationalMap
│ │ │ │ +000459e0: 282e 2e2e 2c44 6f6d 696e 616e 743d 3e2e  (...,Dominant=>.
│ │ │ │ +000459f0: 2e2e 2922 0a20 202a 2022 746f 4d61 7028  ..)".  * "toMap(
│ │ │ │ +00045a00: 2e2e 2e2c 446f 6d69 6e61 6e74 3d3e 2e2e  ...,Dominant=>..
│ │ │ │ +00045a10: 2e29 220a 0a46 6f72 2074 6865 2070 726f  .)"..For the pro
│ │ │ │ +00045a20: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +00045a30: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +00045a40: 6f62 6a65 6374 202a 6e6f 7465 2044 6f6d  object *note Dom
│ │ │ │ +00045a50: 696e 616e 743a 2044 6f6d 696e 616e 742c  inant: Dominant,
│ │ │ │ +00045a60: 2069 7320 6120 2a6e 6f74 6520 7379 6d62   is a *note symb
│ │ │ │ +00045a70: 6f6c 3a20 284d 6163 6175 6c61 7932 446f  ol: (Macaulay2Do
│ │ │ │ +00045a80: 6329 5379 6d62 6f6c 2c2e 0a1f 0a46 696c  c)Symbol,....Fil
│ │ │ │ +00045a90: 653a 2043 7265 6d6f 6e61 2e69 6e66 6f2c  e: Cremona.info,
│ │ │ │ +00045aa0: 204e 6f64 653a 2065 6e74 7269 6573 5f6c   Node: entries_l
│ │ │ │ +00045ab0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +00045ac0: 204e 6578 743a 2045 756c 6572 4368 6172   Next: EulerChar
│ │ │ │ +00045ad0: 6163 7465 7269 7374 6963 2c20 5072 6576  acteristic, Prev
│ │ │ │ +00045ae0: 3a20 446f 6d69 6e61 6e74 2c20 5570 3a20  : Dominant, Up: 
│ │ │ │ +00045af0: 546f 700a 0a65 6e74 7269 6573 2852 6174  Top..entries(Rat
│ │ │ │ +00045b00: 696f 6e61 6c4d 6170 2920 2d2d 2074 6865  ionalMap) -- the
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│ │ │ │ +00045b20: 6d61 7472 6978 2061 7373 6f63 6961 7465  matrix associate
│ │ │ │ +00045b30: 6420 746f 2061 2072 6174 696f 6e61 6c20  d to a rational 
│ │ │ │ +00045b40: 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  map.************
│ │ │ │  00045b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00045b60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00045b70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00045b80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -00045b90: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -00045ba0: 0a20 202a 2046 756e 6374 696f 6e3a 202a  .  * Function: *
│ │ │ │ -00045bb0: 6e6f 7465 2065 6e74 7269 6573 3a20 284d  note entries: (M
│ │ │ │ -00045bc0: 6163 6175 6c61 7932 446f 6329 656e 7472  acaulay2Doc)entr
│ │ │ │ -00045bd0: 6965 732c 0a20 202a 2055 7361 6765 3a20  ies,.  * Usage: 
│ │ │ │ -00045be0: 0a20 2020 2020 2020 2065 6e74 7269 6573  .        entries
│ │ │ │ -00045bf0: 2050 6869 0a20 202a 2049 6e70 7574 733a   Phi.  * Inputs:
│ │ │ │ -00045c00: 0a20 2020 2020 202a 2050 6869 2c20 6120  .      * Phi, a 
│ │ │ │ -00045c10: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ -00045c20: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ -00045c30: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00045c40: 2020 2020 2a20 6120 2a6e 6f74 6520 6c69      * a *note li
│ │ │ │ -00045c50: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ -00045c60: 6329 4c69 7374 2c2c 2074 6865 2065 6e74  c)List,, the ent
│ │ │ │ -00045c70: 7269 6573 206f 6620 7468 6520 6d61 7472  ries of the matr
│ │ │ │ -00045c80: 6978 2061 7373 6f63 6961 7465 640a 2020  ix associated.  
│ │ │ │ -00045c90: 2020 2020 2020 746f 2074 6865 2072 696e        to the rin
│ │ │ │ -00045ca0: 6720 6d61 7020 6465 6669 6e69 6e67 2074  g map defining t
│ │ │ │ -00045cb0: 6865 2072 6174 696f 6e61 6c20 6d61 7020  he rational map 
│ │ │ │ -00045cc0: 5068 690a 0a44 6573 6372 6970 7469 6f6e  Phi..Description
│ │ │ │ -00045cd0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ -00045ce0: 6973 2069 7320 6571 7569 7661 6c65 6e74  is is equivalent
│ │ │ │ -00045cf0: 2074 6f20 666c 6174 7465 6e20 656e 7472   to flatten entr
│ │ │ │ -00045d00: 6965 7320 6d61 7472 6978 2050 6869 2e0a  ies matrix Phi..
│ │ │ │ -00045d10: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -00045d20: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6d61  ==..  * *note ma
│ │ │ │ -00045d30: 7472 6978 2852 6174 696f 6e61 6c4d 6170  trix(RationalMap
│ │ │ │ -00045d40: 293a 206d 6174 7269 785f 6c70 5261 7469  ): matrix_lpRati
│ │ │ │ -00045d50: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2074  onalMap_rp, -- t
│ │ │ │ -00045d60: 6865 206d 6174 7269 780a 2020 2020 6173  he matrix.    as
│ │ │ │ -00045d70: 736f 6369 6174 6564 2074 6f20 6120 7261  sociated to a ra
│ │ │ │ -00045d80: 7469 6f6e 616c 206d 6170 0a0a 5761 7973  tional map..Ways
│ │ │ │ -00045d90: 2074 6f20 7573 6520 7468 6973 206d 6574   to use this met
│ │ │ │ -00045da0: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  hod:.===========
│ │ │ │ -00045db0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -00045dc0: 202a 202a 6e6f 7465 2065 6e74 7269 6573   * *note entries
│ │ │ │ -00045dd0: 2852 6174 696f 6e61 6c4d 6170 293a 2065  (RationalMap): e
│ │ │ │ -00045de0: 6e74 7269 6573 5f6c 7052 6174 696f 6e61  ntries_lpRationa
│ │ │ │ -00045df0: 6c4d 6170 5f72 702c 202d 2d20 7468 6520  lMap_rp, -- the 
│ │ │ │ -00045e00: 656e 7472 6965 7320 6f66 2074 6865 0a20  entries of the. 
│ │ │ │ -00045e10: 2020 206d 6174 7269 7820 6173 736f 6369     matrix associ
│ │ │ │ -00045e20: 6174 6564 2074 6f20 6120 7261 7469 6f6e  ated to a ration
│ │ │ │ -00045e30: 616c 206d 6170 0a1f 0a46 696c 653a 2043  al map...File: C
│ │ │ │ -00045e40: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ -00045e50: 653a 2045 756c 6572 4368 6172 6163 7465  e: EulerCharacte
│ │ │ │ -00045e60: 7269 7374 6963 2c20 4e65 7874 3a20 6578  ristic, Next: ex
│ │ │ │ -00045e70: 6365 7074 696f 6e61 6c4c 6f63 7573 2c20  ceptionalLocus, 
│ │ │ │ -00045e80: 5072 6576 3a20 656e 7472 6965 735f 6c70  Prev: entries_lp
│ │ │ │ -00045e90: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -00045ea0: 5570 3a20 546f 700a 0a45 756c 6572 4368  Up: Top..EulerCh
│ │ │ │ -00045eb0: 6172 6163 7465 7269 7374 6963 202d 2d20  aracteristic -- 
│ │ │ │ -00045ec0: 746f 706f 6c6f 6769 6361 6c20 4575 6c65  topological Eule
│ │ │ │ -00045ed0: 7220 6368 6172 6163 7465 7269 7374 6963  r characteristic
│ │ │ │ -00045ee0: 206f 6620 6120 2873 6d6f 6f74 6829 2070   of a (smooth) p
│ │ │ │ -00045ef0: 726f 6a65 6374 6976 6520 7661 7269 6574  rojective variet
│ │ │ │ -00045f00: 790a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  y.**************
│ │ │ │ +00045b80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00045b90: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +00045ba0: 3d3d 3d3d 3d0a 0a20 202a 2046 756e 6374  =====..  * Funct
│ │ │ │ +00045bb0: 696f 6e3a 202a 6e6f 7465 2065 6e74 7269  ion: *note entri
│ │ │ │ +00045bc0: 6573 3a20 284d 6163 6175 6c61 7932 446f  es: (Macaulay2Do
│ │ │ │ +00045bd0: 6329 656e 7472 6965 732c 0a20 202a 2055  c)entries,.  * U
│ │ │ │ +00045be0: 7361 6765 3a20 0a20 2020 2020 2020 2065  sage: .        e
│ │ │ │ +00045bf0: 6e74 7269 6573 2050 6869 0a20 202a 2049  ntries Phi.  * I
│ │ │ │ +00045c00: 6e70 7574 733a 0a20 2020 2020 202a 2050  nputs:.      * P
│ │ │ │ +00045c10: 6869 2c20 6120 2a6e 6f74 6520 7261 7469  hi, a *note rati
│ │ │ │ +00045c20: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ +00045c30: 616c 4d61 702c 0a20 202a 204f 7574 7075  alMap,.  * Outpu
│ │ │ │ +00045c40: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ +00045c50: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +00045c60: 6c61 7932 446f 6329 4c69 7374 2c2c 2074  lay2Doc)List,, t
│ │ │ │ +00045c70: 6865 2065 6e74 7269 6573 206f 6620 7468  he entries of th
│ │ │ │ +00045c80: 6520 6d61 7472 6978 2061 7373 6f63 6961  e matrix associa
│ │ │ │ +00045c90: 7465 640a 2020 2020 2020 2020 746f 2074  ted.        to t
│ │ │ │ +00045ca0: 6865 2072 696e 6720 6d61 7020 6465 6669  he ring map defi
│ │ │ │ +00045cb0: 6e69 6e67 2074 6865 2072 6174 696f 6e61  ning the rationa
│ │ │ │ +00045cc0: 6c20 6d61 7020 5068 690a 0a44 6573 6372  l map Phi..Descr
│ │ │ │ +00045cd0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00045ce0: 3d3d 0a0a 5468 6973 2069 7320 6571 7569  ==..This is equi
│ │ │ │ +00045cf0: 7661 6c65 6e74 2074 6f20 666c 6174 7465  valent to flatte
│ │ │ │ +00045d00: 6e20 656e 7472 6965 7320 6d61 7472 6978  n entries matrix
│ │ │ │ +00045d10: 2050 6869 2e0a 0a53 6565 2061 6c73 6f0a   Phi...See also.
│ │ │ │ +00045d20: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00045d30: 6f74 6520 6d61 7472 6978 2852 6174 696f  ote matrix(Ratio
│ │ │ │ +00045d40: 6e61 6c4d 6170 293a 206d 6174 7269 785f  nalMap): matrix_
│ │ │ │ +00045d50: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ +00045d60: 2c20 2d2d 2074 6865 206d 6174 7269 780a  , -- the matrix.
│ │ │ │ +00045d70: 2020 2020 6173 736f 6369 6174 6564 2074      associated t
│ │ │ │ +00045d80: 6f20 6120 7261 7469 6f6e 616c 206d 6170  o a rational map
│ │ │ │ +00045d90: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
│ │ │ │ +00045da0: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d  is method:.=====
│ │ │ │ +00045db0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00045dc0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2065  ===..  * *note e
│ │ │ │ +00045dd0: 6e74 7269 6573 2852 6174 696f 6e61 6c4d  ntries(RationalM
│ │ │ │ +00045de0: 6170 293a 2065 6e74 7269 6573 5f6c 7052  ap): entries_lpR
│ │ │ │ +00045df0: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +00045e00: 2d20 7468 6520 656e 7472 6965 7320 6f66  - the entries of
│ │ │ │ +00045e10: 2074 6865 0a20 2020 206d 6174 7269 7820   the.    matrix 
│ │ │ │ +00045e20: 6173 736f 6369 6174 6564 2074 6f20 6120  associated to a 
│ │ │ │ +00045e30: 7261 7469 6f6e 616c 206d 6170 0a1f 0a46  rational map...F
│ │ │ │ +00045e40: 696c 653a 2043 7265 6d6f 6e61 2e69 6e66  ile: Cremona.inf
│ │ │ │ +00045e50: 6f2c 204e 6f64 653a 2045 756c 6572 4368  o, Node: EulerCh
│ │ │ │ +00045e60: 6172 6163 7465 7269 7374 6963 2c20 4e65  aracteristic, Ne
│ │ │ │ +00045e70: 7874 3a20 6578 6365 7074 696f 6e61 6c4c  xt: exceptionalL
│ │ │ │ +00045e80: 6f63 7573 2c20 5072 6576 3a20 656e 7472  ocus, Prev: entr
│ │ │ │ +00045e90: 6965 735f 6c70 5261 7469 6f6e 616c 4d61  ies_lpRationalMa
│ │ │ │ +00045ea0: 705f 7270 2c20 5570 3a20 546f 700a 0a45  p_rp, Up: Top..E
│ │ │ │ +00045eb0: 756c 6572 4368 6172 6163 7465 7269 7374  ulerCharacterist
│ │ │ │ +00045ec0: 6963 202d 2d20 746f 706f 6c6f 6769 6361  ic -- topologica
│ │ │ │ +00045ed0: 6c20 4575 6c65 7220 6368 6172 6163 7465  l Euler characte
│ │ │ │ +00045ee0: 7269 7374 6963 206f 6620 6120 2873 6d6f  ristic of a (smo
│ │ │ │ +00045ef0: 6f74 6829 2070 726f 6a65 6374 6976 6520  oth) projective 
│ │ │ │ +00045f00: 7661 7269 6574 790a 2a2a 2a2a 2a2a 2a2a  variety.********
│ │ │ │  00045f10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00045f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00045f30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00045f40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00045f50: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ -00045f60: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ -00045f70: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -00045f80: 2020 2045 756c 6572 4368 6172 6163 7465     EulerCharacte
│ │ │ │ -00045f90: 7269 7374 6963 2049 0a20 202a 2049 6e70  ristic I.  * Inp
│ │ │ │ -00045fa0: 7574 733a 0a20 2020 2020 202a 2049 2c20  uts:.      * I, 
│ │ │ │ -00045fb0: 616e 202a 6e6f 7465 2069 6465 616c 3a20  an *note ideal: 
│ │ │ │ -00045fc0: 284d 6163 6175 6c61 7932 446f 6329 4964  (Macaulay2Doc)Id
│ │ │ │ -00045fd0: 6561 6c2c 2c20 6120 686f 6d6f 6765 6e65  eal,, a homogene
│ │ │ │ -00045fe0: 6f75 7320 6964 6561 6c20 6465 6669 6e69  ous ideal defini
│ │ │ │ -00045ff0: 6e67 2061 0a20 2020 2020 2020 2073 6d6f  ng a.        smo
│ │ │ │ -00046000: 6f74 6820 7072 6f6a 6563 7469 7665 2076  oth projective v
│ │ │ │ -00046010: 6172 6965 7479 2024 585c 7375 6273 6574  ariety $X\subset
│ │ │ │ -00046020: 5c6d 6174 6862 627b 507d 5e6e 240a 2020  \mathbb{P}^n$.  
│ │ │ │ -00046030: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c  * *note Optional
│ │ │ │ -00046040: 2069 6e70 7574 733a 2028 4d61 6361 756c   inputs: (Macaul
│ │ │ │ -00046050: 6179 3244 6f63 2975 7369 6e67 2066 756e  ay2Doc)using fun
│ │ │ │ -00046060: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ -00046070: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020  onal inputs,:.  
│ │ │ │ -00046080: 2020 2020 2a20 2a6e 6f74 6520 426c 6f77      * *note Blow
│ │ │ │ -00046090: 5570 5374 7261 7465 6779 3a20 426c 6f77  UpStrategy: Blow
│ │ │ │ -000460a0: 5570 5374 7261 7465 6779 2c20 3d3e 202e  UpStrategy, => .
│ │ │ │ -000460b0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -000460c0: 650a 2020 2020 2020 2020 2245 6c69 6d69  e.        "Elimi
│ │ │ │ -000460d0: 6e61 7465 222c 0a20 2020 2020 202a 202a  nate",.      * *
│ │ │ │ -000460e0: 6e6f 7465 2043 6572 7469 6679 3a20 4365  note Certify: Ce
│ │ │ │ -000460f0: 7274 6966 792c 203d 3e20 2e2e 2e2c 2064  rtify, => ..., d
│ │ │ │ -00046100: 6566 6175 6c74 2076 616c 7565 2066 616c  efault value fal
│ │ │ │ -00046110: 7365 2c20 7768 6574 6865 7220 746f 2065  se, whether to e
│ │ │ │ -00046120: 6e73 7572 650a 2020 2020 2020 2020 636f  nsure.        co
│ │ │ │ -00046130: 7272 6563 746e 6573 7320 6f66 206f 7574  rrectness of out
│ │ │ │ -00046140: 7075 740a 2020 2020 2020 2a20 2a6e 6f74  put.      * *not
│ │ │ │ -00046150: 6520 5665 7262 6f73 653a 2069 6e76 6572  e Verbose: inver
│ │ │ │ -00046160: 7365 4d61 705f 6c70 5f70 645f 7064 5f70  seMap_lp_pd_pd_p
│ │ │ │ -00046170: 645f 636d 5665 7262 6f73 653d 3e5f 7064  d_cmVerbose=>_pd
│ │ │ │ -00046180: 5f70 645f 7064 5f72 702c 203d 3e20 2e2e  _pd_pd_rp, => ..
│ │ │ │ -00046190: 2e2c 0a20 2020 2020 2020 2064 6566 6175  .,.        defau
│ │ │ │ -000461a0: 6c74 2076 616c 7565 2074 7275 652c 0a20  lt value true,. 
│ │ │ │ -000461b0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -000461c0: 2020 2a20 616e 202a 6e6f 7465 2069 6e74    * an *note int
│ │ │ │ -000461d0: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
│ │ │ │ -000461e0: 446f 6329 5a5a 2c2c 2074 6865 2074 6f70  Doc)ZZ,, the top
│ │ │ │ -000461f0: 6f6c 6f67 6963 616c 2045 756c 6572 0a20  ological Euler. 
│ │ │ │ -00046200: 2020 2020 2020 2063 6861 7261 6374 6572         character
│ │ │ │ -00046210: 6973 7469 6373 206f 6620 2458 242e 0a0a  istics of $X$...
│ │ │ │ -00046220: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00046230: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6973  =======..This is
│ │ │ │ -00046240: 2061 6e20 6170 706c 6963 6174 696f 6e20   an application 
│ │ │ │ -00046250: 6f66 2074 6865 206d 6574 686f 6420 2a6e  of the method *n
│ │ │ │ -00046260: 6f74 6520 5365 6772 6543 6c61 7373 3a20  ote SegreClass: 
│ │ │ │ -00046270: 5365 6772 6543 6c61 7373 2c2e 2053 6565  SegreClass,. See
│ │ │ │ -00046280: 2061 6c73 6f0a 7468 6520 636f 7272 6573   also.the corres
│ │ │ │ -00046290: 706f 6e64 696e 6720 6d65 7468 6f64 7320  ponding methods 
│ │ │ │ -000462a0: 696e 2074 6865 2070 6163 6b61 6765 7320  in the packages 
│ │ │ │ -000462b0: 4353 4d2d 4120 2873 6565 0a68 7474 703a  CSM-A (see.http:
│ │ │ │ -000462c0: 2f2f 7777 772e 6d61 7468 2e66 7375 2e65  //www.math.fsu.e
│ │ │ │ -000462d0: 6475 2f7e 616c 7566 6669 2f43 534d 2f43  du/~aluffi/CSM/C
│ │ │ │ -000462e0: 534d 2e68 746d 6c20 292c 2062 7920 502e  SM.html ), by P.
│ │ │ │ -000462f0: 2041 6c75 6666 692c 2061 6e64 0a43 6861   Aluffi, and.Cha
│ │ │ │ -00046300: 7261 6374 6572 6973 7469 6343 6c61 7373  racteristicClass
│ │ │ │ -00046310: 6573 2028 7365 650a 6874 7470 3a2f 2f77  es (see.http://w
│ │ │ │ -00046320: 7777 2e6d 6174 682e 7569 7563 2e65 6475  ww.math.uiuc.edu
│ │ │ │ -00046330: 2f4d 6163 6175 6c61 7932 2f64 6f63 2f4d  /Macaulay2/doc/M
│ │ │ │ -00046340: 6163 6175 6c61 7932 2d31 2e31 362f 7368  acaulay2-1.16/sh
│ │ │ │ -00046350: 6172 652f 646f 632f 4d61 6361 756c 6179  are/doc/Macaulay
│ │ │ │ -00046360: 322f 0a43 6861 7261 6374 6572 6973 7469  2/.Characteristi
│ │ │ │ -00046370: 6343 6c61 7373 6573 2f68 746d 6c2f 2029  cClasses/html/ )
│ │ │ │ -00046380: 2c20 6279 204d 2e20 4865 6c6d 6572 2061  , by M. Helmer a
│ │ │ │ -00046390: 6e64 2043 2e20 4a6f 7374 2e0a 0a49 6e20  nd C. Jost...In 
│ │ │ │ -000463a0: 6765 6e65 7261 6c2c 2065 7665 6e20 6966  general, even if
│ │ │ │ -000463b0: 2074 6865 2069 6e70 7574 2069 6465 616c   the input ideal
│ │ │ │ -000463c0: 2064 6566 696e 6573 2061 2073 696e 6775   defines a singu
│ │ │ │ -000463d0: 6c61 7220 7661 7269 6574 7920 2458 242c  lar variety $X$,
│ │ │ │ -000463e0: 2074 6865 0a72 6574 7572 6e65 6420 7661   the.returned va
│ │ │ │ -000463f0: 6c75 6520 6571 7561 6c73 2074 6865 2064  lue equals the d
│ │ │ │ -00046400: 6567 7265 6520 6f66 2074 6865 2063 6f6d  egree of the com
│ │ │ │ -00046410: 706f 6e65 6e74 206f 6620 6469 6d65 6e73  ponent of dimens
│ │ │ │ -00046420: 696f 6e20 3020 6f66 2074 6865 0a43 6865  ion 0 of the.Che
│ │ │ │ -00046430: 726e 2d46 756c 746f 6e20 636c 6173 7320  rn-Fulton class 
│ │ │ │ -00046440: 6f66 2024 5824 2e20 5468 6520 4575 6c65  of $X$. The Eule
│ │ │ │ -00046450: 7220 6368 6172 6163 7465 7269 7374 6963  r characteristic
│ │ │ │ -00046460: 206f 6620 6120 7369 6e67 756c 6172 2076   of a singular v
│ │ │ │ -00046470: 6172 6965 7479 2063 616e 0a62 6520 636f  ariety can.be co
│ │ │ │ -00046480: 6d70 7574 6564 2076 6961 2074 6865 206d  mputed via the m
│ │ │ │ -00046490: 6574 686f 6420 2a6e 6f74 6520 4368 6572  ethod *note Cher
│ │ │ │ -000464a0: 6e53 6368 7761 7274 7a4d 6163 5068 6572  nSchwartzMacPher
│ │ │ │ -000464b0: 736f 6e3a 0a43 6865 726e 5363 6877 6172  son:.ChernSchwar
│ │ │ │ -000464c0: 747a 4d61 6350 6865 7273 6f6e 2c2e 0a0a  tzMacPherson,...
│ │ │ │ -000464d0: 496e 2074 6865 2065 7861 6d70 6c65 2062  In the example b
│ │ │ │ -000464e0: 656c 6f77 2c20 7765 2063 6f6d 7075 7465  elow, we compute
│ │ │ │ -000464f0: 2074 6865 2045 756c 6572 2063 6861 7261   the Euler chara
│ │ │ │ -00046500: 6374 6572 6973 7469 6320 6f66 0a24 5c6d  cteristic of.$\m
│ │ │ │ -00046510: 6174 6862 627b 477d 2831 2c34 295c 7375  athbb{G}(1,4)\su
│ │ │ │ -00046520: 6273 6574 5c6d 6174 6862 627b 507d 5e7b  bset\mathbb{P}^{
│ │ │ │ -00046530: 397d 242c 2075 7369 6e67 2062 6f74 6820  9}$, using both 
│ │ │ │ -00046540: 6120 7072 6f62 6162 696c 6973 7469 6320  a probabilistic 
│ │ │ │ -00046550: 616e 6420 610a 6e6f 6e2d 7072 6f62 6162  and a.non-probab
│ │ │ │ -00046560: 696c 6973 7469 6320 6170 7072 6f61 6368  ilistic approach
│ │ │ │ -00046570: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +00045f50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00045f60: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ +00045f70: 3d3d 3d0a 0a20 202a 2055 7361 6765 3a20  ===..  * Usage: 
│ │ │ │ +00045f80: 0a20 2020 2020 2020 2045 756c 6572 4368  .        EulerCh
│ │ │ │ +00045f90: 6172 6163 7465 7269 7374 6963 2049 0a20  aracteristic I. 
│ │ │ │ +00045fa0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +00045fb0: 202a 2049 2c20 616e 202a 6e6f 7465 2069   * I, an *note i
│ │ │ │ +00045fc0: 6465 616c 3a20 284d 6163 6175 6c61 7932  deal: (Macaulay2
│ │ │ │ +00045fd0: 446f 6329 4964 6561 6c2c 2c20 6120 686f  Doc)Ideal,, a ho
│ │ │ │ +00045fe0: 6d6f 6765 6e65 6f75 7320 6964 6561 6c20  mogeneous ideal 
│ │ │ │ +00045ff0: 6465 6669 6e69 6e67 2061 0a20 2020 2020  defining a.     
│ │ │ │ +00046000: 2020 2073 6d6f 6f74 6820 7072 6f6a 6563     smooth projec
│ │ │ │ +00046010: 7469 7665 2076 6172 6965 7479 2024 585c  tive variety $X\
│ │ │ │ +00046020: 7375 6273 6574 5c6d 6174 6862 627b 507d  subset\mathbb{P}
│ │ │ │ +00046030: 5e6e 240a 2020 2a20 2a6e 6f74 6520 4f70  ^n$.  * *note Op
│ │ │ │ +00046040: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ +00046050: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ +00046060: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ +00046070: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ +00046080: 732c 3a0a 2020 2020 2020 2a20 2a6e 6f74  s,:.      * *not
│ │ │ │ +00046090: 6520 426c 6f77 5570 5374 7261 7465 6779  e BlowUpStrategy
│ │ │ │ +000460a0: 3a20 426c 6f77 5570 5374 7261 7465 6779  : BlowUpStrategy
│ │ │ │ +000460b0: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +000460c0: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ +000460d0: 2245 6c69 6d69 6e61 7465 222c 0a20 2020  "Eliminate",.   
│ │ │ │ +000460e0: 2020 202a 202a 6e6f 7465 2043 6572 7469     * *note Certi
│ │ │ │ +000460f0: 6679 3a20 4365 7274 6966 792c 203d 3e20  fy: Certify, => 
│ │ │ │ +00046100: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00046110: 7565 2066 616c 7365 2c20 7768 6574 6865  ue false, whethe
│ │ │ │ +00046120: 7220 746f 2065 6e73 7572 650a 2020 2020  r to ensure.    
│ │ │ │ +00046130: 2020 2020 636f 7272 6563 746e 6573 7320      correctness 
│ │ │ │ +00046140: 6f66 206f 7574 7075 740a 2020 2020 2020  of output.      
│ │ │ │ +00046150: 2a20 2a6e 6f74 6520 5665 7262 6f73 653a  * *note Verbose:
│ │ │ │ +00046160: 2069 6e76 6572 7365 4d61 705f 6c70 5f70   inverseMap_lp_p
│ │ │ │ +00046170: 645f 7064 5f70 645f 636d 5665 7262 6f73  d_pd_pd_cmVerbos
│ │ │ │ +00046180: 653d 3e5f 7064 5f70 645f 7064 5f72 702c  e=>_pd_pd_pd_rp,
│ │ │ │ +00046190: 203d 3e20 2e2e 2e2c 0a20 2020 2020 2020   => ...,.       
│ │ │ │ +000461a0: 2064 6566 6175 6c74 2076 616c 7565 2074   default value t
│ │ │ │ +000461b0: 7275 652c 0a20 202a 204f 7574 7075 7473  rue,.  * Outputs
│ │ │ │ +000461c0: 3a0a 2020 2020 2020 2a20 616e 202a 6e6f  :.      * an *no
│ │ │ │ +000461d0: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +000461e0: 6175 6c61 7932 446f 6329 5a5a 2c2c 2074  aulay2Doc)ZZ,, t
│ │ │ │ +000461f0: 6865 2074 6f70 6f6c 6f67 6963 616c 2045  he topological E
│ │ │ │ +00046200: 756c 6572 0a20 2020 2020 2020 2063 6861  uler.        cha
│ │ │ │ +00046210: 7261 6374 6572 6973 7469 6373 206f 6620  racteristics of 
│ │ │ │ +00046220: 2458 242e 0a0a 4465 7363 7269 7074 696f  $X$...Descriptio
│ │ │ │ +00046230: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
│ │ │ │ +00046240: 6869 7320 6973 2061 6e20 6170 706c 6963  his is an applic
│ │ │ │ +00046250: 6174 696f 6e20 6f66 2074 6865 206d 6574  ation of the met
│ │ │ │ +00046260: 686f 6420 2a6e 6f74 6520 5365 6772 6543  hod *note SegreC
│ │ │ │ +00046270: 6c61 7373 3a20 5365 6772 6543 6c61 7373  lass: SegreClass
│ │ │ │ +00046280: 2c2e 2053 6565 2061 6c73 6f0a 7468 6520  ,. See also.the 
│ │ │ │ +00046290: 636f 7272 6573 706f 6e64 696e 6720 6d65  corresponding me
│ │ │ │ +000462a0: 7468 6f64 7320 696e 2074 6865 2070 6163  thods in the pac
│ │ │ │ +000462b0: 6b61 6765 7320 4353 4d2d 4120 2873 6565  kages CSM-A (see
│ │ │ │ +000462c0: 0a68 7474 703a 2f2f 7777 772e 6d61 7468  .http://www.math
│ │ │ │ +000462d0: 2e66 7375 2e65 6475 2f7e 616c 7566 6669  .fsu.edu/~aluffi
│ │ │ │ +000462e0: 2f43 534d 2f43 534d 2e68 746d 6c20 292c  /CSM/CSM.html ),
│ │ │ │ +000462f0: 2062 7920 502e 2041 6c75 6666 692c 2061   by P. Aluffi, a
│ │ │ │ +00046300: 6e64 0a43 6861 7261 6374 6572 6973 7469  nd.Characteristi
│ │ │ │ +00046310: 6343 6c61 7373 6573 2028 7365 650a 6874  cClasses (see.ht
│ │ │ │ +00046320: 7470 3a2f 2f77 7777 2e6d 6174 682e 7569  tp://www.math.ui
│ │ │ │ +00046330: 7563 2e65 6475 2f4d 6163 6175 6c61 7932  uc.edu/Macaulay2
│ │ │ │ +00046340: 2f64 6f63 2f4d 6163 6175 6c61 7932 2d31  /doc/Macaulay2-1
│ │ │ │ +00046350: 2e31 362f 7368 6172 652f 646f 632f 4d61  .16/share/doc/Ma
│ │ │ │ +00046360: 6361 756c 6179 322f 0a43 6861 7261 6374  caulay2/.Charact
│ │ │ │ +00046370: 6572 6973 7469 6343 6c61 7373 6573 2f68  eristicClasses/h
│ │ │ │ +00046380: 746d 6c2f 2029 2c20 6279 204d 2e20 4865  tml/ ), by M. He
│ │ │ │ +00046390: 6c6d 6572 2061 6e64 2043 2e20 4a6f 7374  lmer and C. Jost
│ │ │ │ +000463a0: 2e0a 0a49 6e20 6765 6e65 7261 6c2c 2065  ...In general, e
│ │ │ │ +000463b0: 7665 6e20 6966 2074 6865 2069 6e70 7574  ven if the input
│ │ │ │ +000463c0: 2069 6465 616c 2064 6566 696e 6573 2061   ideal defines a
│ │ │ │ +000463d0: 2073 696e 6775 6c61 7220 7661 7269 6574   singular variet
│ │ │ │ +000463e0: 7920 2458 242c 2074 6865 0a72 6574 7572  y $X$, the.retur
│ │ │ │ +000463f0: 6e65 6420 7661 6c75 6520 6571 7561 6c73  ned value equals
│ │ │ │ +00046400: 2074 6865 2064 6567 7265 6520 6f66 2074   the degree of t
│ │ │ │ +00046410: 6865 2063 6f6d 706f 6e65 6e74 206f 6620  he component of 
│ │ │ │ +00046420: 6469 6d65 6e73 696f 6e20 3020 6f66 2074  dimension 0 of t
│ │ │ │ +00046430: 6865 0a43 6865 726e 2d46 756c 746f 6e20  he.Chern-Fulton 
│ │ │ │ +00046440: 636c 6173 7320 6f66 2024 5824 2e20 5468  class of $X$. Th
│ │ │ │ +00046450: 6520 4575 6c65 7220 6368 6172 6163 7465  e Euler characte
│ │ │ │ +00046460: 7269 7374 6963 206f 6620 6120 7369 6e67  ristic of a sing
│ │ │ │ +00046470: 756c 6172 2076 6172 6965 7479 2063 616e  ular variety can
│ │ │ │ +00046480: 0a62 6520 636f 6d70 7574 6564 2076 6961  .be computed via
│ │ │ │ +00046490: 2074 6865 206d 6574 686f 6420 2a6e 6f74   the method *not
│ │ │ │ +000464a0: 6520 4368 6572 6e53 6368 7761 7274 7a4d  e ChernSchwartzM
│ │ │ │ +000464b0: 6163 5068 6572 736f 6e3a 0a43 6865 726e  acPherson:.Chern
│ │ │ │ +000464c0: 5363 6877 6172 747a 4d61 6350 6865 7273  SchwartzMacPhers
│ │ │ │ +000464d0: 6f6e 2c2e 0a0a 496e 2074 6865 2065 7861  on,...In the exa
│ │ │ │ +000464e0: 6d70 6c65 2062 656c 6f77 2c20 7765 2063  mple below, we c
│ │ │ │ +000464f0: 6f6d 7075 7465 2074 6865 2045 756c 6572  ompute the Euler
│ │ │ │ +00046500: 2063 6861 7261 6374 6572 6973 7469 6320   characteristic 
│ │ │ │ +00046510: 6f66 0a24 5c6d 6174 6862 627b 477d 2831  of.$\mathbb{G}(1
│ │ │ │ +00046520: 2c34 295c 7375 6273 6574 5c6d 6174 6862  ,4)\subset\mathb
│ │ │ │ +00046530: 627b 507d 5e7b 397d 242c 2075 7369 6e67  b{P}^{9}$, using
│ │ │ │ +00046540: 2062 6f74 6820 6120 7072 6f62 6162 696c   both a probabil
│ │ │ │ +00046550: 6973 7469 6320 616e 6420 610a 6e6f 6e2d  istic and a.non-
│ │ │ │ +00046560: 7072 6f62 6162 696c 6973 7469 6320 6170  probabilistic ap
│ │ │ │ +00046570: 7072 6f61 6368 2e0a 0a2b 2d2d 2d2d 2d2d  proach...+------
│ │ │ │  00046580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000465a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000465b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000465c0: 2d2b 0a7c 6931 203a 2049 203d 2047 7261  -+.|i1 : I = Gra
│ │ │ │ -000465d0: 7373 6d61 6e6e 6961 6e28 312c 342c 436f  ssmannian(1,4,Co
│ │ │ │ -000465e0: 6566 6669 6369 656e 7452 696e 673d 3e5a  efficientRing=>Z
│ │ │ │ -000465f0: 5a2f 3139 3031 3831 293b 2020 2020 2020  Z/190181);      
│ │ │ │ +000465c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2049  -------+.|i1 : I
│ │ │ │ +000465d0: 203d 2047 7261 7373 6d61 6e6e 6961 6e28   = Grassmannian(
│ │ │ │ +000465e0: 312c 342c 436f 6566 6669 6369 656e 7452  1,4,CoefficientR
│ │ │ │ +000465f0: 696e 673d 3e5a 5a2f 3139 3031 3831 293b  ing=>ZZ/190181);
│ │ │ │  00046600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046610: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00046610: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00046620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00046670: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │ +00046660: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00046670: 2020 2020 2020 2020 2020 5a5a 2020 2020            ZZ    
│ │ │ │  00046680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000466a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000466b0: 207c 0a7c 6f31 203a 2049 6465 616c 206f   |.|o1 : Ideal o
│ │ │ │ -000466c0: 6620 2d2d 2d2d 2d2d 5b70 2020 202e 2e70  f ------[p   ..p
│ │ │ │ -000466d0: 2020 202c 2070 2020 202c 2070 2020 202c     , p   , p   ,
│ │ │ │ +000466b0: 2020 2020 2020 207c 0a7c 6f31 203a 2049         |.|o1 : I
│ │ │ │ +000466c0: 6465 616c 206f 6620 2d2d 2d2d 2d2d 5b70  deal of ------[p
│ │ │ │ +000466d0: 2020 202e 2e70 2020 202c 2070 2020 202c     ..p   , p   ,
│ │ │ │  000466e0: 2070 2020 202c 2070 2020 202c 2070 2020   p   , p   , p  
│ │ │ │ -000466f0: 202c 2070 2020 202c 2070 2020 202c 2020   , p   , p   ,  
│ │ │ │ -00046700: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00046710: 2020 3139 3031 3831 2020 302c 3120 2020    190181  0,1   
│ │ │ │ -00046720: 302c 3220 2020 312c 3220 2020 302c 3320  0,2   1,2   0,3 
│ │ │ │ -00046730: 2020 312c 3320 2020 322c 3320 2020 302c    1,3   2,3   0,
│ │ │ │ -00046740: 3420 2020 312c 3420 2020 322c 3420 2020  4   1,4   2,4   
│ │ │ │ -00046750: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000466f0: 202c 2070 2020 202c 2070 2020 202c 2070   , p   , p   , p
│ │ │ │ +00046700: 2020 202c 2020 207c 0a7c 2020 2020 2020     ,   |.|      
│ │ │ │ +00046710: 2020 2020 2020 2020 3139 3031 3831 2020          190181  
│ │ │ │ +00046720: 302c 3120 2020 302c 3220 2020 312c 3220  0,1   0,2   1,2 
│ │ │ │ +00046730: 2020 302c 3320 2020 312c 3320 2020 322c    0,3   1,3   2,
│ │ │ │ +00046740: 3320 2020 302c 3420 2020 312c 3420 2020  3   0,4   1,4   
│ │ │ │ +00046750: 322c 3420 2020 207c 0a7c 2d2d 2d2d 2d2d  2,4    |.|------
│ │ │ │  00046760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000467a0: 2d7c 0a7c 7020 2020 5d20 2020 2020 2020  -|.|p   ]       
│ │ │ │ +000467a0: 2d2d 2d2d 2d2d 2d7c 0a7c 7020 2020 5d20  -------|.|p   ] 
│ │ │ │  000467b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000467c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000467d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000467e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000467f0: 207c 0a7c 2033 2c34 2020 2020 2020 2020   |.| 3,4        
│ │ │ │ +000467f0: 2020 2020 2020 207c 0a7c 2033 2c34 2020         |.| 3,4  
│ │ │ │  00046800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046840: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00046840: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00046850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00046890: 2d2b 0a7c 6932 203a 2074 696d 6520 4575  -+.|i2 : time Eu
│ │ │ │ -000468a0: 6c65 7243 6861 7261 6374 6572 6973 7469  lerCharacteristi
│ │ │ │ -000468b0: 6320 4920 2020 2020 2020 2020 2020 2020  c I             
│ │ │ │ +00046890: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2074  -------+.|i2 : t
│ │ │ │ +000468a0: 696d 6520 4575 6c65 7243 6861 7261 6374  ime EulerCharact
│ │ │ │ +000468b0: 6572 6973 7469 6320 4920 2020 2020 2020  eristic I       
│ │ │ │  000468c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000468d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000468e0: 207c 0a7c 202d 2d20 7573 6564 2030 2e33   |.| -- used 0.3
│ │ │ │ -000468f0: 3033 3435 3973 2028 6370 7529 3b20 302e  03459s (cpu); 0.
│ │ │ │ -00046900: 3135 3531 3173 2028 7468 7265 6164 293b  15511s (thread);
│ │ │ │ -00046910: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +000468e0: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ +000468f0: 6564 2030 2e33 3133 3135 3473 2028 6370  ed 0.313154s (cp
│ │ │ │ +00046900: 7529 3b20 302e 3136 3034 3439 7320 2874  u); 0.160449s (t
│ │ │ │ +00046910: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00046920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046930: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00046930: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00046940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046980: 207c 0a7c 6f32 203d 2031 3020 2020 2020   |.|o2 = 10     
│ │ │ │ -00046990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046980: 2020 2020 2020 207c 0a7c 6f32 203d 2031         |.|o2 = 1
│ │ │ │ +00046990: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  000469a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000469b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000469c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000469d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000469d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  000469e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000469f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00046a20: 2d2b 0a7c 6933 203a 2074 696d 6520 4575  -+.|i3 : time Eu
│ │ │ │ -00046a30: 6c65 7243 6861 7261 6374 6572 6973 7469  lerCharacteristi
│ │ │ │ -00046a40: 6328 492c 4365 7274 6966 793d 3e74 7275  c(I,Certify=>tru
│ │ │ │ -00046a50: 6529 2020 2020 2020 2020 2020 2020 2020  e)              
│ │ │ │ +00046a20: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074  -------+.|i3 : t
│ │ │ │ +00046a30: 696d 6520 4575 6c65 7243 6861 7261 6374  ime EulerCharact
│ │ │ │ +00046a40: 6572 6973 7469 6328 492c 4365 7274 6966  eristic(I,Certif
│ │ │ │ +00046a50: 793d 3e74 7275 6529 2020 2020 2020 2020  y=>true)        
│ │ │ │  00046a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046a70: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ -00046a80: 3130 3138 3873 2028 6370 7529 3b20 302e  10188s (cpu); 0.
│ │ │ │ -00046a90: 3031 3135 3932 7320 2874 6872 6561 6429  011592s (thread)
│ │ │ │ -00046aa0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ -00046ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046ac0: 207c 0a7c 4365 7274 6966 793a 206f 7574   |.|Certify: out
│ │ │ │ -00046ad0: 7075 7420 6365 7274 6966 6965 6421 2020  put certified!  
│ │ │ │ -00046ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046a70: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ +00046a80: 6564 2030 2e30 3235 3137 3834 7320 2863  ed 0.0251784s (c
│ │ │ │ +00046a90: 7075 293b 2030 2e30 3133 3034 3338 7320  pu); 0.0130438s 
│ │ │ │ +00046aa0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00046ab0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00046ac0: 2020 2020 2020 207c 0a7c 4365 7274 6966         |.|Certif
│ │ │ │ +00046ad0: 793a 206f 7574 7075 7420 6365 7274 6966  y: output certif
│ │ │ │ +00046ae0: 6965 6421 2020 2020 2020 2020 2020 2020  ied!            
│ │ │ │  00046af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046b10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00046b10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00046b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046b60: 207c 0a7c 6f33 203d 2031 3020 2020 2020   |.|o3 = 10     
│ │ │ │ -00046b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046b60: 2020 2020 2020 207c 0a7c 6f33 203d 2031         |.|o3 = 1
│ │ │ │ +00046b70: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  00046b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046bb0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00046bb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00046bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00046c00: 2d2b 0a0a 4361 7665 6174 0a3d 3d3d 3d3d  -+..Caveat.=====
│ │ │ │ -00046c10: 3d0a 0a4e 6f20 7465 7374 2069 7320 6d61  =..No test is ma
│ │ │ │ -00046c20: 6465 2074 6f20 7365 6520 6966 2074 6865  de to see if the
│ │ │ │ -00046c30: 2070 726f 6a65 6374 6976 6520 7661 7269   projective vari
│ │ │ │ -00046c40: 6574 7920 6973 2073 6d6f 6f74 682e 0a0a  ety is smooth...
│ │ │ │ -00046c50: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -00046c60: 3d0a 0a20 202a 202a 6e6f 7465 2065 756c  =..  * *note eul
│ │ │ │ -00046c70: 6572 2850 726f 6a65 6374 6976 6556 6172  er(ProjectiveVar
│ │ │ │ -00046c80: 6965 7479 293a 2028 5661 7269 6574 6965  iety): (Varietie
│ │ │ │ -00046c90: 7329 6575 6c65 725f 6c70 5072 6f6a 6563  s)euler_lpProjec
│ │ │ │ -00046ca0: 7469 7665 5661 7269 6574 795f 7270 2c20  tiveVariety_rp, 
│ │ │ │ -00046cb0: 2d2d 0a20 2020 2074 6f70 6f6c 6f67 6963  --.    topologic
│ │ │ │ -00046cc0: 616c 2045 756c 6572 2063 6861 7261 6374  al Euler charact
│ │ │ │ -00046cd0: 6572 6973 7469 6320 6f66 2061 2028 736d  eristic of a (sm
│ │ │ │ -00046ce0: 6f6f 7468 2920 7072 6f6a 6563 7469 7665  ooth) projective
│ │ │ │ -00046cf0: 2076 6172 6965 7479 0a20 202a 202a 6e6f   variety.  * *no
│ │ │ │ -00046d00: 7465 2043 6865 726e 5363 6877 6172 747a  te ChernSchwartz
│ │ │ │ -00046d10: 4d61 6350 6865 7273 6f6e 3a20 4368 6572  MacPherson: Cher
│ │ │ │ -00046d20: 6e53 6368 7761 7274 7a4d 6163 5068 6572  nSchwartzMacPher
│ │ │ │ -00046d30: 736f 6e2c 202d 2d0a 2020 2020 4368 6572  son, --.    Cher
│ │ │ │ -00046d40: 6e2d 5363 6877 6172 747a 2d4d 6163 5068  n-Schwartz-MacPh
│ │ │ │ -00046d50: 6572 736f 6e20 636c 6173 7320 6f66 2061  erson class of a
│ │ │ │ -00046d60: 2070 726f 6a65 6374 6976 6520 7363 6865   projective sche
│ │ │ │ -00046d70: 6d65 0a20 202a 202a 6e6f 7465 2053 6567  me.  * *note Seg
│ │ │ │ -00046d80: 7265 436c 6173 733a 2053 6567 7265 436c  reClass: SegreCl
│ │ │ │ -00046d90: 6173 732c 202d 2d20 5365 6772 6520 636c  ass, -- Segre cl
│ │ │ │ -00046da0: 6173 7320 6f66 2061 2063 6c6f 7365 6420  ass of a closed 
│ │ │ │ -00046db0: 7375 6273 6368 656d 6520 6f66 2061 0a20  subscheme of a. 
│ │ │ │ -00046dc0: 2020 2070 726f 6a65 6374 6976 6520 7661     projective va
│ │ │ │ -00046dd0: 7269 6574 790a 0a57 6179 7320 746f 2075  riety..Ways to u
│ │ │ │ -00046de0: 7365 2045 756c 6572 4368 6172 6163 7465  se EulerCharacte
│ │ │ │ -00046df0: 7269 7374 6963 3a0a 3d3d 3d3d 3d3d 3d3d  ristic:.========
│ │ │ │ +00046c00: 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 6174  -------+..Caveat
│ │ │ │ +00046c10: 0a3d 3d3d 3d3d 3d0a 0a4e 6f20 7465 7374  .======..No test
│ │ │ │ +00046c20: 2069 7320 6d61 6465 2074 6f20 7365 6520   is made to see 
│ │ │ │ +00046c30: 6966 2074 6865 2070 726f 6a65 6374 6976  if the projectiv
│ │ │ │ +00046c40: 6520 7661 7269 6574 7920 6973 2073 6d6f  e variety is smo
│ │ │ │ +00046c50: 6f74 682e 0a0a 5365 6520 616c 736f 0a3d  oth...See also.=
│ │ │ │ +00046c60: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ +00046c70: 7465 2065 756c 6572 2850 726f 6a65 6374  te euler(Project
│ │ │ │ +00046c80: 6976 6556 6172 6965 7479 293a 2028 5661  iveVariety): (Va
│ │ │ │ +00046c90: 7269 6574 6965 7329 6575 6c65 725f 6c70  rieties)euler_lp
│ │ │ │ +00046ca0: 5072 6f6a 6563 7469 7665 5661 7269 6574  ProjectiveVariet
│ │ │ │ +00046cb0: 795f 7270 2c20 2d2d 0a20 2020 2074 6f70  y_rp, --.    top
│ │ │ │ +00046cc0: 6f6c 6f67 6963 616c 2045 756c 6572 2063  ological Euler c
│ │ │ │ +00046cd0: 6861 7261 6374 6572 6973 7469 6320 6f66  haracteristic of
│ │ │ │ +00046ce0: 2061 2028 736d 6f6f 7468 2920 7072 6f6a   a (smooth) proj
│ │ │ │ +00046cf0: 6563 7469 7665 2076 6172 6965 7479 0a20  ective variety. 
│ │ │ │ +00046d00: 202a 202a 6e6f 7465 2043 6865 726e 5363   * *note ChernSc
│ │ │ │ +00046d10: 6877 6172 747a 4d61 6350 6865 7273 6f6e  hwartzMacPherson
│ │ │ │ +00046d20: 3a20 4368 6572 6e53 6368 7761 7274 7a4d  : ChernSchwartzM
│ │ │ │ +00046d30: 6163 5068 6572 736f 6e2c 202d 2d0a 2020  acPherson, --.  
│ │ │ │ +00046d40: 2020 4368 6572 6e2d 5363 6877 6172 747a    Chern-Schwartz
│ │ │ │ +00046d50: 2d4d 6163 5068 6572 736f 6e20 636c 6173  -MacPherson clas
│ │ │ │ +00046d60: 7320 6f66 2061 2070 726f 6a65 6374 6976  s of a projectiv
│ │ │ │ +00046d70: 6520 7363 6865 6d65 0a20 202a 202a 6e6f  e scheme.  * *no
│ │ │ │ +00046d80: 7465 2053 6567 7265 436c 6173 733a 2053  te SegreClass: S
│ │ │ │ +00046d90: 6567 7265 436c 6173 732c 202d 2d20 5365  egreClass, -- Se
│ │ │ │ +00046da0: 6772 6520 636c 6173 7320 6f66 2061 2063  gre class of a c
│ │ │ │ +00046db0: 6c6f 7365 6420 7375 6273 6368 656d 6520  losed subscheme 
│ │ │ │ +00046dc0: 6f66 2061 0a20 2020 2070 726f 6a65 6374  of a.    project
│ │ │ │ +00046dd0: 6976 6520 7661 7269 6574 790a 0a57 6179  ive variety..Way
│ │ │ │ +00046de0: 7320 746f 2075 7365 2045 756c 6572 4368  s to use EulerCh
│ │ │ │ +00046df0: 6172 6163 7465 7269 7374 6963 3a0a 3d3d  aracteristic:.==
│ │ │ │  00046e00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00046e10: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245  ========..  * "E
│ │ │ │ -00046e20: 756c 6572 4368 6172 6163 7465 7269 7374  ulerCharacterist
│ │ │ │ -00046e30: 6963 2849 6465 616c 2922 0a0a 466f 7220  ic(Ideal)"..For 
│ │ │ │ -00046e40: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -00046e50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00046e60: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -00046e70: 6f74 6520 4575 6c65 7243 6861 7261 6374  ote EulerCharact
│ │ │ │ -00046e80: 6572 6973 7469 633a 2045 756c 6572 4368  eristic: EulerCh
│ │ │ │ -00046e90: 6172 6163 7465 7269 7374 6963 2c20 6973  aracteristic, is
│ │ │ │ -00046ea0: 2061 202a 6e6f 7465 206d 6574 686f 640a   a *note method.
│ │ │ │ -00046eb0: 6675 6e63 7469 6f6e 2077 6974 6820 6f70  function with op
│ │ │ │ -00046ec0: 7469 6f6e 733a 2028 4d61 6361 756c 6179  tions: (Macaulay
│ │ │ │ -00046ed0: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ -00046ee0: 696f 6e57 6974 684f 7074 696f 6e73 2c2e  ionWithOptions,.
│ │ │ │ -00046ef0: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ -00046f00: 2e69 6e66 6f2c 204e 6f64 653a 2065 7863  .info, Node: exc
│ │ │ │ -00046f10: 6570 7469 6f6e 616c 4c6f 6375 732c 204e  eptionalLocus, N
│ │ │ │ -00046f20: 6578 743a 2066 6c61 7474 656e 5f6c 7052  ext: flatten_lpR
│ │ │ │ -00046f30: 6174 696f 6e61 6c4d 6170 5f72 702c 2050  ationalMap_rp, P
│ │ │ │ -00046f40: 7265 763a 2045 756c 6572 4368 6172 6163  rev: EulerCharac
│ │ │ │ -00046f50: 7465 7269 7374 6963 2c20 5570 3a20 546f  teristic, Up: To
│ │ │ │ -00046f60: 700a 0a65 7863 6570 7469 6f6e 616c 4c6f  p..exceptionalLo
│ │ │ │ -00046f70: 6375 7320 2d2d 2065 7863 6570 7469 6f6e  cus -- exception
│ │ │ │ -00046f80: 616c 206c 6f63 7573 206f 6620 6120 6269  al locus of a bi
│ │ │ │ -00046f90: 7261 7469 6f6e 616c 206d 6170 0a2a 2a2a  rational map.***
│ │ │ │ -00046fa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00046e10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00046e20: 2020 2a20 2245 756c 6572 4368 6172 6163    * "EulerCharac
│ │ │ │ +00046e30: 7465 7269 7374 6963 2849 6465 616c 2922  teristic(Ideal)"
│ │ │ │ +00046e40: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +00046e50: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +00046e60: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +00046e70: 6563 7420 2a6e 6f74 6520 4575 6c65 7243  ect *note EulerC
│ │ │ │ +00046e80: 6861 7261 6374 6572 6973 7469 633a 2045  haracteristic: E
│ │ │ │ +00046e90: 756c 6572 4368 6172 6163 7465 7269 7374  ulerCharacterist
│ │ │ │ +00046ea0: 6963 2c20 6973 2061 202a 6e6f 7465 206d  ic, is a *note m
│ │ │ │ +00046eb0: 6574 686f 640a 6675 6e63 7469 6f6e 2077  ethod.function w
│ │ │ │ +00046ec0: 6974 6820 6f70 7469 6f6e 733a 2028 4d61  ith options: (Ma
│ │ │ │ +00046ed0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +00046ee0: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
│ │ │ │ +00046ef0: 696f 6e73 2c2e 0a1f 0a46 696c 653a 2043  ions,....File: C
│ │ │ │ +00046f00: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ +00046f10: 653a 2065 7863 6570 7469 6f6e 616c 4c6f  e: exceptionalLo
│ │ │ │ +00046f20: 6375 732c 204e 6578 743a 2066 6c61 7474  cus, Next: flatt
│ │ │ │ +00046f30: 656e 5f6c 7052 6174 696f 6e61 6c4d 6170  en_lpRationalMap
│ │ │ │ +00046f40: 5f72 702c 2050 7265 763a 2045 756c 6572  _rp, Prev: Euler
│ │ │ │ +00046f50: 4368 6172 6163 7465 7269 7374 6963 2c20  Characteristic, 
│ │ │ │ +00046f60: 5570 3a20 546f 700a 0a65 7863 6570 7469  Up: Top..excepti
│ │ │ │ +00046f70: 6f6e 616c 4c6f 6375 7320 2d2d 2065 7863  onalLocus -- exc
│ │ │ │ +00046f80: 6570 7469 6f6e 616c 206c 6f63 7573 206f  eptional locus o
│ │ │ │ +00046f90: 6620 6120 6269 7261 7469 6f6e 616c 206d  f a birational m
│ │ │ │ +00046fa0: 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ap.*************
│ │ │ │  00046fb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00046fc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00046fd0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ -00046fe0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055  .========..  * U
│ │ │ │ -00046ff0: 7361 6765 3a20 0a20 2020 2020 2020 2065  sage: .        e
│ │ │ │ -00047000: 7863 6570 7469 6f6e 616c 4c6f 6375 7320  xceptionalLocus 
│ │ │ │ -00047010: 7068 690a 2020 2a20 496e 7075 7473 3a0a  phi.  * Inputs:.
│ │ │ │ -00047020: 2020 2020 2020 2a20 7068 692c 2061 202a        * phi, a *
│ │ │ │ -00047030: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ -00047040: 703a 2052 6174 696f 6e61 6c4d 6170 2c2c  p: RationalMap,,
│ │ │ │ -00047050: 2061 2062 6972 6174 696f 6e61 6c20 6d61   a birational ma
│ │ │ │ -00047060: 700a 2020 2020 2020 2020 2458 5c64 6173  p.        $X\das
│ │ │ │ -00047070: 6872 6967 6874 6172 726f 7720 5924 0a20  hrightarrow Y$. 
│ │ │ │ -00047080: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ -00047090: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ -000470a0: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ -000470b0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -000470c0: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ -000470d0: 2020 2020 202a 202a 6e6f 7465 2043 6572       * *note Cer
│ │ │ │ -000470e0: 7469 6679 3a20 4365 7274 6966 792c 203d  tify: Certify, =
│ │ │ │ -000470f0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -00047100: 616c 7565 2066 616c 7365 2c20 7768 6574  alue false, whet
│ │ │ │ -00047110: 6865 7220 746f 2065 6e73 7572 650a 2020  her to ensure.  
│ │ │ │ -00047120: 2020 2020 2020 636f 7272 6563 746e 6573        correctnes
│ │ │ │ -00047130: 7320 6f66 206f 7574 7075 740a 2020 2a20  s of output.  * 
│ │ │ │ -00047140: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00047150: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a   an *note ideal:
│ │ │ │ -00047160: 2028 4d61 6361 756c 6179 3244 6f63 2949   (Macaulay2Doc)I
│ │ │ │ -00047170: 6465 616c 2c2c 2074 6865 2069 6465 616c  deal,, the ideal
│ │ │ │ -00047180: 2064 6566 696e 696e 6720 7468 6520 636c   defining the cl
│ │ │ │ -00047190: 6f73 7572 6520 696e 0a20 2020 2020 2020  osure in.       
│ │ │ │ -000471a0: 2058 206f 6620 7468 6520 6c6f 6375 7320   X of the locus 
│ │ │ │ -000471b0: 7768 6572 6520 7068 6920 6973 206e 6f74  where phi is not
│ │ │ │ -000471c0: 2061 206c 6f63 616c 2069 736f 6d6f 7270   a local isomorp
│ │ │ │ -000471d0: 6869 736d 0a0a 4465 7363 7269 7074 696f  hism..Descriptio
│ │ │ │ -000471e0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
│ │ │ │ -000471f0: 6869 7320 6d65 7468 6f64 2073 696d 706c  his method simpl
│ │ │ │ -00047200: 7920 6361 6c63 756c 6174 6573 2074 6865  y calculates the
│ │ │ │ -00047210: 202a 6e6f 7465 2069 6e76 6572 7365 2069   *note inverse i
│ │ │ │ -00047220: 6d61 6765 3a20 5261 7469 6f6e 616c 4d61  mage: RationalMa
│ │ │ │ -00047230: 7020 5e5f 7374 5f73 740a 4964 6561 6c2c  p ^_st_st.Ideal,
│ │ │ │ -00047240: 206f 6620 7468 6520 2a6e 6f74 6520 6261   of the *note ba
│ │ │ │ -00047250: 7365 206c 6f63 7573 3a20 6964 6561 6c5f  se locus: ideal_
│ │ │ │ -00047260: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00047270: 2c20 6f66 2074 6865 2069 6e76 6572 7365  , of the inverse
│ │ │ │ -00047280: 206d 6170 2c0a 7768 6963 6820 696e 2074   map,.which in t
│ │ │ │ -00047290: 7572 6e20 6973 2064 6574 6572 6d69 6e65  urn is determine
│ │ │ │ -000472a0: 6420 7468 726f 7567 6820 7468 6520 6d65  d through the me
│ │ │ │ -000472b0: 7468 6f64 202a 6e6f 7465 2069 6e76 6572  thod *note inver
│ │ │ │ -000472c0: 7365 3a0a 696e 7665 7273 655f 6c70 5261  se:.inverse_lpRa
│ │ │ │ -000472d0: 7469 6f6e 616c 4d61 705f 7270 2c2e 0a0a  tionalMap_rp,...
│ │ │ │ -000472e0: 4265 6c6f 772c 2077 6520 636f 6d70 7574  Below, we comput
│ │ │ │ -000472f0: 6520 7468 6520 6578 6365 7074 696f 6e61  e the exceptiona
│ │ │ │ -00047300: 6c20 6c6f 6375 7320 6f66 2074 6865 206d  l locus of the m
│ │ │ │ -00047310: 6170 2064 6566 696e 6564 2062 7920 7468  ap defined by th
│ │ │ │ -00047320: 6520 6c69 6e65 6172 2073 7973 7465 6d0a  e linear system.
│ │ │ │ -00047330: 6f66 2071 7561 6472 6963 7320 7468 726f  of quadrics thro
│ │ │ │ -00047340: 7567 6820 7468 6520 7175 696e 7469 6320  ugh the quintic 
│ │ │ │ -00047350: 7261 7469 6f6e 616c 206e 6f72 6d61 6c20  rational normal 
│ │ │ │ -00047360: 6375 7276 6520 696e 2024 5c6d 6174 6862  curve in $\mathb
│ │ │ │ -00047370: 627b 507d 5e35 242e 0a0a 2b2d 2d2d 2d2d  b{P}^5$...+-----
│ │ │ │ -00047380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00046fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ +00046fe0: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ +00046ff0: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00047000: 2020 2020 2065 7863 6570 7469 6f6e 616c       exceptional
│ │ │ │ +00047010: 4c6f 6375 7320 7068 690a 2020 2a20 496e  Locus phi.  * In
│ │ │ │ +00047020: 7075 7473 3a0a 2020 2020 2020 2a20 7068  puts:.      * ph
│ │ │ │ +00047030: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ +00047040: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ +00047050: 6c4d 6170 2c2c 2061 2062 6972 6174 696f  lMap,, a biratio
│ │ │ │ +00047060: 6e61 6c20 6d61 700a 2020 2020 2020 2020  nal map.        
│ │ │ │ +00047070: 2458 5c64 6173 6872 6967 6874 6172 726f  $X\dashrightarro
│ │ │ │ +00047080: 7720 5924 0a20 202a 202a 6e6f 7465 204f  w Y$.  * *note O
│ │ │ │ +00047090: 7074 696f 6e61 6c20 696e 7075 7473 3a20  ptional inputs: 
│ │ │ │ +000470a0: 284d 6163 6175 6c61 7932 446f 6329 7573  (Macaulay2Doc)us
│ │ │ │ +000470b0: 696e 6720 6675 6e63 7469 6f6e 7320 7769  ing functions wi
│ │ │ │ +000470c0: 7468 206f 7074 696f 6e61 6c20 696e 7075  th optional inpu
│ │ │ │ +000470d0: 7473 2c3a 0a20 2020 2020 202a 202a 6e6f  ts,:.      * *no
│ │ │ │ +000470e0: 7465 2043 6572 7469 6679 3a20 4365 7274  te Certify: Cert
│ │ │ │ +000470f0: 6966 792c 203d 3e20 2e2e 2e2c 2064 6566  ify, => ..., def
│ │ │ │ +00047100: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ +00047110: 2c20 7768 6574 6865 7220 746f 2065 6e73  , whether to ens
│ │ │ │ +00047120: 7572 650a 2020 2020 2020 2020 636f 7272  ure.        corr
│ │ │ │ +00047130: 6563 746e 6573 7320 6f66 206f 7574 7075  ectness of outpu
│ │ │ │ +00047140: 740a 2020 2a20 4f75 7470 7574 733a 0a20  t.  * Outputs:. 
│ │ │ │ +00047150: 2020 2020 202a 2061 6e20 2a6e 6f74 6520       * an *note 
│ │ │ │ +00047160: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ +00047170: 3244 6f63 2949 6465 616c 2c2c 2074 6865  2Doc)Ideal,, the
│ │ │ │ +00047180: 2069 6465 616c 2064 6566 696e 696e 6720   ideal defining 
│ │ │ │ +00047190: 7468 6520 636c 6f73 7572 6520 696e 0a20  the closure in. 
│ │ │ │ +000471a0: 2020 2020 2020 2058 206f 6620 7468 6520         X of the 
│ │ │ │ +000471b0: 6c6f 6375 7320 7768 6572 6520 7068 6920  locus where phi 
│ │ │ │ +000471c0: 6973 206e 6f74 2061 206c 6f63 616c 2069  is not a local i
│ │ │ │ +000471d0: 736f 6d6f 7270 6869 736d 0a0a 4465 7363  somorphism..Desc
│ │ │ │ +000471e0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +000471f0: 3d3d 3d0a 0a54 6869 7320 6d65 7468 6f64  ===..This method
│ │ │ │ +00047200: 2073 696d 706c 7920 6361 6c63 756c 6174   simply calculat
│ │ │ │ +00047210: 6573 2074 6865 202a 6e6f 7465 2069 6e76  es the *note inv
│ │ │ │ +00047220: 6572 7365 2069 6d61 6765 3a20 5261 7469  erse image: Rati
│ │ │ │ +00047230: 6f6e 616c 4d61 7020 5e5f 7374 5f73 740a  onalMap ^_st_st.
│ │ │ │ +00047240: 4964 6561 6c2c 206f 6620 7468 6520 2a6e  Ideal, of the *n
│ │ │ │ +00047250: 6f74 6520 6261 7365 206c 6f63 7573 3a20  ote base locus: 
│ │ │ │ +00047260: 6964 6561 6c5f 6c70 5261 7469 6f6e 616c  ideal_lpRational
│ │ │ │ +00047270: 4d61 705f 7270 2c20 6f66 2074 6865 2069  Map_rp, of the i
│ │ │ │ +00047280: 6e76 6572 7365 206d 6170 2c0a 7768 6963  nverse map,.whic
│ │ │ │ +00047290: 6820 696e 2074 7572 6e20 6973 2064 6574  h in turn is det
│ │ │ │ +000472a0: 6572 6d69 6e65 6420 7468 726f 7567 6820  ermined through 
│ │ │ │ +000472b0: 7468 6520 6d65 7468 6f64 202a 6e6f 7465  the method *note
│ │ │ │ +000472c0: 2069 6e76 6572 7365 3a0a 696e 7665 7273   inverse:.invers
│ │ │ │ +000472d0: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +000472e0: 7270 2c2e 0a0a 4265 6c6f 772c 2077 6520  rp,...Below, we 
│ │ │ │ +000472f0: 636f 6d70 7574 6520 7468 6520 6578 6365  compute the exce
│ │ │ │ +00047300: 7074 696f 6e61 6c20 6c6f 6375 7320 6f66  ptional locus of
│ │ │ │ +00047310: 2074 6865 206d 6170 2064 6566 696e 6564   the map defined
│ │ │ │ +00047320: 2062 7920 7468 6520 6c69 6e65 6172 2073   by the linear s
│ │ │ │ +00047330: 7973 7465 6d0a 6f66 2071 7561 6472 6963  ystem.of quadric
│ │ │ │ +00047340: 7320 7468 726f 7567 6820 7468 6520 7175  s through the qu
│ │ │ │ +00047350: 696e 7469 6320 7261 7469 6f6e 616c 206e  intic rational n
│ │ │ │ +00047360: 6f72 6d61 6c20 6375 7276 6520 696e 2024  ormal curve in $
│ │ │ │ +00047370: 5c6d 6174 6862 627b 507d 5e35 242e 0a0a  \mathbb{P}^5$...
│ │ │ │ +00047380: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00047390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000473a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000473b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000473c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -000473d0: 5035 203a 3d20 5a5a 2f31 3030 3030 335b  P5 := ZZ/100003[
│ │ │ │ -000473e0: 785f 302e 2e78 5f35 5d3b 2020 2020 2020  x_0..x_5];      
│ │ │ │ +000473c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000473d0: 7c69 3120 3a20 5035 203a 3d20 5a5a 2f31  |i1 : P5 := ZZ/1
│ │ │ │ +000473e0: 3030 3030 335b 785f 302e 2e78 5f35 5d3b  00003[x_0..x_5];
│ │ │ │  000473f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047410: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00047420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00047410: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047420: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00047430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047460: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -00047470: 7068 6920 3d20 7261 7469 6f6e 616c 4d61  phi = rationalMa
│ │ │ │ -00047480: 7028 6d69 6e6f 7273 2832 2c6d 6174 7269  p(minors(2,matri
│ │ │ │ -00047490: 787b 7b78 5f30 2c78 5f31 2c78 5f32 2c78  x{{x_0,x_1,x_2,x
│ │ │ │ -000474a0: 5f33 2c78 5f34 7d2c 7b78 5f31 2c78 5f20  _3,x_4},{x_1,x_ 
│ │ │ │ -000474b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000474c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00047470: 7c69 3220 3a20 7068 6920 3d20 7261 7469  |i2 : phi = rati
│ │ │ │ +00047480: 6f6e 616c 4d61 7028 6d69 6e6f 7273 2832  onalMap(minors(2
│ │ │ │ +00047490: 2c6d 6174 7269 787b 7b78 5f30 2c78 5f31  ,matrix{{x_0,x_1
│ │ │ │ +000474a0: 2c78 5f32 2c78 5f33 2c78 5f34 7d2c 7b78  ,x_2,x_3,x_4},{x
│ │ │ │ +000474b0: 5f31 2c78 5f20 2020 2020 2020 2020 7c0a  _1,x_         |.
│ │ │ │ +000474c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000474d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000474e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000474f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047500: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -00047510: 5261 7469 6f6e 616c 4d61 7020 2871 7561  RationalMap (qua
│ │ │ │ -00047520: 6472 6174 6963 2072 6174 696f 6e61 6c20  dratic rational 
│ │ │ │ -00047530: 6d61 7020 6672 6f6d 2050 505e 3520 746f  map from PP^5 to
│ │ │ │ -00047540: 2035 2d64 696d 656e 7369 6f6e 616c 2020   5-dimensional  
│ │ │ │ -00047550: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ -00047560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00047500: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047510: 7c6f 3220 3a20 5261 7469 6f6e 616c 4d61  |o2 : RationalMa
│ │ │ │ +00047520: 7020 2871 7561 6472 6174 6963 2072 6174  p (quadratic rat
│ │ │ │ +00047530: 696f 6e61 6c20 6d61 7020 6672 6f6d 2050  ional map from P
│ │ │ │ +00047540: 505e 3520 746f 2035 2d64 696d 656e 7369  P^5 to 5-dimensi
│ │ │ │ +00047550: 6f6e 616c 2020 2020 2020 2020 2020 7c0a  onal          |.
│ │ │ │ +00047560: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00047570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000475a0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c32 2c78 5f33  --------|.|2,x_3
│ │ │ │ -000475b0: 2c78 5f34 2c78 5f35 7d7d 292c 446f 6d69  ,x_4,x_5}}),Domi
│ │ │ │ -000475c0: 6e61 6e74 3d3e 3229 3b20 2020 2020 2020  nant=>2);       
│ │ │ │ +000475a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000475b0: 7c32 2c78 5f33 2c78 5f34 2c78 5f35 7d7d  |2,x_3,x_4,x_5}}
│ │ │ │ +000475c0: 292c 446f 6d69 6e61 6e74 3d3e 3229 3b20  ),Dominant=>2); 
│ │ │ │  000475d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000475e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000475f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00047600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000475f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047600: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00047610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047640: 2020 2020 2020 2020 7c0a 7c73 7562 7661          |.|subva
│ │ │ │ -00047650: 7269 6574 7920 6f66 2050 505e 3929 2020  riety of PP^9)  
│ │ │ │ -00047660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047640: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047650: 7c73 7562 7661 7269 6574 7920 6f66 2050  |subvariety of P
│ │ │ │ +00047660: 505e 3929 2020 2020 2020 2020 2020 2020  P^9)            
│ │ │ │  00047670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047690: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -000476a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00047690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000476a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000476b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000476c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000476d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000476e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -000476f0: 4520 3d20 6578 6365 7074 696f 6e61 6c4c  E = exceptionalL
│ │ │ │ -00047700: 6f63 7573 2070 6869 3b20 2020 2020 2020  ocus phi;       
│ │ │ │ +000476e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000476f0: 7c69 3320 3a20 4520 3d20 6578 6365 7074  |i3 : E = except
│ │ │ │ +00047700: 696f 6e61 6c4c 6f63 7573 2070 6869 3b20  ionalLocus phi; 
│ │ │ │  00047710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00047740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047740: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00047750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047780: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00047790: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │ -000477a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047780: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047790: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000477a0: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │  000477b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000477c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000477d0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -000477e0: 4964 6561 6c20 6f66 202d 2d2d 2d2d 2d5b  Ideal of ------[
│ │ │ │ -000477f0: 7820 2e2e 7820 5d20 2020 2020 2020 2020  x ..x ]         
│ │ │ │ +000477d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000477e0: 7c6f 3320 3a20 4964 6561 6c20 6f66 202d  |o3 : Ideal of -
│ │ │ │ +000477f0: 2d2d 2d2d 2d5b 7820 2e2e 7820 5d20 2020  -----[x ..x ]   
│ │ │ │  00047800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047820: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00047830: 2020 2020 2020 2020 2031 3030 3030 3320           100003 
│ │ │ │ -00047840: 2030 2020 2035 2020 2020 2020 2020 2020   0   5          
│ │ │ │ +00047820: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047830: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ +00047840: 3030 3030 3320 2030 2020 2035 2020 2020  00003  0   5    
│ │ │ │  00047850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047870: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00047880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00047870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047880: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00047890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000478a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000478b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000478c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -000478d0: 6173 7365 7274 2845 203d 3d20 7068 695e  assert(E == phi^
│ │ │ │ -000478e0: 2a20 6964 6561 6c20 7068 695e 2d31 2920  * ideal phi^-1) 
│ │ │ │ -000478f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000478c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000478d0: 7c69 3420 3a20 6173 7365 7274 2845 203d  |i4 : assert(E =
│ │ │ │ +000478e0: 3d20 7068 695e 2a20 6964 6561 6c20 7068  = phi^* ideal ph
│ │ │ │ +000478f0: 695e 2d31 2920 2020 2020 2020 2020 2020  i^-1)           
│ │ │ │  00047900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047910: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00047920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00047910: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047920: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00047930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047960: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -00047970: 6173 7365 7274 2845 203d 3d20 6d69 6e6f  assert(E == mino
│ │ │ │ -00047980: 7273 2833 2c6d 6174 7269 787b 7b78 5f30  rs(3,matrix{{x_0
│ │ │ │ -00047990: 2c78 5f31 2c78 5f32 2c78 5f33 7d2c 7b78  ,x_1,x_2,x_3},{x
│ │ │ │ -000479a0: 5f31 2c78 5f32 2c78 5f33 2c78 5f34 7d2c  _1,x_2,x_3,x_4},
│ │ │ │ -000479b0: 7b78 5f32 2c78 5f33 7c0a 7c2d 2d2d 2d2d  {x_2,x_3|.|-----
│ │ │ │ -000479c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00047960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00047970: 7c69 3520 3a20 6173 7365 7274 2845 203d  |i5 : assert(E =
│ │ │ │ +00047980: 3d20 6d69 6e6f 7273 2833 2c6d 6174 7269  = minors(3,matri
│ │ │ │ +00047990: 787b 7b78 5f30 2c78 5f31 2c78 5f32 2c78  x{{x_0,x_1,x_2,x
│ │ │ │ +000479a0: 5f33 7d2c 7b78 5f31 2c78 5f32 2c78 5f33  _3},{x_1,x_2,x_3
│ │ │ │ +000479b0: 2c78 5f34 7d2c 7b78 5f32 2c78 5f33 7c0a  ,x_4},{x_2,x_3|.
│ │ │ │ +000479c0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  000479d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000479e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000479f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00047a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00047a10: 7c2c 785f 342c 785f 357d 7d29 2920 2020  |,x_4,x_5}}))   
│ │ │ │  00047a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047a50: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00047a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00047a50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00047a60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00047a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047aa0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -00047ab0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -00047ac0: 2a20 2a6e 6f74 6520 6964 6561 6c28 5261  * *note ideal(Ra
│ │ │ │ -00047ad0: 7469 6f6e 616c 4d61 7029 3a20 6964 6561  tionalMap): idea
│ │ │ │ -00047ae0: 6c5f 6c70 5261 7469 6f6e 616c 4d61 705f  l_lpRationalMap_
│ │ │ │ -00047af0: 7270 2c20 2d2d 2062 6173 6520 6c6f 6375  rp, -- base locu
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│ │ │ │ -00047b10: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74  nal map.  * *not
│ │ │ │ -00047b20: 6520 696e 7665 7273 654d 6170 2852 6174  e inverseMap(Rat
│ │ │ │ -00047b30: 696f 6e61 6c4d 6170 293a 2069 6e76 6572  ionalMap): inver
│ │ │ │ -00047b40: 7365 4d61 702c 202d 2d20 696e 7665 7273  seMap, -- invers
│ │ │ │ -00047b50: 6520 6f66 2061 2062 6972 6174 696f 6e61  e of a birationa
│ │ │ │ -00047b60: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ -00047b70: 5261 7469 6f6e 616c 4d61 7020 5e2a 2a20  RationalMap ^** 
│ │ │ │ -00047b80: 4964 6561 6c3a 2052 6174 696f 6e61 6c4d  Ideal: RationalM
│ │ │ │ -00047b90: 6170 205e 5f73 745f 7374 2049 6465 616c  ap ^_st_st Ideal
│ │ │ │ -00047ba0: 2c20 2d2d 2069 6e76 6572 7365 2069 6d61  , -- inverse ima
│ │ │ │ -00047bb0: 6765 0a20 2020 2076 6961 2061 2072 6174  ge.    via a rat
│ │ │ │ -00047bc0: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ -00047bd0: 6f74 6520 6973 4973 6f6d 6f72 7068 6973  ote isIsomorphis
│ │ │ │ -00047be0: 6d28 5261 7469 6f6e 616c 4d61 7029 3a20  m(RationalMap): 
│ │ │ │ -00047bf0: 6973 4973 6f6d 6f72 7068 6973 6d5f 6c70  isIsomorphism_lp
│ │ │ │ -00047c00: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -00047c10: 2d2d 0a20 2020 2077 6865 7468 6572 2061  --.    whether a
│ │ │ │ -00047c20: 2062 6972 6174 696f 6e61 6c20 6d61 7020   birational map 
│ │ │ │ -00047c30: 6973 2061 6e20 6973 6f6d 6f72 7068 6973  is an isomorphis
│ │ │ │ -00047c40: 6d0a 2020 2a20 2a6e 6f74 6520 666f 7263  m.  * *note forc
│ │ │ │ -00047c50: 6549 6e76 6572 7365 4d61 703a 2066 6f72  eInverseMap: for
│ │ │ │ -00047c60: 6365 496e 7665 7273 654d 6170 2c20 2d2d  ceInverseMap, --
│ │ │ │ -00047c70: 2064 6563 6c61 7265 2074 6861 7420 7477   declare that tw
│ │ │ │ -00047c80: 6f20 7261 7469 6f6e 616c 206d 6170 730a  o rational maps.
│ │ │ │ -00047c90: 2020 2020 6172 6520 6f6e 6520 7468 6520      are one the 
│ │ │ │ -00047ca0: 696e 7665 7273 6520 6f66 2074 6865 206f  inverse of the o
│ │ │ │ -00047cb0: 7468 6572 0a0a 5761 7973 2074 6f20 7573  ther..Ways to us
│ │ │ │ -00047cc0: 6520 6578 6365 7074 696f 6e61 6c4c 6f63  e exceptionalLoc
│ │ │ │ -00047cd0: 7573 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  us:.============
│ │ │ │ +00047aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00047ab0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00047ac0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6964  ==..  * *note id
│ │ │ │ +00047ad0: 6561 6c28 5261 7469 6f6e 616c 4d61 7029  eal(RationalMap)
│ │ │ │ +00047ae0: 3a20 6964 6561 6c5f 6c70 5261 7469 6f6e  : ideal_lpRation
│ │ │ │ +00047af0: 616c 4d61 705f 7270 2c20 2d2d 2062 6173  alMap_rp, -- bas
│ │ │ │ +00047b00: 6520 6c6f 6375 7320 6f66 2061 0a20 2020  e locus of a.   
│ │ │ │ +00047b10: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ +00047b20: 2a20 2a6e 6f74 6520 696e 7665 7273 654d  * *note inverseM
│ │ │ │ +00047b30: 6170 2852 6174 696f 6e61 6c4d 6170 293a  ap(RationalMap):
│ │ │ │ +00047b40: 2069 6e76 6572 7365 4d61 702c 202d 2d20   inverseMap, -- 
│ │ │ │ +00047b50: 696e 7665 7273 6520 6f66 2061 2062 6972  inverse of a bir
│ │ │ │ +00047b60: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ +00047b70: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ +00047b80: 7020 5e2a 2a20 4964 6561 6c3a 2052 6174  p ^** Ideal: Rat
│ │ │ │ +00047b90: 696f 6e61 6c4d 6170 205e 5f73 745f 7374  ionalMap ^_st_st
│ │ │ │ +00047ba0: 2049 6465 616c 2c20 2d2d 2069 6e76 6572   Ideal, -- inver
│ │ │ │ +00047bb0: 7365 2069 6d61 6765 0a20 2020 2076 6961  se image.    via
│ │ │ │ +00047bc0: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +00047bd0: 2020 2a20 2a6e 6f74 6520 6973 4973 6f6d    * *note isIsom
│ │ │ │ +00047be0: 6f72 7068 6973 6d28 5261 7469 6f6e 616c  orphism(Rational
│ │ │ │ +00047bf0: 4d61 7029 3a20 6973 4973 6f6d 6f72 7068  Map): isIsomorph
│ │ │ │ +00047c00: 6973 6d5f 6c70 5261 7469 6f6e 616c 4d61  ism_lpRationalMa
│ │ │ │ +00047c10: 705f 7270 2c20 2d2d 0a20 2020 2077 6865  p_rp, --.    whe
│ │ │ │ +00047c20: 7468 6572 2061 2062 6972 6174 696f 6e61  ther a birationa
│ │ │ │ +00047c30: 6c20 6d61 7020 6973 2061 6e20 6973 6f6d  l map is an isom
│ │ │ │ +00047c40: 6f72 7068 6973 6d0a 2020 2a20 2a6e 6f74  orphism.  * *not
│ │ │ │ +00047c50: 6520 666f 7263 6549 6e76 6572 7365 4d61  e forceInverseMa
│ │ │ │ +00047c60: 703a 2066 6f72 6365 496e 7665 7273 654d  p: forceInverseM
│ │ │ │ +00047c70: 6170 2c20 2d2d 2064 6563 6c61 7265 2074  ap, -- declare t
│ │ │ │ +00047c80: 6861 7420 7477 6f20 7261 7469 6f6e 616c  hat two rational
│ │ │ │ +00047c90: 206d 6170 730a 2020 2020 6172 6520 6f6e   maps.    are on
│ │ │ │ +00047ca0: 6520 7468 6520 696e 7665 7273 6520 6f66  e the inverse of
│ │ │ │ +00047cb0: 2074 6865 206f 7468 6572 0a0a 5761 7973   the other..Ways
│ │ │ │ +00047cc0: 2074 6f20 7573 6520 6578 6365 7074 696f   to use exceptio
│ │ │ │ +00047cd0: 6e61 6c4c 6f63 7573 3a0a 3d3d 3d3d 3d3d  nalLocus:.======
│ │ │ │  00047ce0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00047cf0: 3d0a 0a20 202a 2022 6578 6365 7074 696f  =..  * "exceptio
│ │ │ │ -00047d00: 6e61 6c4c 6f63 7573 2852 6174 696f 6e61  nalLocus(Rationa
│ │ │ │ -00047d10: 6c4d 6170 2922 0a0a 466f 7220 7468 6520  lMap)"..For the 
│ │ │ │ -00047d20: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00047d30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -00047d40: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -00047d50: 6578 6365 7074 696f 6e61 6c4c 6f63 7573  exceptionalLocus
│ │ │ │ -00047d60: 3a20 6578 6365 7074 696f 6e61 6c4c 6f63  : exceptionalLoc
│ │ │ │ -00047d70: 7573 2c20 6973 2061 202a 6e6f 7465 206d  us, is a *note m
│ │ │ │ -00047d80: 6574 686f 6420 6675 6e63 7469 6f6e 0a77  ethod function.w
│ │ │ │ -00047d90: 6974 6820 6f70 7469 6f6e 733a 2028 4d61  ith options: (Ma
│ │ │ │ -00047da0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00047db0: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
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│ │ │ │ -00047dd0: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ -00047de0: 653a 2066 6c61 7474 656e 5f6c 7052 6174  e: flatten_lpRat
│ │ │ │ -00047df0: 696f 6e61 6c4d 6170 5f72 702c 204e 6578  ionalMap_rp, Nex
│ │ │ │ -00047e00: 743a 2066 6f72 6365 496d 6167 652c 2050  t: forceImage, P
│ │ │ │ -00047e10: 7265 763a 2065 7863 6570 7469 6f6e 616c  rev: exceptional
│ │ │ │ -00047e20: 4c6f 6375 732c 2055 703a 2054 6f70 0a0a  Locus, Up: Top..
│ │ │ │ -00047e30: 666c 6174 7465 6e28 5261 7469 6f6e 616c  flatten(Rational
│ │ │ │ -00047e40: 4d61 7029 202d 2d20 7772 6974 6520 736f  Map) -- write so
│ │ │ │ -00047e50: 7572 6365 2061 6e64 2074 6172 6765 7420  urce and target 
│ │ │ │ -00047e60: 6173 206e 6f6e 6465 6765 6e65 7261 7465  as nondegenerate
│ │ │ │ -00047e70: 2076 6172 6965 7469 6573 0a2a 2a2a 2a2a   varieties.*****
│ │ │ │ -00047e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00047cf0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6578  =======..  * "ex
│ │ │ │ +00047d00: 6365 7074 696f 6e61 6c4c 6f63 7573 2852  ceptionalLocus(R
│ │ │ │ +00047d10: 6174 696f 6e61 6c4d 6170 2922 0a0a 466f  ationalMap)"..Fo
│ │ │ │ +00047d20: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00047d30: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
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│ │ │ │ +00047d50: 2a6e 6f74 6520 6578 6365 7074 696f 6e61  *note exceptiona
│ │ │ │ +00047d60: 6c4c 6f63 7573 3a20 6578 6365 7074 696f  lLocus: exceptio
│ │ │ │ +00047d70: 6e61 6c4c 6f63 7573 2c20 6973 2061 202a  nalLocus, is a *
│ │ │ │ +00047d80: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +00047d90: 7469 6f6e 0a77 6974 6820 6f70 7469 6f6e  tion.with option
│ │ │ │ +00047da0: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ +00047db0: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
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│ │ │ │ +00047dd0: 696c 653a 2043 7265 6d6f 6e61 2e69 6e66  ile: Cremona.inf
│ │ │ │ +00047de0: 6f2c 204e 6f64 653a 2066 6c61 7474 656e  o, Node: flatten
│ │ │ │ +00047df0: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +00047e00: 702c 204e 6578 743a 2066 6f72 6365 496d  p, Next: forceIm
│ │ │ │ +00047e10: 6167 652c 2050 7265 763a 2065 7863 6570  age, Prev: excep
│ │ │ │ +00047e20: 7469 6f6e 616c 4c6f 6375 732c 2055 703a  tionalLocus, Up:
│ │ │ │ +00047e30: 2054 6f70 0a0a 666c 6174 7465 6e28 5261   Top..flatten(Ra
│ │ │ │ +00047e40: 7469 6f6e 616c 4d61 7029 202d 2d20 7772  tionalMap) -- wr
│ │ │ │ +00047e50: 6974 6520 736f 7572 6365 2061 6e64 2074  ite source and t
│ │ │ │ +00047e60: 6172 6765 7420 6173 206e 6f6e 6465 6765  arget as nondege
│ │ │ │ +00047e70: 6e65 7261 7465 2076 6172 6965 7469 6573  nerate varieties
│ │ │ │ +00047e80: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  00047e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00047ea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00047eb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00047ec0: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ -00047ed0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675  ========..  * Fu
│ │ │ │ -00047ee0: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 666c  nction: *note fl
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│ │ │ │ -00047f00: 3244 6f63 2966 6c61 7474 656e 2c0a 2020  2Doc)flatten,.  
│ │ │ │ -00047f10: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -00047f20: 2020 666c 6174 7465 6e20 7068 690a 2020    flatten phi.  
│ │ │ │ -00047f30: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00047f40: 2a20 7068 692c 2061 202a 6e6f 7465 2072  * phi, a *note r
│ │ │ │ -00047f50: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ -00047f60: 696f 6e61 6c4d 6170 2c0a 2020 2a20 4f75  ionalMap,.  * Ou
│ │ │ │ -00047f70: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ -00047f80: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ -00047f90: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ -00047fa0: 2c2c 2061 2072 6174 696f 6e61 6c20 6d61  ,, a rational ma
│ │ │ │ -00047fb0: 7020 6973 6f6d 6f72 7068 6963 2074 6f20  p isomorphic to 
│ │ │ │ -00047fc0: 7468 650a 2020 2020 2020 2020 6f72 6967  the.        orig
│ │ │ │ -00047fd0: 696e 616c 206d 6170 2c20 666c 6174 7465  inal map, flatte
│ │ │ │ -00047fe0: 6e65 6420 696e 2074 6865 2073 656e 7365  ned in the sense
│ │ │ │ -00047ff0: 2074 6861 7420 7468 6520 6964 6561 6c73   that the ideals
│ │ │ │ -00048000: 206f 6620 736f 7572 6365 2061 6e64 0a20   of source and. 
│ │ │ │ -00048010: 2020 2020 2020 2074 6172 6765 7420 636f         target co
│ │ │ │ -00048020: 6e74 6169 6e20 6e6f 206c 696e 6561 7220  ntain no linear 
│ │ │ │ -00048030: 666f 726d 730a 0a44 6573 6372 6970 7469  forms..Descripti
│ │ │ │ -00048040: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00048050: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00047ec0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ +00047ed0: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ +00047ee0: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e    * Function: *n
│ │ │ │ +00047ef0: 6f74 6520 666c 6174 7465 6e3a 2028 4d61  ote flatten: (Ma
│ │ │ │ +00047f00: 6361 756c 6179 3244 6f63 2966 6c61 7474  caulay2Doc)flatt
│ │ │ │ +00047f10: 656e 2c0a 2020 2a20 5573 6167 653a 200a  en,.  * Usage: .
│ │ │ │ +00047f20: 2020 2020 2020 2020 666c 6174 7465 6e20          flatten 
│ │ │ │ +00047f30: 7068 690a 2020 2a20 496e 7075 7473 3a0a  phi.  * Inputs:.
│ │ │ │ +00047f40: 2020 2020 2020 2a20 7068 692c 2061 202a        * phi, a *
│ │ │ │ +00047f50: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ +00047f60: 703a 2052 6174 696f 6e61 6c4d 6170 2c0a  p: RationalMap,.
│ │ │ │ +00047f70: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +00047f80: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ +00047f90: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +00047fa0: 6e61 6c4d 6170 2c2c 2061 2072 6174 696f  nalMap,, a ratio
│ │ │ │ +00047fb0: 6e61 6c20 6d61 7020 6973 6f6d 6f72 7068  nal map isomorph
│ │ │ │ +00047fc0: 6963 2074 6f20 7468 650a 2020 2020 2020  ic to the.      
│ │ │ │ +00047fd0: 2020 6f72 6967 696e 616c 206d 6170 2c20    original map, 
│ │ │ │ +00047fe0: 666c 6174 7465 6e65 6420 696e 2074 6865  flattened in the
│ │ │ │ +00047ff0: 2073 656e 7365 2074 6861 7420 7468 6520   sense that the 
│ │ │ │ +00048000: 6964 6561 6c73 206f 6620 736f 7572 6365  ideals of source
│ │ │ │ +00048010: 2061 6e64 0a20 2020 2020 2020 2074 6172   and.        tar
│ │ │ │ +00048020: 6765 7420 636f 6e74 6169 6e20 6e6f 206c  get contain no l
│ │ │ │ +00048030: 696e 6561 7220 666f 726d 730a 0a44 6573  inear forms..Des
│ │ │ │ +00048040: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00048050: 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ====..+---------
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│ │ │ │  00048070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000480a0: 7c69 3120 3a20 5035 203d 2051 515b 745f  |i1 : P5 = QQ[t_
│ │ │ │ -000480b0: 302e 2e74 5f35 5d3b 2070 6869 203d 2072  0..t_5]; phi = r
│ │ │ │ -000480c0: 6174 696f 6e61 6c4d 6170 2850 352f 2833  ationalMap(P5/(3
│ │ │ │ -000480d0: 352a 745f 312b 3435 2a74 5f32 2b32 312a  5*t_1+45*t_2+21*
│ │ │ │ -000480e0: 7420 2020 2020 2020 2020 2020 2020 7c0a  t             |.
│ │ │ │ -000480f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00048090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000480a0: 2d2d 2d2d 2b0a 7c69 3120 3a20 5035 203d  ----+.|i1 : P5 =
│ │ │ │ +000480b0: 2051 515b 745f 302e 2e74 5f35 5d3b 2070   QQ[t_0..t_5]; p
│ │ │ │ +000480c0: 6869 203d 2072 6174 696f 6e61 6c4d 6170  hi = rationalMap
│ │ │ │ +000480d0: 2850 352f 2833 352a 745f 312b 3435 2a74  (P5/(35*t_1+45*t
│ │ │ │ +000480e0: 5f32 2b32 312a 7420 2020 2020 2020 2020  _2+21*t         
│ │ │ │ +000480f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00048100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048130: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00048140: 7c6f 3220 3a20 5261 7469 6f6e 616c 4d61  |o2 : RationalMa
│ │ │ │ -00048150: 7020 286c 696e 6561 7220 7261 7469 6f6e  p (linear ration
│ │ │ │ -00048160: 616c 206d 6170 2066 726f 6d20 7468 7265  al map from thre
│ │ │ │ -00048170: 6566 6f6c 6420 696e 2050 505e 3520 746f  efold in PP^5 to
│ │ │ │ -00048180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00048190: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00048130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048140: 2020 2020 7c0a 7c6f 3220 3a20 5261 7469      |.|o2 : Rati
│ │ │ │ +00048150: 6f6e 616c 4d61 7020 286c 696e 6561 7220  onalMap (linear 
│ │ │ │ +00048160: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ +00048170: 6d20 7468 7265 6566 6f6c 6420 696e 2050  m threefold in P
│ │ │ │ +00048180: 505e 3520 746f 2020 2020 2020 2020 2020  P^5 to          
│ │ │ │ +00048190: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │  000481a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000481b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000481c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000481d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -000481e0: 7c5f 332b 3532 352a 745f 342b 3133 3635  |_3+525*t_4+1365
│ │ │ │ -000481f0: 2a74 5f35 2c31 3537 352a 745f 302a 745f  *t_5,1575*t_0*t_
│ │ │ │ -00048200: 322d 3332 3530 2a74 5f32 5e32 2b37 3335  2-3250*t_2^2+735
│ │ │ │ -00048210: 2a74 5f30 2a74 5f33 2d31 3839 302a 745f  *t_0*t_3-1890*t_
│ │ │ │ -00048220: 322a 745f 332d 3136 3636 2a74 5f33 7c0a  2*t_3-1666*t_3|.
│ │ │ │ -00048230: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000481d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000481e0: 2d2d 2d2d 7c0a 7c5f 332b 3532 352a 745f  ----|.|_3+525*t_
│ │ │ │ +000481f0: 342b 3133 3635 2a74 5f35 2c31 3537 352a  4+1365*t_5,1575*
│ │ │ │ +00048200: 745f 302a 745f 322d 3332 3530 2a74 5f32  t_0*t_2-3250*t_2
│ │ │ │ +00048210: 5e32 2b37 3335 2a74 5f30 2a74 5f33 2d31  ^2+735*t_0*t_3-1
│ │ │ │ +00048220: 3839 302a 745f 322a 745f 332d 3136 3636  890*t_2*t_3-1666
│ │ │ │ +00048230: 2a74 5f33 7c0a 7c20 2020 2020 2020 2020  *t_3|.|         
│ │ │ │  00048240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00048280: 7c68 7970 6572 7375 7266 6163 6520 696e  |hypersurface in
│ │ │ │ -00048290: 2050 505e 3529 2020 2020 2020 2020 2020   PP^5)          
│ │ │ │ +00048270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048280: 2020 2020 7c0a 7c68 7970 6572 7375 7266      |.|hypersurf
│ │ │ │ +00048290: 6163 6520 696e 2050 505e 3529 2020 2020  ace in PP^5)    
│ │ │ │  000482a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000482b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000482c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000482d0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +000482c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000482d0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │  000482e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000482f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00048320: 7c5e 322b 3137 3135 302a 745f 302a 745f  |^2+17150*t_0*t_
│ │ │ │ -00048330: 342d 3437 3235 302a 745f 322a 745f 342d  4-47250*t_2*t_4-
│ │ │ │ -00048340: 3232 3035 302a 745f 332a 745f 342d 3237  22050*t_3*t_4-27
│ │ │ │ -00048350: 3638 3530 2a74 5f34 5e32 2b34 3635 3530  6850*t_4^2+46550
│ │ │ │ -00048360: 2a74 5f30 2a74 5f35 2d20 2020 2020 7c0a  *t_0*t_5-     |.
│ │ │ │ -00048370: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00048310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00048320: 2d2d 2d2d 7c0a 7c5e 322b 3137 3135 302a  ----|.|^2+17150*
│ │ │ │ +00048330: 745f 302a 745f 342d 3437 3235 302a 745f  t_0*t_4-47250*t_
│ │ │ │ +00048340: 322a 745f 342d 3232 3035 302a 745f 332a  2*t_4-22050*t_3*
│ │ │ │ +00048350: 745f 342d 3237 3638 3530 2a74 5f34 5e32  t_4-276850*t_4^2
│ │ │ │ +00048360: 2b34 3635 3530 2a74 5f30 2a74 5f35 2d20  +46550*t_0*t_5- 
│ │ │ │ +00048370: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │  00048380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000483a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000483b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -000483c0: 7c31 3232 3835 302a 745f 322a 745f 352d  |122850*t_2*t_5-
│ │ │ │ -000483d0: 3537 3333 302a 745f 332a 745f 352d 3134  57330*t_3*t_5-14
│ │ │ │ -000483e0: 3333 3235 302a 745f 342a 745f 352d 3138  33250*t_4*t_5-18
│ │ │ │ -000483f0: 3634 3435 302a 745f 355e 3229 2c50 352f  64450*t_5^2),P5/
│ │ │ │ -00048400: 2833 3135 2a74 5f30 2b32 3830 2a74 7c0a  (315*t_0+280*t|.
│ │ │ │ -00048410: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +000483b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000483c0: 2d2d 2d2d 7c0a 7c31 3232 3835 302a 745f  ----|.|122850*t_
│ │ │ │ +000483d0: 322a 745f 352d 3537 3333 302a 745f 332a  2*t_5-57330*t_3*
│ │ │ │ +000483e0: 745f 352d 3134 3333 3235 302a 745f 342a  t_5-1433250*t_4*
│ │ │ │ +000483f0: 745f 352d 3138 3634 3435 302a 745f 355e  t_5-1864450*t_5^
│ │ │ │ +00048400: 3229 2c50 352f 2833 3135 2a74 5f30 2b32  2),P5/(315*t_0+2
│ │ │ │ +00048410: 3830 2a74 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  80*t|.|---------
│ │ │ │  00048420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00048460: 7c5f 312b 3435 2a74 5f32 2b32 312a 745f  |_1+45*t_2+21*t_
│ │ │ │ -00048470: 332b 3231 302a 745f 342b 3130 3530 2a74  3+210*t_4+1050*t
│ │ │ │ -00048480: 5f35 292c 7b2d 3435 2a74 5f32 2d32 312a  _5),{-45*t_2-21*
│ │ │ │ -00048490: 745f 332d 3439 302a 745f 342d 3133 3330  t_3-490*t_4-1330
│ │ │ │ -000484a0: 2a74 5f35 2c20 2020 2020 2020 2020 7c0a  *t_5,         |.
│ │ │ │ -000484b0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00048450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00048460: 2d2d 2d2d 7c0a 7c5f 312b 3435 2a74 5f32  ----|.|_1+45*t_2
│ │ │ │ +00048470: 2b32 312a 745f 332b 3231 302a 745f 342b  +21*t_3+210*t_4+
│ │ │ │ +00048480: 3130 3530 2a74 5f35 292c 7b2d 3435 2a74  1050*t_5),{-45*t
│ │ │ │ +00048490: 5f32 2d32 312a 745f 332d 3439 302a 745f  _2-21*t_3-490*t_
│ │ │ │ +000484a0: 342d 3133 3330 2a74 5f35 2c20 2020 2020  4-1330*t_5,     
│ │ │ │ +000484b0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │  000484c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000484d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000484e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000484f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00048500: 7c34 352a 745f 322b 3231 2a74 5f33 2b35  |45*t_2+21*t_3+5
│ │ │ │ -00048510: 3235 2a74 5f34 2b31 3336 352a 745f 352c  25*t_4+1365*t_5,
│ │ │ │ -00048520: 2033 352a 745f 322c 2033 352a 745f 332c   35*t_2, 35*t_3,
│ │ │ │ -00048530: 2033 352a 745f 342c 2033 352a 745f 357d   35*t_4, 35*t_5}
│ │ │ │ -00048540: 293b 2020 2020 2020 2020 2020 2020 7c0a  );            |.
│ │ │ │ -00048550: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000484f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00048500: 2d2d 2d2d 7c0a 7c34 352a 745f 322b 3231  ----|.|45*t_2+21
│ │ │ │ +00048510: 2a74 5f33 2b35 3235 2a74 5f34 2b31 3336  *t_3+525*t_4+136
│ │ │ │ +00048520: 352a 745f 352c 2033 352a 745f 322c 2033  5*t_5, 35*t_2, 3
│ │ │ │ +00048530: 352a 745f 332c 2033 352a 745f 342c 2033  5*t_3, 35*t_4, 3
│ │ │ │ +00048540: 352a 745f 357d 293b 2020 2020 2020 2020  5*t_5});        
│ │ │ │ +00048550: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00048560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000485a0: 7c69 3320 3a20 6465 7363 7269 6265 2070  |i3 : describe p
│ │ │ │ -000485b0: 6869 2020 2020 2020 2020 2020 2020 2020  hi              
│ │ │ │ +00048590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000485a0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6465 7363  ----+.|i3 : desc
│ │ │ │ +000485b0: 7269 6265 2070 6869 2020 2020 2020 2020  ribe phi        
│ │ │ │  000485c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000485d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000485e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000485f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000485e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000485f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00048600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048630: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00048640: 7c6f 3320 3d20 7261 7469 6f6e 616c 206d  |o3 = rational m
│ │ │ │ -00048650: 6170 2064 6566 696e 6564 2062 7920 666f  ap defined by fo
│ │ │ │ -00048660: 726d 7320 6f66 2064 6567 7265 6520 3120  rms of degree 1 
│ │ │ │ -00048670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048680: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00048690: 7c20 2020 2020 736f 7572 6365 2076 6172  |     source var
│ │ │ │ -000486a0: 6965 7479 3a20 736d 6f6f 7468 2063 6f6d  iety: smooth com
│ │ │ │ -000486b0: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ -000486c0: 6f6e 206f 6620 7479 7065 2028 312c 3229  on of type (1,2)
│ │ │ │ -000486d0: 2069 6e20 5050 5e35 2020 2020 2020 7c0a   in PP^5      |.
│ │ │ │ -000486e0: 7c20 2020 2020 7461 7267 6574 2076 6172  |     target var
│ │ │ │ -000486f0: 6965 7479 3a20 6879 7065 7270 6c61 6e65  iety: hyperplane
│ │ │ │ -00048700: 2069 6e20 5050 5e35 2020 2020 2020 2020   in PP^5        
│ │ │ │ +00048630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048640: 2020 2020 7c0a 7c6f 3320 3d20 7261 7469      |.|o3 = rati
│ │ │ │ +00048650: 6f6e 616c 206d 6170 2064 6566 696e 6564  onal map defined
│ │ │ │ +00048660: 2062 7920 666f 726d 7320 6f66 2064 6567   by forms of deg
│ │ │ │ +00048670: 7265 6520 3120 2020 2020 2020 2020 2020  ree 1           
│ │ │ │ +00048680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048690: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ +000486a0: 6365 2076 6172 6965 7479 3a20 736d 6f6f  ce variety: smoo
│ │ │ │ +000486b0: 7468 2063 6f6d 706c 6574 6520 696e 7465  th complete inte
│ │ │ │ +000486c0: 7273 6563 7469 6f6e 206f 6620 7479 7065  rsection of type
│ │ │ │ +000486d0: 2028 312c 3229 2069 6e20 5050 5e35 2020   (1,2) in PP^5  
│ │ │ │ +000486e0: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ +000486f0: 6574 2076 6172 6965 7479 3a20 6879 7065  et variety: hype
│ │ │ │ +00048700: 7270 6c61 6e65 2069 6e20 5050 5e35 2020  rplane in PP^5  
│ │ │ │  00048710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048720: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00048730: 7c20 2020 2020 636f 6566 6669 6369 656e  |     coefficien
│ │ │ │ -00048740: 7420 7269 6e67 3a20 5151 2020 2020 2020  t ring: QQ      
│ │ │ │ +00048720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048730: 2020 2020 7c0a 7c20 2020 2020 636f 6566      |.|     coef
│ │ │ │ +00048740: 6669 6369 656e 7420 7269 6e67 3a20 5151  ficient ring: QQ
│ │ │ │  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 7c0a                |.
│ │ │ │ -00048780: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00048770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048780: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00048790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000487a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000487b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000487c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000487d0: 7c69 3420 3a20 7073 6920 3d20 666c 6174  |i4 : psi = flat
│ │ │ │ -000487e0: 7465 6e20 7068 693b 2020 2020 2020 2020  ten phi;        
│ │ │ │ +000487c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000487d0: 2d2d 2d2d 2b0a 7c69 3420 3a20 7073 6920  ----+.|i4 : psi 
│ │ │ │ +000487e0: 3d20 666c 6174 7465 6e20 7068 693b 2020  = flatten phi;  
│ │ │ │  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 7c0a                |.
│ │ │ │ -00048820: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00048810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048820: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00048830: 2020 2020 2020 2020 2020 2020 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 7c0a                |.
│ │ │ │ -00048870: 7c6f 3420 3a20 5261 7469 6f6e 616c 4d61  |o4 : RationalMa
│ │ │ │ -00048880: 7020 286c 696e 6561 7220 7261 7469 6f6e  p (linear ration
│ │ │ │ -00048890: 616c 206d 6170 2066 726f 6d20 6879 7065  al map from hype
│ │ │ │ -000488a0: 7273 7572 6661 6365 2069 6e20 5050 5e34  rsurface in PP^4
│ │ │ │ -000488b0: 2074 6f20 5050 5e34 2920 2020 2020 7c0a   to PP^4)     |.
│ │ │ │ -000488c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00048860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048870: 2020 2020 7c0a 7c6f 3420 3a20 5261 7469      |.|o4 : Rati
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│ │ │ │ +00048890: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ +000488a0: 6d20 6879 7065 7273 7572 6661 6365 2069  m hypersurface i
│ │ │ │ +000488b0: 6e20 5050 5e34 2074 6f20 5050 5e34 2920  n PP^4 to PP^4) 
│ │ │ │ +000488c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000488d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000488e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000488f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
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│ │ │ │ -00048920: 7369 2020 2020 2020 2020 2020 2020 2020  si              
│ │ │ │ +00048900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00048920: 7269 6265 2070 7369 2020 2020 2020 2020  ribe psi        
│ │ │ │  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 7c0a                |.
│ │ │ │ -00048960: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00048950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048960: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00048970: 2020 2020 2020 2020 2020 2020 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 7c0a                |.
│ │ │ │ -000489b0: 7c6f 3520 3d20 7261 7469 6f6e 616c 206d  |o5 = rational m
│ │ │ │ -000489c0: 6170 2064 6566 696e 6564 2062 7920 666f  ap defined by fo
│ │ │ │ -000489d0: 726d 7320 6f66 2064 6567 7265 6520 3120  rms of degree 1 
│ │ │ │ -000489e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000489f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00048a00: 7c20 2020 2020 736f 7572 6365 2076 6172  |     source var
│ │ │ │ -00048a10: 6965 7479 3a20 736d 6f6f 7468 2071 7561  iety: smooth qua
│ │ │ │ -00048a20: 6472 6963 2068 7970 6572 7375 7266 6163  dric hypersurfac
│ │ │ │ -00048a30: 6520 696e 2050 505e 3420 2020 2020 2020  e in PP^4       
│ │ │ │ -00048a40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00048a50: 7c20 2020 2020 7461 7267 6574 2076 6172  |     target var
│ │ │ │ -00048a60: 6965 7479 3a20 5050 5e34 2020 2020 2020  iety: PP^4      
│ │ │ │ +000489a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000489b0: 2020 2020 7c0a 7c6f 3520 3d20 7261 7469      |.|o5 = rati
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│ │ │ │ +000489d0: 2062 7920 666f 726d 7320 6f66 2064 6567   by forms of deg
│ │ │ │ +000489e0: 7265 6520 3120 2020 2020 2020 2020 2020  ree 1           
│ │ │ │ +000489f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048a00: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ +00048a10: 6365 2076 6172 6965 7479 3a20 736d 6f6f  ce variety: smoo
│ │ │ │ +00048a20: 7468 2071 7561 6472 6963 2068 7970 6572  th quadric hyper
│ │ │ │ +00048a30: 7375 7266 6163 6520 696e 2050 505e 3420  surface in PP^4 
│ │ │ │ +00048a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048a50: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ +00048a60: 6574 2076 6172 6965 7479 3a20 5050 5e34  et variety: PP^4
│ │ │ │  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 7c0a                |.
│ │ │ │ -00048aa0: 7c20 2020 2020 636f 6566 6669 6369 656e  |     coefficien
│ │ │ │ -00048ab0: 7420 7269 6e67 3a20 5151 2020 2020 2020  t ring: QQ      
│ │ │ │ +00048a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048aa0: 2020 2020 7c0a 7c20 2020 2020 636f 6566      |.|     coef
│ │ │ │ +00048ab0: 6669 6369 656e 7420 7269 6e67 3a20 5151  ficient ring: QQ
│ │ │ │  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 7c0a                |.
│ │ │ │ -00048af0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00048ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048af0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00048b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00048b40: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ -00048b50: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ +00048b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00048b40: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
│ │ │ │ +00048b50: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │  00048b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00048b70: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 666c  ==..  * *note fl
│ │ │ │ -00048b80: 6174 7465 6e28 5261 7469 6f6e 616c 4d61  atten(RationalMa
│ │ │ │ -00048b90: 7029 3a20 666c 6174 7465 6e5f 6c70 5261  p): flatten_lpRa
│ │ │ │ -00048ba0: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -00048bb0: 2077 7269 7465 2073 6f75 7263 6520 616e   write source an
│ │ │ │ -00048bc0: 640a 2020 2020 7461 7267 6574 2061 7320  d.    target as 
│ │ │ │ -00048bd0: 6e6f 6e64 6567 656e 6572 6174 6520 7661  nondegenerate va
│ │ │ │ -00048be0: 7269 6574 6965 730a 1f0a 4669 6c65 3a20  rieties...File: 
│ │ │ │ -00048bf0: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ -00048c00: 6465 3a20 666f 7263 6549 6d61 6765 2c20  de: forceImage, 
│ │ │ │ -00048c10: 4e65 7874 3a20 666f 7263 6549 6e76 6572  Next: forceInver
│ │ │ │ -00048c20: 7365 4d61 702c 2050 7265 763a 2066 6c61  seMap, Prev: fla
│ │ │ │ -00048c30: 7474 656e 5f6c 7052 6174 696f 6e61 6c4d  tten_lpRationalM
│ │ │ │ -00048c40: 6170 5f72 702c 2055 703a 2054 6f70 0a0a  ap_rp, Up: Top..
│ │ │ │ -00048c50: 666f 7263 6549 6d61 6765 202d 2d20 6465  forceImage -- de
│ │ │ │ -00048c60: 636c 6172 6520 7768 6963 6820 6973 2074  clare which is t
│ │ │ │ -00048c70: 6865 2069 6d61 6765 206f 6620 6120 7261  he image of a ra
│ │ │ │ -00048c80: 7469 6f6e 616c 206d 6170 0a2a 2a2a 2a2a  tional map.*****
│ │ │ │ -00048c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00048b70: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00048b80: 6f74 6520 666c 6174 7465 6e28 5261 7469  ote flatten(Rati
│ │ │ │ +00048b90: 6f6e 616c 4d61 7029 3a20 666c 6174 7465  onalMap): flatte
│ │ │ │ +00048ba0: 6e5f 6c70 5261 7469 6f6e 616c 4d61 705f  n_lpRationalMap_
│ │ │ │ +00048bb0: 7270 2c20 2d2d 2077 7269 7465 2073 6f75  rp, -- write sou
│ │ │ │ +00048bc0: 7263 6520 616e 640a 2020 2020 7461 7267  rce and.    targ
│ │ │ │ +00048bd0: 6574 2061 7320 6e6f 6e64 6567 656e 6572  et as nondegener
│ │ │ │ +00048be0: 6174 6520 7661 7269 6574 6965 730a 1f0a  ate varieties...
│ │ │ │ +00048bf0: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ +00048c00: 666f 2c20 4e6f 6465 3a20 666f 7263 6549  fo, Node: forceI
│ │ │ │ +00048c10: 6d61 6765 2c20 4e65 7874 3a20 666f 7263  mage, Next: forc
│ │ │ │ +00048c20: 6549 6e76 6572 7365 4d61 702c 2050 7265  eInverseMap, Pre
│ │ │ │ +00048c30: 763a 2066 6c61 7474 656e 5f6c 7052 6174  v: flatten_lpRat
│ │ │ │ +00048c40: 696f 6e61 6c4d 6170 5f72 702c 2055 703a  ionalMap_rp, Up:
│ │ │ │ +00048c50: 2054 6f70 0a0a 666f 7263 6549 6d61 6765   Top..forceImage
│ │ │ │ +00048c60: 202d 2d20 6465 636c 6172 6520 7768 6963   -- declare whic
│ │ │ │ +00048c70: 6820 6973 2074 6865 2069 6d61 6765 206f  h is the image o
│ │ │ │ +00048c80: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ +00048c90: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  00048ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00048cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00048cc0: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ -00048cd0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ -00048ce0: 6167 653a 200a 2020 2020 2020 2020 666f  age: .        fo
│ │ │ │ -00048cf0: 7263 6549 6d61 6765 2850 6869 2c49 290a  rceImage(Phi,I).
│ │ │ │ -00048d00: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00048d10: 2020 2a20 5068 692c 2061 202a 6e6f 7465    * Phi, a *note
│ │ │ │ -00048d20: 2072 6174 696f 6e61 6c20 6d61 703a 2052   rational map: R
│ │ │ │ -00048d30: 6174 696f 6e61 6c4d 6170 2c0a 2020 2020  ationalMap,.    
│ │ │ │ -00048d40: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520    * I, an *note 
│ │ │ │ -00048d50: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ -00048d60: 3244 6f63 2949 6465 616c 2c0a 2020 2a20  2Doc)Ideal,.  * 
│ │ │ │ -00048d70: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00048d80: 202a 6e6f 7465 206e 756c 6c3a 2028 4d61   *note null: (Ma
│ │ │ │ -00048d90: 6361 756c 6179 3244 6f63 296e 756c 6c2c  caulay2Doc)null,
│ │ │ │ -00048da0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
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│ │ │ │ -00048e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00048cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
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│ │ │ │ +00048ce0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +00048cf0: 2020 2020 666f 7263 6549 6d61 6765 2850      forceImage(P
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│ │ │ │ +00048d10: 3a0a 2020 2020 2020 2a20 5068 692c 2061  :.      * Phi, a
│ │ │ │ +00048d20: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +00048d30: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
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│ │ │ │ +00048d50: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ +00048d60: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ +00048d70: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ +00048d80: 2020 2020 202a 202a 6e6f 7465 206e 756c       * *note nul
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│ │ │ │ +00048e80: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
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│ │ │ │  00048ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ -00048ed0: 3a20 5036 203d 2051 515b 745f 302e 2e74  : P6 = QQ[t_0..t
│ │ │ │ -00048ee0: 5f36 5d3b 2058 203d 2020 2020 2020 2020  _6]; X =        
│ │ │ │ +00048ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00048ed0: 2b0a 7c69 3120 3a20 5036 203d 2051 515b  +.|i1 : P6 = QQ[
│ │ │ │ +00048ee0: 745f 302e 2e74 5f36 5d3b 2058 203d 2020  t_0..t_6]; X =  
│ │ │ │  00048ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048f10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00048f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048f20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00048f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00048f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00048fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00049180: 2d64 696d 656e 7369 6f6e 616c 2073 7562  -dimensional sub
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│ │ │ │  000491c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000491d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00049210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049220: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00049270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000492d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
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│ │ │ │ -00049320: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00049330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000492d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000492e0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030  |.| -- used 0.00
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│ │ │ │  00049360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000493a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000493b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000493c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000493d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000493c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000493d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000493e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000493f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049410: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
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│ │ │ │ -00049470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00049470: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │  00049480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000494a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000494f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049500: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00049510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000495b0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +000495c0: 3d0a 0a20 202a 202a 6e6f 7465 2069 6d61  =..  * *note ima
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│ │ │ │ +000495e0: 2069 6d61 6765 5f6c 7052 6174 696f 6e61   image_lpRationa
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│ │ │ │ +00049810: 7468 6520 696e 7665 7273 6520 6f66 2074  the inverse of t
│ │ │ │ +00049820: 6865 206f 7468 6572 0a2a 2a2a 2a2a 2a2a  he other.*******
│ │ │ │  00049830: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00049840: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00049850: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00049860: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00049870: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ -00049880: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ -00049890: 6167 653a 200a 2020 2020 2020 2020 666f  age: .        fo
│ │ │ │ -000498a0: 7263 6549 6e76 6572 7365 4d61 7028 5068  rceInverseMap(Ph
│ │ │ │ -000498b0: 692c 5073 6929 0a20 202a 2049 6e70 7574  i,Psi).  * Input
│ │ │ │ -000498c0: 733a 0a20 2020 2020 202a 2050 6869 2c20  s:.      * Phi, 
│ │ │ │ -000498d0: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ -000498e0: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ -000498f0: 702c 0a20 2020 2020 202a 2050 7369 2c20  p,.      * Psi, 
│ │ │ │ -00049900: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ -00049910: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ -00049920: 702c 0a20 202a 204f 7574 7075 7473 3a0a  p,.  * Outputs:.
│ │ │ │ -00049930: 2020 2020 2020 2a20 2a6e 6f74 6520 6e75        * *note nu
│ │ │ │ -00049940: 6c6c 3a20 284d 6163 6175 6c61 7932 446f  ll: (Macaulay2Do
│ │ │ │ -00049950: 6329 6e75 6c6c 2c0a 0a44 6573 6372 6970  c)null,..Descrip
│ │ │ │ -00049960: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00049970: 0a0a 5468 6973 206d 6574 686f 6420 616c  ..This method al
│ │ │ │ -00049980: 6c6f 7773 2074 6f20 696e 666f 726d 2074  lows to inform t
│ │ │ │ -00049990: 6865 2073 7973 7465 6d20 7468 6174 2074  he system that t
│ │ │ │ -000499a0: 776f 206d 6170 7320 6172 6520 6f6e 6520  wo maps are one 
│ │ │ │ -000499b0: 7468 6520 696e 7665 7273 6520 6f66 0a74  the inverse of.t
│ │ │ │ -000499c0: 6865 206f 7468 6572 2077 6974 686f 7574  he other without
│ │ │ │ -000499d0: 2070 6572 666f 726d 696e 6720 616e 7920   performing any 
│ │ │ │ -000499e0: 636f 6d70 7574 6174 696f 6e2e 2054 6869  computation. Thi
│ │ │ │ -000499f0: 7320 6973 2075 7365 6675 6c20 696e 2070  s is useful in p
│ │ │ │ -00049a00: 6172 7469 6375 6c61 7220 6966 0a79 6f75  articular if.you
│ │ │ │ -00049a10: 2063 616c 6375 6c61 7465 2074 6865 2069   calculate the i
│ │ │ │ -00049a20: 6e76 6572 7365 206d 6170 2075 7369 6e67  nverse map using
│ │ │ │ -00049a30: 2079 6f75 7220 6f77 6e20 6d65 7468 6f64   your own method
│ │ │ │ -00049a40: 2e0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d  ...Caveat.======
│ │ │ │ -00049a50: 0a0a 4966 2074 6865 2064 6563 6c61 7261  ..If the declara
│ │ │ │ -00049a60: 7469 6f6e 2069 7320 6661 6c73 652c 206e  tion is false, n
│ │ │ │ -00049a70: 6f6e 7365 6e73 6963 616c 2061 6e73 7765  onsensical answe
│ │ │ │ -00049a80: 7273 206d 6179 2072 6573 756c 742e 0a0a  rs may result...
│ │ │ │ -00049a90: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -00049aa0: 3d0a 0a20 202a 202a 6e6f 7465 2069 7349  =..  * *note isI
│ │ │ │ -00049ab0: 6e76 6572 7365 4d61 7028 5261 7469 6f6e  nverseMap(Ration
│ │ │ │ -00049ac0: 616c 4d61 702c 5261 7469 6f6e 616c 4d61  alMap,RationalMa
│ │ │ │ -00049ad0: 7029 3a0a 2020 2020 6973 496e 7665 7273  p):.    isInvers
│ │ │ │ -00049ae0: 654d 6170 5f6c 7052 6174 696f 6e61 6c4d  eMap_lpRationalM
│ │ │ │ -00049af0: 6170 5f63 6d52 6174 696f 6e61 6c4d 6170  ap_cmRationalMap
│ │ │ │ -00049b00: 5f72 702c 202d 2d20 6368 6563 6b73 2077  _rp, -- checks w
│ │ │ │ -00049b10: 6865 7468 6572 2074 776f 2072 6174 696f  hether two ratio
│ │ │ │ -00049b20: 6e61 6c0a 2020 2020 6d61 7073 2061 7265  nal.    maps are
│ │ │ │ -00049b30: 206f 6e65 2074 6865 2069 6e76 6572 7365   one the inverse
│ │ │ │ -00049b40: 206f 6620 7468 6520 6f74 6865 720a 2020   of the other.  
│ │ │ │ -00049b50: 2a20 2a6e 6f74 6520 696e 7665 7273 6528  * *note inverse(
│ │ │ │ -00049b60: 5261 7469 6f6e 616c 4d61 7029 3a20 696e  RationalMap): in
│ │ │ │ -00049b70: 7665 7273 655f 6c70 5261 7469 6f6e 616c  verse_lpRational
│ │ │ │ -00049b80: 4d61 705f 7270 2c20 2d2d 2069 6e76 6572  Map_rp, -- inver
│ │ │ │ -00049b90: 7365 206f 6620 610a 2020 2020 6269 7261  se of a.    bira
│ │ │ │ -00049ba0: 7469 6f6e 616c 206d 6170 0a0a 5761 7973  tional map..Ways
│ │ │ │ -00049bb0: 2074 6f20 7573 6520 666f 7263 6549 6e76   to use forceInv
│ │ │ │ -00049bc0: 6572 7365 4d61 703a 0a3d 3d3d 3d3d 3d3d  erseMap:.=======
│ │ │ │ +00049870: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ +00049880: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ +00049890: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +000498a0: 2020 2020 666f 7263 6549 6e76 6572 7365      forceInverse
│ │ │ │ +000498b0: 4d61 7028 5068 692c 5073 6929 0a20 202a  Map(Phi,Psi).  *
│ │ │ │ +000498c0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +000498d0: 2050 6869 2c20 6120 2a6e 6f74 6520 7261   Phi, a *note ra
│ │ │ │ +000498e0: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ +000498f0: 6f6e 616c 4d61 702c 0a20 2020 2020 202a  onalMap,.      *
│ │ │ │ +00049900: 2050 7369 2c20 6120 2a6e 6f74 6520 7261   Psi, a *note ra
│ │ │ │ +00049910: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ +00049920: 6f6e 616c 4d61 702c 0a20 202a 204f 7574  onalMap,.  * Out
│ │ │ │ +00049930: 7075 7473 3a0a 2020 2020 2020 2a20 2a6e  puts:.      * *n
│ │ │ │ +00049940: 6f74 6520 6e75 6c6c 3a20 284d 6163 6175  ote null: (Macau
│ │ │ │ +00049950: 6c61 7932 446f 6329 6e75 6c6c 2c0a 0a44  lay2Doc)null,..D
│ │ │ │ +00049960: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00049970: 3d3d 3d3d 3d3d 0a0a 5468 6973 206d 6574  ======..This met
│ │ │ │ +00049980: 686f 6420 616c 6c6f 7773 2074 6f20 696e  hod allows to in
│ │ │ │ +00049990: 666f 726d 2074 6865 2073 7973 7465 6d20  form the system 
│ │ │ │ +000499a0: 7468 6174 2074 776f 206d 6170 7320 6172  that two maps ar
│ │ │ │ +000499b0: 6520 6f6e 6520 7468 6520 696e 7665 7273  e one the invers
│ │ │ │ +000499c0: 6520 6f66 0a74 6865 206f 7468 6572 2077  e of.the other w
│ │ │ │ +000499d0: 6974 686f 7574 2070 6572 666f 726d 696e  ithout performin
│ │ │ │ +000499e0: 6720 616e 7920 636f 6d70 7574 6174 696f  g any computatio
│ │ │ │ +000499f0: 6e2e 2054 6869 7320 6973 2075 7365 6675  n. This is usefu
│ │ │ │ +00049a00: 6c20 696e 2070 6172 7469 6375 6c61 7220  l in particular 
│ │ │ │ +00049a10: 6966 0a79 6f75 2063 616c 6375 6c61 7465  if.you calculate
│ │ │ │ +00049a20: 2074 6865 2069 6e76 6572 7365 206d 6170   the inverse map
│ │ │ │ +00049a30: 2075 7369 6e67 2079 6f75 7220 6f77 6e20   using your own 
│ │ │ │ +00049a40: 6d65 7468 6f64 2e0a 0a43 6176 6561 740a  method...Caveat.
│ │ │ │ +00049a50: 3d3d 3d3d 3d3d 0a0a 4966 2074 6865 2064  ======..If the d
│ │ │ │ +00049a60: 6563 6c61 7261 7469 6f6e 2069 7320 6661  eclaration is fa
│ │ │ │ +00049a70: 6c73 652c 206e 6f6e 7365 6e73 6963 616c  lse, nonsensical
│ │ │ │ +00049a80: 2061 6e73 7765 7273 206d 6179 2072 6573   answers may res
│ │ │ │ +00049a90: 756c 742e 0a0a 5365 6520 616c 736f 0a3d  ult...See also.=
│ │ │ │ +00049aa0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ +00049ab0: 7465 2069 7349 6e76 6572 7365 4d61 7028  te isInverseMap(
│ │ │ │ +00049ac0: 5261 7469 6f6e 616c 4d61 702c 5261 7469  RationalMap,Rati
│ │ │ │ +00049ad0: 6f6e 616c 4d61 7029 3a0a 2020 2020 6973  onalMap):.    is
│ │ │ │ +00049ae0: 496e 7665 7273 654d 6170 5f6c 7052 6174  InverseMap_lpRat
│ │ │ │ +00049af0: 696f 6e61 6c4d 6170 5f63 6d52 6174 696f  ionalMap_cmRatio
│ │ │ │ +00049b00: 6e61 6c4d 6170 5f72 702c 202d 2d20 6368  nalMap_rp, -- ch
│ │ │ │ +00049b10: 6563 6b73 2077 6865 7468 6572 2074 776f  ecks whether two
│ │ │ │ +00049b20: 2072 6174 696f 6e61 6c0a 2020 2020 6d61   rational.    ma
│ │ │ │ +00049b30: 7073 2061 7265 206f 6e65 2074 6865 2069  ps are one the i
│ │ │ │ +00049b40: 6e76 6572 7365 206f 6620 7468 6520 6f74  nverse of the ot
│ │ │ │ +00049b50: 6865 720a 2020 2a20 2a6e 6f74 6520 696e  her.  * *note in
│ │ │ │ +00049b60: 7665 7273 6528 5261 7469 6f6e 616c 4d61  verse(RationalMa
│ │ │ │ +00049b70: 7029 3a20 696e 7665 7273 655f 6c70 5261  p): inverse_lpRa
│ │ │ │ +00049b80: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ +00049b90: 2069 6e76 6572 7365 206f 6620 610a 2020   inverse of a.  
│ │ │ │ +00049ba0: 2020 6269 7261 7469 6f6e 616c 206d 6170    birational map
│ │ │ │ +00049bb0: 0a0a 5761 7973 2074 6f20 7573 6520 666f  ..Ways to use fo
│ │ │ │ +00049bc0: 7263 6549 6e76 6572 7365 4d61 703a 0a3d  rceInverseMap:.=
│ │ │ │  00049bd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00049be0: 3d3d 3d3d 3d0a 0a20 202a 2022 666f 7263  =====..  * "forc
│ │ │ │ -00049bf0: 6549 6e76 6572 7365 4d61 7028 5261 7469  eInverseMap(Rati
│ │ │ │ -00049c00: 6f6e 616c 4d61 702c 5261 7469 6f6e 616c  onalMap,Rational
│ │ │ │ -00049c10: 4d61 7029 220a 0a46 6f72 2074 6865 2070  Map)"..For the p
│ │ │ │ -00049c20: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -00049c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -00049c40: 6520 6f62 6a65 6374 202a 6e6f 7465 2066  e object *note f
│ │ │ │ -00049c50: 6f72 6365 496e 7665 7273 654d 6170 3a20  orceInverseMap: 
│ │ │ │ -00049c60: 666f 7263 6549 6e76 6572 7365 4d61 702c  forceInverseMap,
│ │ │ │ -00049c70: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ -00049c80: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ -00049c90: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00049ca0: 6446 756e 6374 696f 6e2c 2e0a 1f0a 4669  dFunction,....Fi
│ │ │ │ -00049cb0: 6c65 3a20 4372 656d 6f6e 612e 696e 666f  le: Cremona.info
│ │ │ │ -00049cc0: 2c20 4e6f 6465 3a20 6772 6170 682c 204e  , Node: graph, N
│ │ │ │ -00049cd0: 6578 743a 2067 7261 7068 5f6c 7052 696e  ext: graph_lpRin
│ │ │ │ -00049ce0: 674d 6170 5f72 702c 2050 7265 763a 2066  gMap_rp, Prev: f
│ │ │ │ -00049cf0: 6f72 6365 496e 7665 7273 654d 6170 2c20  orceInverseMap, 
│ │ │ │ -00049d00: 5570 3a20 546f 700a 0a67 7261 7068 202d  Up: Top..graph -
│ │ │ │ -00049d10: 2d20 636c 6f73 7572 6520 6f66 2074 6865  - closure of the
│ │ │ │ -00049d20: 2067 7261 7068 206f 6620 6120 7261 7469   graph of a rati
│ │ │ │ -00049d30: 6f6e 616c 206d 6170 0a2a 2a2a 2a2a 2a2a  onal map.*******
│ │ │ │ +00049be0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00049bf0: 2022 666f 7263 6549 6e76 6572 7365 4d61   "forceInverseMa
│ │ │ │ +00049c00: 7028 5261 7469 6f6e 616c 4d61 702c 5261  p(RationalMap,Ra
│ │ │ │ +00049c10: 7469 6f6e 616c 4d61 7029 220a 0a46 6f72  tionalMap)"..For
│ │ │ │ +00049c20: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +00049c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00049c40: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +00049c50: 6e6f 7465 2066 6f72 6365 496e 7665 7273  note forceInvers
│ │ │ │ +00049c60: 654d 6170 3a20 666f 7263 6549 6e76 6572  eMap: forceInver
│ │ │ │ +00049c70: 7365 4d61 702c 2069 7320 6120 2a6e 6f74  seMap, is a *not
│ │ │ │ +00049c80: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ +00049c90: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63  n:.(Macaulay2Doc
│ │ │ │ +00049ca0: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +00049cb0: 2e0a 1f0a 4669 6c65 3a20 4372 656d 6f6e  ....File: Cremon
│ │ │ │ +00049cc0: 612e 696e 666f 2c20 4e6f 6465 3a20 6772  a.info, Node: gr
│ │ │ │ +00049cd0: 6170 682c 204e 6578 743a 2067 7261 7068  aph, Next: graph
│ │ │ │ +00049ce0: 5f6c 7052 696e 674d 6170 5f72 702c 2050  _lpRingMap_rp, P
│ │ │ │ +00049cf0: 7265 763a 2066 6f72 6365 496e 7665 7273  rev: forceInvers
│ │ │ │ +00049d00: 654d 6170 2c20 5570 3a20 546f 700a 0a67  eMap, Up: Top..g
│ │ │ │ +00049d10: 7261 7068 202d 2d20 636c 6f73 7572 6520  raph -- closure 
│ │ │ │ +00049d20: 6f66 2074 6865 2067 7261 7068 206f 6620  of the graph of 
│ │ │ │ +00049d30: 6120 7261 7469 6f6e 616c 206d 6170 0a2a  a rational map.*
│ │ │ │  00049d40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00049d50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00049d60: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ -00049d70: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ -00049d80: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00049d90: 2067 7261 7068 2070 6869 0a20 202a 2049   graph phi.  * I
│ │ │ │ -00049da0: 6e70 7574 733a 0a20 2020 2020 202a 2070  nputs:.      * p
│ │ │ │ -00049db0: 6869 2c20 6120 2a6e 6f74 6520 7261 7469  hi, a *note rati
│ │ │ │ -00049dc0: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -00049dd0: 616c 4d61 702c 0a20 202a 202a 6e6f 7465  alMap,.  * *note
│ │ │ │ -00049de0: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ -00049df0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00049e00: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ -00049e10: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
│ │ │ │ -00049e20: 7075 7473 2c3a 0a20 2020 2020 202a 202a  puts,:.      * *
│ │ │ │ -00049e30: 6e6f 7465 2042 6c6f 7755 7053 7472 6174  note BlowUpStrat
│ │ │ │ -00049e40: 6567 793a 2042 6c6f 7755 7053 7472 6174  egy: BlowUpStrat
│ │ │ │ -00049e50: 6567 792c 203d 3e20 2e2e 2e2c 2064 6566  egy, => ..., def
│ │ │ │ -00049e60: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -00049e70: 2020 2022 456c 696d 696e 6174 6522 2c0a     "Eliminate",.
│ │ │ │ -00049e80: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00049e90: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ -00049ea0: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00049eb0: 6e61 6c4d 6170 2c2c 2074 6865 2066 6972  nalMap,, the fir
│ │ │ │ -00049ec0: 7374 2070 726f 6a65 6374 696f 6e0a 2020  st projection.  
│ │ │ │ -00049ed0: 2020 2020 2a20 6120 2a6e 6f74 6520 7261      * a *note ra
│ │ │ │ -00049ee0: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ -00049ef0: 6f6e 616c 4d61 702c 2c20 7468 6520 7365  onalMap,, the se
│ │ │ │ -00049f00: 636f 6e64 2070 726f 6a65 6374 696f 6e0a  cond projection.
│ │ │ │ -00049f10: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ -00049f20: 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d  ========..+-----
│ │ │ │ -00049f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00049d60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00049d70: 5379 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d  Synopsis.=======
│ │ │ │ +00049d80: 3d0a 0a20 202a 2055 7361 6765 3a20 0a20  =..  * Usage: . 
│ │ │ │ +00049d90: 2020 2020 2020 2067 7261 7068 2070 6869         graph phi
│ │ │ │ +00049da0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +00049db0: 2020 202a 2070 6869 2c20 6120 2a6e 6f74     * phi, a *not
│ │ │ │ +00049dc0: 6520 7261 7469 6f6e 616c 206d 6170 3a20  e rational map: 
│ │ │ │ +00049dd0: 5261 7469 6f6e 616c 4d61 702c 0a20 202a  RationalMap,.  *
│ │ │ │ +00049de0: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ +00049df0: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ +00049e00: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ +00049e10: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ +00049e20: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ +00049e30: 2020 202a 202a 6e6f 7465 2042 6c6f 7755     * *note BlowU
│ │ │ │ +00049e40: 7053 7472 6174 6567 793a 2042 6c6f 7755  pStrategy: BlowU
│ │ │ │ +00049e50: 7053 7472 6174 6567 792c 203d 3e20 2e2e  pStrategy, => ..
│ │ │ │ +00049e60: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +00049e70: 0a20 2020 2020 2020 2022 456c 696d 696e  .        "Elimin
│ │ │ │ +00049e80: 6174 6522 2c0a 2020 2a20 4f75 7470 7574  ate",.  * Output
│ │ │ │ +00049e90: 733a 0a20 2020 2020 202a 2061 202a 6e6f  s:.      * a *no
│ │ │ │ +00049ea0: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ +00049eb0: 2052 6174 696f 6e61 6c4d 6170 2c2c 2074   RationalMap,, t
│ │ │ │ +00049ec0: 6865 2066 6972 7374 2070 726f 6a65 6374  he first project
│ │ │ │ +00049ed0: 696f 6e0a 2020 2020 2020 2a20 6120 2a6e  ion.      * a *n
│ │ │ │ +00049ee0: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ +00049ef0: 3a20 5261 7469 6f6e 616c 4d61 702c 2c20  : RationalMap,, 
│ │ │ │ +00049f00: 7468 6520 7365 636f 6e64 2070 726f 6a65  the second proje
│ │ │ │ +00049f10: 6374 696f 6e0a 0a44 6573 6372 6970 7469  ction..Descripti
│ │ │ │ +00049f20: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +00049f30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00049f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049f70: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00049f80: 285a 5a2f 3139 3031 3831 295b 785f 302e  (ZZ/190181)[x_0.
│ │ │ │ -00049f90: 2e78 5f34 5d3b 2070 6869 203d 2072 6174  .x_4]; phi = rat
│ │ │ │ -00049fa0: 696f 6e61 6c4d 6170 286d 696e 6f72 7328  ionalMap(minors(
│ │ │ │ -00049fb0: 322c 6d61 7472 6978 7b7b 785f 302e 2e78  2,matrix{{x_0..x
│ │ │ │ -00049fc0: 5f33 7d2c 7b78 5f20 7c0a 7c20 2020 2020  _3},{x_ |.|     
│ │ │ │ -00049fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00049f80: 7c69 3120 3a20 285a 5a2f 3139 3031 3831  |i1 : (ZZ/190181
│ │ │ │ +00049f90: 295b 785f 302e 2e78 5f34 5d3b 2070 6869  )[x_0..x_4]; phi
│ │ │ │ +00049fa0: 203d 2072 6174 696f 6e61 6c4d 6170 286d   = rationalMap(m
│ │ │ │ +00049fb0: 696e 6f72 7328 322c 6d61 7472 6978 7b7b  inors(2,matrix{{
│ │ │ │ +00049fc0: 785f 302e 2e78 5f33 7d2c 7b78 5f20 7c0a  x_0..x_3},{x_ |.
│ │ │ │ +00049fd0: 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a010: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -0004a020: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -0004a030: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +0004a010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a020: 7c6f 3220 3d20 2d2d 2072 6174 696f 6e61  |o2 = -- rationa
│ │ │ │ +0004a030: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │  0004a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a060: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a070: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ -0004a080: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │ +0004a060: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a070: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a080: 2020 2020 205a 5a20 2020 2020 2020 2020       ZZ         
│ │ │ │  0004a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a0b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a0c0: 736f 7572 6365 3a20 5072 6f6a 282d 2d2d  source: Proj(---
│ │ │ │ -0004a0d0: 2d2d 2d5b 7820 2c20 7820 2c20 7820 2c20  ---[x , x , x , 
│ │ │ │ -0004a0e0: 7820 2c20 7820 5d29 2020 2020 2020 2020  x , x ])        
│ │ │ │ +0004a0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a0c0: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ +0004a0d0: 6f6a 282d 2d2d 2d2d 2d5b 7820 2c20 7820  oj(------[x , x 
│ │ │ │ +0004a0e0: 2c20 7820 2c20 7820 2c20 7820 5d29 2020  , x , x , x ])  
│ │ │ │  0004a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a100: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a110: 2020 2020 2020 2020 2020 2020 2031 3930               190
│ │ │ │ -0004a120: 3138 3120 2030 2020 2031 2020 2032 2020  181  0   1   2  
│ │ │ │ -0004a130: 2033 2020 2034 2020 2020 2020 2020 2020   3   4          
│ │ │ │ +0004a100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a120: 2020 2031 3930 3138 3120 2030 2020 2031     190181  0   1
│ │ │ │ +0004a130: 2020 2032 2020 2033 2020 2034 2020 2020     2   3   4    
│ │ │ │  0004a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a150: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a170: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ -0004a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a160: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a180: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  0004a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a1a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a1b0: 7461 7267 6574 3a20 7375 6276 6172 6965  target: subvarie
│ │ │ │ -0004a1c0: 7479 206f 6620 5072 6f6a 282d 2d2d 2d2d  ty of Proj(-----
│ │ │ │ -0004a1d0: 2d5b 7920 2c20 7920 2c20 7920 2c20 7920  -[y , y , y , y 
│ │ │ │ -0004a1e0: 2c20 7920 2c20 7920 5d29 2064 6566 696e  , y , y ]) defin
│ │ │ │ -0004a1f0: 6564 2062 7920 2020 7c0a 7c20 2020 2020  ed by   |.|     
│ │ │ │ -0004a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a210: 2020 2020 2020 2020 2020 2031 3930 3138             19018
│ │ │ │ -0004a220: 3120 2030 2020 2031 2020 2032 2020 2033  1  0   1   2   3
│ │ │ │ -0004a230: 2020 2034 2020 2035 2020 2020 2020 2020     4   5        
│ │ │ │ -0004a240: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a250: 2020 2020 2020 2020 7b20 2020 2020 2020          {       
│ │ │ │ +0004a1a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a1b0: 7c20 2020 2020 7461 7267 6574 3a20 7375  |     target: su
│ │ │ │ +0004a1c0: 6276 6172 6965 7479 206f 6620 5072 6f6a  bvariety of Proj
│ │ │ │ +0004a1d0: 282d 2d2d 2d2d 2d5b 7920 2c20 7920 2c20  (------[y , y , 
│ │ │ │ +0004a1e0: 7920 2c20 7920 2c20 7920 2c20 7920 5d29  y , y , y , y ])
│ │ │ │ +0004a1f0: 2064 6566 696e 6564 2062 7920 2020 7c0a   defined by   |.
│ │ │ │ +0004a200: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a220: 2031 3930 3138 3120 2030 2020 2031 2020   190181  0   1  
│ │ │ │ +0004a230: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ +0004a240: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a250: 7c20 2020 2020 2020 2020 2020 2020 7b20  |             { 
│ │ │ │  0004a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a290: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a2a0: 2020 2020 2020 2020 2079 2079 2020 2d20           y y  - 
│ │ │ │ -0004a2b0: 7920 7920 202b 2079 2079 2020 2020 2020  y y  + y y      
│ │ │ │ +0004a290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a2a0: 7c20 2020 2020 2020 2020 2020 2020 2079  |              y
│ │ │ │ +0004a2b0: 2079 2020 2d20 7920 7920 202b 2079 2079   y  - y y  + y y
│ │ │ │  0004a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a2e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a2f0: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -0004a300: 2031 2034 2020 2020 3020 3520 2020 2020   1 4    0 5     
│ │ │ │ -0004a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a2e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a2f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a300: 3220 3320 2020 2031 2034 2020 2020 3020  2 3    1 4    0 
│ │ │ │ +0004a310: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0004a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a330: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a340: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │ +0004a330: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a340: 7c20 2020 2020 2020 2020 2020 2020 7d20  |             } 
│ │ │ │  0004a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a380: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a390: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -0004a3a0: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +0004a380: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a390: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ +0004a3a0: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │  0004a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a3d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a3f0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0004a3d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a3e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a3f0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0004a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a420: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a440: 202d 2078 2020 2b20 7820 7820 2c20 2020   - x  + x x ,   
│ │ │ │ -0004a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a420: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a430: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a440: 2020 2020 2020 202d 2078 2020 2b20 7820         - x  + x 
│ │ │ │ +0004a450: 7820 2c20 2020 2020 2020 2020 2020 2020  x ,             
│ │ │ │  0004a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a470: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a490: 2020 2020 3120 2020 2030 2032 2020 2020      1    0 2    
│ │ │ │ -0004a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a480: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a490: 2020 2020 2020 2020 2020 3120 2020 2030            1    0
│ │ │ │ +0004a4a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0004a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a4c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a4c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a4d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a530: 202d 2078 2078 2020 2b20 7820 7820 2c20   - x x  + x x , 
│ │ │ │ -0004a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a520: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a530: 2020 2020 2020 202d 2078 2078 2020 2b20         - x x  + 
│ │ │ │ +0004a540: 7820 7820 2c20 2020 2020 2020 2020 2020  x x ,           
│ │ │ │  0004a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a560: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a580: 2020 2020 3120 3220 2020 2030 2033 2020      1 2    0 3  
│ │ │ │ -0004a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a570: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a580: 2020 2020 2020 2020 2020 3120 3220 2020            1 2   
│ │ │ │ +0004a590: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
│ │ │ │  0004a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a5b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a5b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a5c0: 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a600: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a620: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0004a600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a610: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a620: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0004a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a650: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a670: 202d 2078 2020 2b20 7820 7820 2c20 2020   - x  + x x ,   
│ │ │ │ -0004a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a650: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a660: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a670: 2020 2020 2020 202d 2078 2020 2b20 7820         - x  + x 
│ │ │ │ +0004a680: 7820 2c20 2020 2020 2020 2020 2020 2020  x ,             
│ │ │ │  0004a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a6a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a6c0: 2020 2020 3220 2020 2031 2033 2020 2020      2    1 3    
│ │ │ │ -0004a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a6a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a6b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a6c0: 2020 2020 2020 2020 2020 3220 2020 2031            2    1
│ │ │ │ +0004a6d0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0004a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a6f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a6f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a700: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a740: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a760: 202d 2078 2078 2020 2b20 7820 7820 2c20   - x x  + x x , 
│ │ │ │ -0004a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a750: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a760: 2020 2020 2020 202d 2078 2078 2020 2b20         - x x  + 
│ │ │ │ +0004a770: 7820 7820 2c20 2020 2020 2020 2020 2020  x x ,           
│ │ │ │  0004a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a790: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a7b0: 2020 2020 3120 3320 2020 2030 2034 2020      1 3    0 4  
│ │ │ │ -0004a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a7a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a7b0: 2020 2020 2020 2020 2020 3120 3320 2020            1 3   
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│ │ │ │  0004a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a7e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a7f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a830: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a850: 202d 2078 2078 2020 2b20 7820 7820 2c20   - x x  + x x , 
│ │ │ │ -0004a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a830: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a840: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004a850: 2020 2020 2020 202d 2078 2078 2020 2b20         - x x  + 
│ │ │ │ +0004a860: 7820 7820 2c20 2020 2020 2020 2020 2020  x x ,           
│ │ │ │  0004a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a880: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a8a0: 2020 2020 3220 3320 2020 2031 2034 2020      2 3    1 4  
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│ │ │ │  0004a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a8d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a8d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004a8e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │  0004a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a920: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a940: 2020 2020 3220 2020 2020 2020 2020 2020      2           
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│ │ │ │ +0004aac0: 7c6f 3220 3a20 5261 7469 6f6e 616c 4d61  |o2 : RationalMa
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│ │ │ │  0004ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004abb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004abf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004abf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ac00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │  0004ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ac40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ac90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ac90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004aca0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004ace0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ace0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004acf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ad30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ad40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ad80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ad80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ad90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004add0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004add0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ade0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004ae20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ae20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ae30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ae70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ae80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004aec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004aec0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004aed0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004af10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004af20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004af60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004af60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004af70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004afb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004afb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004afc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b000: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b000: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b010: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b060: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004b0a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b0a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b0b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b0f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b100: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b1a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004b1e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b1f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b230: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b240: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b280: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b290: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004b2d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b2e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b330: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b370: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b380: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004b3c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b3d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b410: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b420: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b460: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b470: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004b4b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b4c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b500: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b500: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b510: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b550: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b560: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b5a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b5a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b5b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b5f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b5f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b600: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b640: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b640: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b650: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b690: 2020 2020 2020 2020 7c0a 7c69 6e20 5050          |.|in PP
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│ │ │ │  0004b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004b6e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
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│ │ │ │ +0004b6e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b6f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0004b700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004b710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004b720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004b730: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -0004b740: 7469 6d65 2028 7031 2c70 3229 203d 2067  time (p1,p2) = g
│ │ │ │ -0004b750: 7261 7068 2070 6869 3b20 2020 2020 2020  raph phi;       
│ │ │ │ +0004b730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0004b740: 7c69 3320 3a20 7469 6d65 2028 7031 2c70  |i3 : time (p1,p
│ │ │ │ +0004b750: 3229 203d 2067 7261 7068 2070 6869 3b20  2) = graph phi; 
│ │ │ │  0004b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b780: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -0004b790: 7365 6420 302e 3130 3832 3535 7320 2863  sed 0.108255s (c
│ │ │ │ -0004b7a0: 7075 293b 2030 2e30 3334 3338 3435 7320  pu); 0.0343845s 
│ │ │ │ -0004b7b0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -0004b7c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0004b7d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0004b7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0004b780: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b790: 7c20 2d2d 2075 7365 6420 302e 3033 3338  | -- used 0.0338
│ │ │ │ +0004b7a0: 3939 3973 2028 6370 7529 3b20 302e 3031  999s (cpu); 0.01
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│ │ │ │ +0004b7c0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +0004b7d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b7e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0004b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004b820: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -0004b830: 7031 2020 2020 2020 2020 2020 2020 2020  p1              
│ │ │ │ +0004b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0004b830: 7c69 3420 3a20 7031 2020 2020 2020 2020  |i4 : p1        
│ │ │ │  0004b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b870: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b880: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b8c0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -0004b8d0: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -0004b8e0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +0004b8c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b8d0: 7c6f 3420 3d20 2d2d 2072 6174 696f 6e61  |o4 = -- rationa
│ │ │ │ +0004b8e0: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │  0004b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b910: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b930: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ -0004b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b910: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b920: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b940: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  0004b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b960: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b970: 736f 7572 6365 3a20 7375 6276 6172 6965  source: subvarie
│ │ │ │ -0004b980: 7479 206f 6620 5072 6f6a 282d 2d2d 2d2d  ty of Proj(-----
│ │ │ │ -0004b990: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820  -[x , x , x , x 
│ │ │ │ -0004b9a0: 2c20 7820 2020 2020 2020 2020 2020 2020  , x             
│ │ │ │ -0004b9b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b9d0: 2020 2020 2020 2020 2020 2031 3930 3138             19018
│ │ │ │ -0004b9e0: 3120 2030 2020 2031 2020 2032 2020 2033  1  0   1   2   3
│ │ │ │ -0004b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ba10: 2020 2020 2020 2020 7b20 2020 2020 2020          {       
│ │ │ │ +0004b960: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004b970: 7c20 2020 2020 736f 7572 6365 3a20 7375  |     source: su
│ │ │ │ +0004b980: 6276 6172 6965 7479 206f 6620 5072 6f6a  bvariety of Proj
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│ │ │ │ +0004b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004ba00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ba10: 7c20 2020 2020 2020 2020 2020 2020 7b20  |             { 
│ │ │ │  0004ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ba60: 2020 2020 2020 2020 2079 2079 2020 2d20           y y  - 
│ │ │ │ -0004ba70: 7920 7920 202b 2079 2079 202c 2020 2020  y y  + y y ,    
│ │ │ │ -0004ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ba50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ba60: 7c20 2020 2020 2020 2020 2020 2020 2079  |              y
│ │ │ │ +0004ba70: 2079 2020 2d20 7920 7920 202b 2079 2079   y  - y y  + y y
│ │ │ │ +0004ba80: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0004ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004baa0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bab0: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -0004bac0: 2031 2034 2020 2020 3020 3520 2020 2020   1 4    0 5     
│ │ │ │ -0004bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004baa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bab0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004bac0: 3220 3320 2020 2031 2034 2020 2020 3020  2 3    1 4    0 
│ │ │ │ +0004bad0: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0004bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004baf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004baf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bb00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bb50: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -0004bb60: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ -0004bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bb40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bb50: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004bb60: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ +0004bb70: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0004bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bba0: 2020 2020 2020 2020 2020 3420 3220 2020            4 2   
│ │ │ │ -0004bbb0: 2033 2034 2020 2020 3220 3520 2020 2020   3 4    2 5     
│ │ │ │ -0004bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bb90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bba0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004bbb0: 3420 3220 2020 2033 2034 2020 2020 3220  4 2    3 4    2 
│ │ │ │ +0004bbc0: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0004bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bbe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bbe0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bbf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bc40: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -0004bc50: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ -0004bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bc30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bc40: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004bc50: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ +0004bc60: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0004bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bc90: 2020 2020 2020 2020 2020 3320 3220 2020            3 2   
│ │ │ │ -0004bca0: 2032 2034 2020 2020 3120 3520 2020 2020   2 4    1 5     
│ │ │ │ -0004bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bc80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bc90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004bca0: 3320 3220 2020 2032 2034 2020 2020 3120  3 2    2 4    1 
│ │ │ │ +0004bcb0: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0004bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bcd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bcd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bce0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bd20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bd30: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -0004bd40: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ -0004bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bd20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bd30: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004bd40: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ +0004bd50: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0004bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bd70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bd80: 2020 2020 2020 2020 2020 3420 3120 2020            4 1   
│ │ │ │ -0004bd90: 2033 2033 2020 2020 3120 3520 2020 2020   3 3    1 5     
│ │ │ │ -0004bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bd70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bd80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004bd90: 3420 3120 2020 2033 2033 2020 2020 3120  4 1    3 3    1 
│ │ │ │ +0004bda0: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0004bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bdc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bdc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bdd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004be10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004be20: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -0004be30: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ -0004be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004be10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004be20: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004be30: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ +0004be40: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0004be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004be60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004be70: 2020 2020 2020 2020 2020 3320 3120 2020            3 1   
│ │ │ │ -0004be80: 2032 2033 2020 2020 3020 3520 2020 2020   2 3    0 5     
│ │ │ │ -0004be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004be60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004be70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004be80: 3320 3120 2020 2032 2033 2020 2020 3020  3 1    2 3    0 
│ │ │ │ +0004be90: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0004bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004beb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004beb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bec0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bf00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bf10: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -0004bf20: 7820 7920 202d 2078 2079 2020 2b20 7820  x y  - x y  + x 
│ │ │ │ -0004bf30: 7920 2c20 2020 2020 2020 2020 2020 2020  y ,             
│ │ │ │ +0004bf00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bf10: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004bf20: 2079 2020 2d20 7820 7920 202d 2078 2079   y  - x y  - x y
│ │ │ │ +0004bf30: 2020 2b20 7820 7920 2c20 2020 2020 2020    + x y ,       
│ │ │ │  0004bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bf50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bf60: 2020 2020 2020 2020 2020 3220 3120 2020            2 1   
│ │ │ │ -0004bf70: 2031 2032 2020 2020 3120 3320 2020 2030   1 2    1 3    0
│ │ │ │ -0004bf80: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0004bf50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bf60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004bf70: 3220 3120 2020 2031 2032 2020 2020 3120  2 1    1 2    1 
│ │ │ │ +0004bf80: 3320 2020 2030 2034 2020 2020 2020 2020  3    0 4        
│ │ │ │  0004bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bfa0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bfa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004bfb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bff0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004c000: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -0004c010: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ -0004c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bff0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004c000: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004c010: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ +0004c020: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0004c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c040: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004c050: 2020 2020 2020 2020 2020 3420 3020 2020            4 0   
│ │ │ │ -0004c060: 2032 2033 2020 2020 3120 3420 2020 2020   2 3    1 4     
│ │ │ │ -0004c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004c050: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004c060: 3420 3020 2020 2032 2033 2020 2020 3120  4 0    2 3    1 
│ │ │ │ +0004c070: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │  0004c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c090: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c090: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004c0a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c0e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004c0f0: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -0004c100: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ -0004c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c0e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004c0f0: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004c100: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ +0004c110: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0004c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c130: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004c140: 2020 2020 2020 2020 2020 3320 3020 2020            3 0   
│ │ │ │ -0004c150: 2031 2033 2020 2020 3020 3420 2020 2020   1 3    0 4     
│ │ │ │ -0004c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c130: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004c140: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004c150: 3320 3020 2020 2031 2033 2020 2020 3020  3 0    1 3    0 
│ │ │ │ +0004c160: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │  0004c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c180: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004c190: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004d540: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ -0004d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004d630: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +0004d710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004d720: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +0004d800: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004d810: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │  0004d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004d850: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004d860: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004d8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  0004d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004d900: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d940: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d940: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004d950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d990: 2020 2020 2020 2020 7c0a 7c64 696d 656e          |.|dimen
│ │ │ │ -0004d9a0: 7369 6f6e 616c 2073 7562 7661 7269 6574  sional subvariet
│ │ │ │ -0004d9b0: 7920 6f66 2050 505e 3420 7820 5050 5e35  y of PP^4 x PP^5
│ │ │ │ -0004d9c0: 2074 6f20 5050 5e34 2920 2020 2020 2020   to PP^4)       
│ │ │ │ +0004d990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004d9a0: 7c64 696d 656e 7369 6f6e 616c 2073 7562  |dimensional sub
│ │ │ │ +0004d9b0: 7661 7269 6574 7920 6f66 2050 505e 3420  variety of PP^4 
│ │ │ │ +0004d9c0: 7820 5050 5e35 2074 6f20 5050 5e34 2920  x PP^5 to PP^4) 
│ │ │ │  0004d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d9e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0004d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0004d9e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004d9f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0004da00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004da10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004da30: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -0004da40: 7032 2020 2020 2020 2020 2020 2020 2020  p2              
│ │ │ │ +0004da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0004da40: 7c69 3520 3a20 7032 2020 2020 2020 2020  |i5 : p2        
│ │ │ │  0004da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004da80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004da80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004da90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dad0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -0004dae0: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -0004daf0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +0004dad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004dae0: 7c6f 3520 3d20 2d2d 2072 6174 696f 6e61  |o5 = -- rationa
│ │ │ │ +0004daf0: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │  0004db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004db20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004db40: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ -0004db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004db20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004db30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004db50: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
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│ │ │ │ -0004db70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004db80: 736f 7572 6365 3a20 7375 6276 6172 6965  source: subvarie
│ │ │ │ -0004db90: 7479 206f 6620 5072 6f6a 282d 2d2d 2d2d  ty of Proj(-----
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│ │ │ │ +0004db70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004db80: 7c20 2020 2020 736f 7572 6365 3a20 7375  |     source: su
│ │ │ │ +0004db90: 6276 6172 6965 7479 206f 6620 5072 6f6a  bvariety of Proj
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│ │ │ │ -0004dbc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dbe0: 2020 2020 2020 2020 2020 2031 3930 3138             19018
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│ │ │ │ +0004dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dbf0: 2031 3930 3138 3120 2030 2020 2020 2020   190181  0      
│ │ │ │  0004dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dc10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004dc20: 2020 2020 2020 2020 7b20 2020 2020 2020          {       
│ │ │ │ +0004dc10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -0004df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004df30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004df40: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004df50: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ +0004df60: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
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│ │ │ │ -0004dfa0: 2033 2033 2020 2020 3120 3520 2020 2020   3 3    1 5     
│ │ │ │ -0004dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004df80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004df90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004dfa0: 3420 3120 2020 2033 2033 2020 2020 3120  4 1    3 3    1 
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│ │ │ │ +0004dfd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ +0004e040: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ +0004e050: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
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│ │ │ │ -0004e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004e0c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -0004e130: 7820 7920 202d 2078 2079 2020 2b20 7820  x y  - x y  + x 
│ │ │ │ -0004e140: 7920 2c20 2020 2020 2020 2020 2020 2020  y ,             
│ │ │ │ +0004e110: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e120: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004e130: 2079 2020 2d20 7820 7920 202d 2078 2079   y  - x y  - x y
│ │ │ │ +0004e140: 2020 2b20 7820 7920 2c20 2020 2020 2020    + x y ,       
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│ │ │ │ -0004e160: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ -0004e180: 2031 2032 2020 2020 3120 3320 2020 2030   1 2    1 3    0
│ │ │ │ -0004e190: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0004e160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e170: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004e180: 3220 3120 2020 2031 2032 2020 2020 3120  2 1    1 2    1 
│ │ │ │ +0004e190: 3320 2020 2030 2034 2020 2020 2020 2020  3    0 4        
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│ │ │ │ -0004e1b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e1b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e1c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +0004e210: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
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│ │ │ │ +0004e230: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
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│ │ │ │ +0004e2b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +0004e2f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e300: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
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│ │ │ │ +0004e360: 3320 3020 2020 2031 2033 2020 2020 3020  3 0    1 3    0 
│ │ │ │ +0004e370: 3420 2020 2020 2020 2020 2020 2020 2020  4               
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│ │ │ │ +0004e3a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ -0004e3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e3f0: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -0004e400: 7820 7920 202b 2078 2079 2020 2020 2020  x y  + x y      
│ │ │ │ +0004e3e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e3f0: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ +0004e400: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │  0004e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004e430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ +0004e430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e440: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004e450: 3220 3020 2020 2031 2031 2020 2020 3020  2 0    1 1    0 
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│ │ │ │ +0004e480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  0004e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e4d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e4f0: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ -0004e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e4d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e4e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e500: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  0004e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e530: 7461 7267 6574 3a20 7375 6276 6172 6965  target: subvarie
│ │ │ │ -0004e540: 7479 206f 6620 5072 6f6a 282d 2d2d 2d2d  ty of Proj(-----
│ │ │ │ -0004e550: 2d5b 7920 2c20 2020 2020 2020 2020 2020  -[y ,           
│ │ │ │ +0004e520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e530: 7c20 2020 2020 7461 7267 6574 3a20 7375  |     target: su
│ │ │ │ +0004e540: 6276 6172 6965 7479 206f 6620 5072 6f6a  bvariety of Proj
│ │ │ │ +0004e550: 282d 2d2d 2d2d 2d5b 7920 2c20 2020 2020  (------[y ,     
│ │ │ │  0004e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e590: 2020 2020 2020 2020 2020 2031 3930 3138             19018
│ │ │ │ -0004e5a0: 3120 2030 2020 2020 2020 2020 2020 2020  1  0            
│ │ │ │ +0004e570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e580: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e5a0: 2031 3930 3138 3120 2030 2020 2020 2020   190181  0      
│ │ │ │  0004e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e5c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e5d0: 2020 2020 2020 2020 7b20 2020 2020 2020          {       
│ │ │ │ +0004e5c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e5d0: 7c20 2020 2020 2020 2020 2020 2020 7b20  |             { 
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│ │ │ │ -0004e610: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e620: 2020 2020 2020 2020 2079 2079 2020 2d20           y y  - 
│ │ │ │ -0004e630: 7920 7920 202b 2079 2079 2020 2020 2020  y y  + y y      
│ │ │ │ +0004e610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e620: 7c20 2020 2020 2020 2020 2020 2020 2079  |              y
│ │ │ │ +0004e630: 2079 2020 2d20 7920 7920 202b 2079 2079   y  - y y  + y y
│ │ │ │  0004e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e660: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e670: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -0004e680: 2031 2034 2020 2020 3020 3520 2020 2020   1 4    0 5     
│ │ │ │ -0004e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e670: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0004e680: 3220 3320 2020 2031 2034 2020 2020 3020  2 3    1 4    0 
│ │ │ │ +0004e690: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0004e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e6b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e6c0: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │ +0004e6b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  0004e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e710: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -0004e720: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +0004e700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004e710: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
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│ │ │ │  0004e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e750: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f4c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f4d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f520: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f560: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f570: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f5b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f5b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f5c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f600: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f610: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f650: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f650: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f660: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f6a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f6a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f6b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f6f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f6f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f700: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f740: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f750: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f790: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f7a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f7e0: 2020 2020 2020 2020 7c0a 7c79 202c 2079          |.|y , y
│ │ │ │ -0004f7f0: 202c 2079 202c 2079 202c 2079 205d 2920   , y , y , y ]) 
│ │ │ │ -0004f800: 6465 6669 6e65 6420 6279 2020 2020 2020  defined by      
│ │ │ │ +0004f7e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f7f0: 7c79 202c 2079 202c 2079 202c 2079 202c  |y , y , y , y ,
│ │ │ │ +0004f800: 2079 205d 2920 6465 6669 6e65 6420 6279   y ]) defined by
│ │ │ │  0004f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f830: 2020 2020 2020 2020 7c0a 7c20 3120 2020          |.| 1   
│ │ │ │ -0004f840: 3220 2020 3320 2020 3420 2020 3520 2020  2   3   4   5   
│ │ │ │ -0004f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f830: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f840: 7c20 3120 2020 3220 2020 3320 2020 3420  | 1   2   3   4 
│ │ │ │ +0004f850: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │  0004f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f880: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f8d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f8d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f8e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f920: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f930: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f970: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f980: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f9c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f9c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004f9d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fa10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fa10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fa20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fa60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fa60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fa70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fac0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fb00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fb00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fb10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fb50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fb50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fb60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fba0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fbb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fbf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fbf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fc00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fc40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fc40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fc50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fc90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fc90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fca0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fce0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fce0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fcf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fd30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fd30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fd40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fd80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fd80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fd90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fdd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fdd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fde0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fe20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fe20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fe30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fe70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fe70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fe80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fec0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004fed0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ff10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ff10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ff20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ff60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ff60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ff70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ffa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ffb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0004ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ffb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0004ffc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050000: 2020 2020 2020 2020 7c0a 7c72 6174 696f          |.|ratio
│ │ │ │ -00050010: 6e61 6c20 6d61 7020 6672 6f6d 2034 2d64  nal map from 4-d
│ │ │ │ -00050020: 696d 656e 7369 6f6e 616c 2073 7562 7661  imensional subva
│ │ │ │ -00050030: 7269 6574 7920 6f66 2050 505e 3420 7820  riety of PP^4 x 
│ │ │ │ -00050040: 5050 5e35 2074 6f20 6879 7065 7273 7572  PP^5 to hypersur
│ │ │ │ -00050050: 6661 6365 2069 6e20 7c0a 7c2d 2d2d 2d2d  face in |.|-----
│ │ │ │ -00050060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00050000: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050010: 7c72 6174 696f 6e61 6c20 6d61 7020 6672  |rational map fr
│ │ │ │ +00050020: 6f6d 2034 2d64 696d 656e 7369 6f6e 616c  om 4-dimensional
│ │ │ │ +00050030: 2073 7562 7661 7269 6574 7920 6f66 2050   subvariety of P
│ │ │ │ +00050040: 505e 3420 7820 5050 5e35 2074 6f20 6879  P^4 x PP^5 to hy
│ │ │ │ +00050050: 7065 7273 7572 6661 6365 2069 6e20 7c0a  persurface in |.
│ │ │ │ +00050060: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00050070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000500a0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c50 505e 3529  --------|.|PP^5)
│ │ │ │ -000500b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000500a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000500b0: 7c50 505e 3529 2020 2020 2020 2020 2020  |PP^5)          
│ │ │ │  000500c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000500d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000500e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000500f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00050100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000500f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050100: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00050110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050140: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00050150: 6173 7365 7274 2870 3120 2a20 7068 6920  assert(p1 * phi 
│ │ │ │ -00050160: 3d3d 2070 3220 616e 6420 7032 202a 2070  == p2 and p2 * p
│ │ │ │ -00050170: 6869 5e2d 3120 3d3d 2070 3129 2020 2020  hi^-1 == p1)    
│ │ │ │ -00050180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050190: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -000501a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00050140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00050150: 7c69 3620 3a20 6173 7365 7274 2870 3120  |i6 : assert(p1 
│ │ │ │ +00050160: 2a20 7068 6920 3d3d 2070 3220 616e 6420  * phi == p2 and 
│ │ │ │ +00050170: 7032 202a 2070 6869 5e2d 3120 3d3d 2070  p2 * phi^-1 == p
│ │ │ │ +00050180: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │ +00050190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000501a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000501b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000501c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000501d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000501e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ -000501f0: 6465 7363 7269 6265 2070 3220 2020 2020  describe p2     
│ │ │ │ -00050200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000501e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000501f0: 7c69 3720 3a20 6465 7363 7269 6265 2070  |i7 : describe p
│ │ │ │ +00050200: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00050210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00050240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050230: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050240: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00050250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050280: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ -00050290: 7261 7469 6f6e 616c 206d 6170 2064 6566  rational map def
│ │ │ │ -000502a0: 696e 6564 2062 7920 6d75 6c74 6966 6f72  ined by multifor
│ │ │ │ -000502b0: 6d73 206f 6620 6465 6772 6565 207b 302c  ms of degree {0,
│ │ │ │ -000502c0: 2031 7d20 2020 2020 2020 2020 2020 2020   1}             
│ │ │ │ -000502d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000502e0: 736f 7572 6365 2076 6172 6965 7479 3a20  source variety: 
│ │ │ │ -000502f0: 342d 6469 6d65 6e73 696f 6e61 6c20 7375  4-dimensional su
│ │ │ │ -00050300: 6276 6172 6965 7479 206f 6620 5050 5e34  bvariety of PP^4
│ │ │ │ -00050310: 2078 2050 505e 3520 6375 7420 6f75 7420   x PP^5 cut out 
│ │ │ │ -00050320: 6279 2039 2020 2020 7c0a 7c20 2020 2020  by 9    |.|     
│ │ │ │ -00050330: 7461 7267 6574 2076 6172 6965 7479 3a20  target variety: 
│ │ │ │ -00050340: 736d 6f6f 7468 2071 7561 6472 6963 2068  smooth quadric h
│ │ │ │ -00050350: 7970 6572 7375 7266 6163 6520 696e 2050  ypersurface in P
│ │ │ │ -00050360: 505e 3520 2020 2020 2020 2020 2020 2020  P^5             
│ │ │ │ -00050370: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00050380: 646f 6d69 6e61 6e63 653a 2074 7275 6520  dominance: true 
│ │ │ │ -00050390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050280: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050290: 7c6f 3720 3d20 7261 7469 6f6e 616c 206d  |o7 = rational m
│ │ │ │ +000502a0: 6170 2064 6566 696e 6564 2062 7920 6d75  ap defined by mu
│ │ │ │ +000502b0: 6c74 6966 6f72 6d73 206f 6620 6465 6772  ltiforms of degr
│ │ │ │ +000502c0: 6565 207b 302c 2031 7d20 2020 2020 2020  ee {0, 1}       
│ │ │ │ +000502d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000502e0: 7c20 2020 2020 736f 7572 6365 2076 6172  |     source var
│ │ │ │ +000502f0: 6965 7479 3a20 342d 6469 6d65 6e73 696f  iety: 4-dimensio
│ │ │ │ +00050300: 6e61 6c20 7375 6276 6172 6965 7479 206f  nal subvariety o
│ │ │ │ +00050310: 6620 5050 5e34 2078 2050 505e 3520 6375  f PP^4 x PP^5 cu
│ │ │ │ +00050320: 7420 6f75 7420 6279 2039 2020 2020 7c0a  t out by 9    |.
│ │ │ │ +00050330: 7c20 2020 2020 7461 7267 6574 2076 6172  |     target var
│ │ │ │ +00050340: 6965 7479 3a20 736d 6f6f 7468 2071 7561  iety: smooth qua
│ │ │ │ +00050350: 6472 6963 2068 7970 6572 7375 7266 6163  dric hypersurfac
│ │ │ │ +00050360: 6520 696e 2050 505e 3520 2020 2020 2020  e in PP^5       
│ │ │ │ +00050370: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050380: 7c20 2020 2020 646f 6d69 6e61 6e63 653a  |     dominance:
│ │ │ │ +00050390: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │  000503a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000503b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000503c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000503d0: 636f 6566 6669 6369 656e 7420 7269 6e67  coefficient ring
│ │ │ │ -000503e0: 3a20 5a5a 2f31 3930 3138 3120 2020 2020  : ZZ/190181     
│ │ │ │ -000503f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000503c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000503d0: 7c20 2020 2020 636f 6566 6669 6369 656e  |     coefficien
│ │ │ │ +000503e0: 7420 7269 6e67 3a20 5a5a 2f31 3930 3138  t ring: ZZ/19018
│ │ │ │ +000503f0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00050400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050410: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ -00050420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00050410: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050420: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00050430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050460: 2d2d 2d2d 2d2d 2d2d 7c0a 7c68 7970 6572  --------|.|hyper
│ │ │ │ -00050470: 7375 7266 6163 6573 206f 6620 6465 6772  surfaces of degr
│ │ │ │ -00050480: 6565 7320 287b 302c 2032 7d2c 7b31 2c20  ees ({0, 2},{1, 
│ │ │ │ -00050490: 317d 2c7b 312c 2031 7d2c 7b31 2c20 317d  1},{1, 1},{1, 1}
│ │ │ │ -000504a0: 2c7b 312c 2031 7d2c 7b31 2c20 317d 2c7b  ,{1, 1},{1, 1},{
│ │ │ │ -000504b0: 312c 2020 2020 2020 7c0a 7c2d 2d2d 2d2d  1,      |.|-----
│ │ │ │ -000504c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00050460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00050470: 7c68 7970 6572 7375 7266 6163 6573 206f  |hypersurfaces o
│ │ │ │ +00050480: 6620 6465 6772 6565 7320 287b 302c 2032  f degrees ({0, 2
│ │ │ │ +00050490: 7d2c 7b31 2c20 317d 2c7b 312c 2031 7d2c  },{1, 1},{1, 1},
│ │ │ │ +000504a0: 7b31 2c20 317d 2c7b 312c 2031 7d2c 7b31  {1, 1},{1, 1},{1
│ │ │ │ +000504b0: 2c20 317d 2c7b 312c 2020 2020 2020 7c0a  , 1},{1,      |.
│ │ │ │ +000504c0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  000504d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000504e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000504f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050500: 2d2d 2d2d 2d2d 2d2d 7c0a 7c31 7d2c 7b31  --------|.|1},{1
│ │ │ │ -00050510: 2c20 317d 2c7b 312c 2031 7d29 2020 2020  , 1},{1, 1})    
│ │ │ │ -00050520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00050510: 7c31 7d2c 7b31 2c20 317d 2c7b 312c 2031  |1},{1, 1},{1, 1
│ │ │ │ +00050520: 7d29 2020 2020 2020 2020 2020 2020 2020  })              
│ │ │ │  00050530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050550: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00050560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00050550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050560: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00050570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000505a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -000505b0: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -000505c0: 7320 7032 2020 2020 2020 2020 2020 2020  s p2            
│ │ │ │ +000505a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000505b0: 7c69 3820 3a20 7072 6f6a 6563 7469 7665  |i8 : projective
│ │ │ │ +000505c0: 4465 6772 6565 7320 7032 2020 2020 2020  Degrees p2      
│ │ │ │  000505d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000505e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000505f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00050600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000505f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050600: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00050610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050640: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ -00050650: 7b35 312c 2032 382c 2031 342c 2036 2c20  {51, 28, 14, 6, 
│ │ │ │ -00050660: 327d 2020 2020 2020 2020 2020 2020 2020  2}              
│ │ │ │ +00050640: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050650: 7c6f 3820 3d20 7b35 312c 2032 382c 2031  |o8 = {51, 28, 1
│ │ │ │ +00050660: 342c 2036 2c20 327d 2020 2020 2020 2020  4, 6, 2}        
│ │ │ │  00050670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050690: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000506a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000506a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000506b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000506c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000506d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000506e0: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ -000506f0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +000506e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000506f0: 7c6f 3820 3a20 4c69 7374 2020 2020 2020  |o8 : List      
│ │ │ │  00050700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050730: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00050740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00050730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00050740: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00050750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050780: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6865 6e20  --------+..When 
│ │ │ │ -00050790: 7468 6520 736f 7572 6365 206f 6620 7468  the source of th
│ │ │ │ -000507a0: 6520 7261 7469 6f6e 616c 206d 6170 2069  e rational map i
│ │ │ │ -000507b0: 7320 6120 6d75 6c74 692d 7072 6f6a 6563  s a multi-projec
│ │ │ │ -000507c0: 7469 7665 2076 6172 6965 7479 2c20 7468  tive variety, th
│ │ │ │ -000507d0: 6520 6d65 7468 6f64 0a72 6574 7572 6e73  e method.returns
│ │ │ │ -000507e0: 2061 6c6c 2074 6865 2070 726f 6a65 6374   all the project
│ │ │ │ -000507f0: 696f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ions...+--------
│ │ │ │ +00050780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00050790: 0a57 6865 6e20 7468 6520 736f 7572 6365  .When the source
│ │ │ │ +000507a0: 206f 6620 7468 6520 7261 7469 6f6e 616c   of the rational
│ │ │ │ +000507b0: 206d 6170 2069 7320 6120 6d75 6c74 692d   map is a multi-
│ │ │ │ +000507c0: 7072 6f6a 6563 7469 7665 2076 6172 6965  projective varie
│ │ │ │ +000507d0: 7479 2c20 7468 6520 6d65 7468 6f64 0a72  ty, the method.r
│ │ │ │ +000507e0: 6574 7572 6e73 2061 6c6c 2074 6865 2070  eturns all the p
│ │ │ │ +000507f0: 726f 6a65 6374 696f 6e73 2e0a 0a2b 2d2d  rojections...+--
│ │ │ │  00050800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050840: 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 696d  -----+.|i9 : tim
│ │ │ │ -00050850: 6520 6720 3d20 6772 6170 6820 7032 3b20  e g = graph p2; 
│ │ │ │ -00050860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939  -----------+.|i9
│ │ │ │ +00050850: 203a 2074 696d 6520 6720 3d20 6772 6170   : time g = grap
│ │ │ │ +00050860: 6820 7032 3b20 2020 2020 2020 2020 2020  h p2;           
│ │ │ │  00050870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050890: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000508a0: 2030 2e30 3238 3339 3631 7320 2863 7075   0.0283961s (cpu
│ │ │ │ -000508b0: 293b 2030 2e30 3237 3532 3635 7320 2874  ); 0.0275265s (t
│ │ │ │ -000508c0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -000508d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000508e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00050890: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +000508a0: 2d20 7573 6564 2030 2e30 3936 3139 3439  - used 0.0961949
│ │ │ │ +000508b0: 7320 2863 7075 293b 2030 2e30 3339 3734  s (cpu); 0.03974
│ │ │ │ +000508c0: 3231 7320 2874 6872 6561 6429 3b20 3073  21s (thread); 0s
│ │ │ │ +000508d0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +000508e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  000508f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050930: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 675f  -----+.|i10 : g_
│ │ │ │ -00050940: 303b 2020 2020 2020 2020 2020 2020 2020  0;              
│ │ │ │ +00050930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00050940: 3020 3a20 675f 303b 2020 2020 2020 2020  0 : g_0;        
│ │ │ │  00050950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00050980: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00050990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000509a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000509b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000509c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000509d0: 2020 2020 207c 0a7c 6f31 3020 3a20 4d75       |.|o10 : Mu
│ │ │ │ -000509e0: 6c74 6968 6f6d 6f67 656e 656f 7573 5261  ltihomogeneousRa
│ │ │ │ -000509f0: 7469 6f6e 616c 4d61 7020 2872 6174 696f  tionalMap (ratio
│ │ │ │ -00050a00: 6e61 6c20 6d61 7020 6672 6f6d 2034 2d64  nal map from 4-d
│ │ │ │ -00050a10: 696d 656e 7369 6f6e 616c 2073 7562 7661  imensional subva
│ │ │ │ -00050a20: 7269 6574 797c 0a7c 2d2d 2d2d 2d2d 2d2d  riety|.|--------
│ │ │ │ +000509d0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000509e0: 3020 3a20 4d75 6c74 6968 6f6d 6f67 656e  0 : Multihomogen
│ │ │ │ +000509f0: 656f 7573 5261 7469 6f6e 616c 4d61 7020  eousRationalMap 
│ │ │ │ +00050a00: 2872 6174 696f 6e61 6c20 6d61 7020 6672  (rational map fr
│ │ │ │ +00050a10: 6f6d 2034 2d64 696d 656e 7369 6f6e 616c  om 4-dimensional
│ │ │ │ +00050a20: 2073 7562 7661 7269 6574 797c 0a7c 2d2d   subvariety|.|--
│ │ │ │  00050a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050a70: 2d2d 2d2d 2d7c 0a7c 6f66 2050 505e 3420  -----|.|of PP^4 
│ │ │ │ -00050a80: 7820 5050 5e35 2078 2050 505e 3520 746f  x PP^5 x PP^5 to
│ │ │ │ -00050a90: 2050 505e 3429 2020 2020 2020 2020 2020   PP^4)          
│ │ │ │ +00050a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6f66  -----------|.|of
│ │ │ │ +00050a80: 2050 505e 3420 7820 5050 5e35 2078 2050   PP^4 x PP^5 x P
│ │ │ │ +00050a90: 505e 3520 746f 2050 505e 3429 2020 2020  P^5 to PP^4)    
│ │ │ │  00050aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050ac0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00050ac0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00050ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050b10: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 675f  -----+.|i11 : g_
│ │ │ │ -00050b20: 313b 2020 2020 2020 2020 2020 2020 2020  1;              
│ │ │ │ +00050b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00050b20: 3120 3a20 675f 313b 2020 2020 2020 2020  1 : g_1;        
│ │ │ │  00050b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050b60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00050b60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00050b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050bb0: 2020 2020 207c 0a7c 6f31 3120 3a20 4d75       |.|o11 : Mu
│ │ │ │ -00050bc0: 6c74 6968 6f6d 6f67 656e 656f 7573 5261  ltihomogeneousRa
│ │ │ │ -00050bd0: 7469 6f6e 616c 4d61 7020 2872 6174 696f  tionalMap (ratio
│ │ │ │ -00050be0: 6e61 6c20 6d61 7020 6672 6f6d 2034 2d64  nal map from 4-d
│ │ │ │ -00050bf0: 696d 656e 7369 6f6e 616c 2073 7562 7661  imensional subva
│ │ │ │ -00050c00: 7269 6574 797c 0a7c 2d2d 2d2d 2d2d 2d2d  riety|.|--------
│ │ │ │ +00050bb0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00050bc0: 3120 3a20 4d75 6c74 6968 6f6d 6f67 656e  1 : Multihomogen
│ │ │ │ +00050bd0: 656f 7573 5261 7469 6f6e 616c 4d61 7020  eousRationalMap 
│ │ │ │ +00050be0: 2872 6174 696f 6e61 6c20 6d61 7020 6672  (rational map fr
│ │ │ │ +00050bf0: 6f6d 2034 2d64 696d 656e 7369 6f6e 616c  om 4-dimensional
│ │ │ │ +00050c00: 2073 7562 7661 7269 6574 797c 0a7c 2d2d   subvariety|.|--
│ │ │ │  00050c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050c50: 2d2d 2d2d 2d7c 0a7c 6f66 2050 505e 3420  -----|.|of PP^4 
│ │ │ │ -00050c60: 7820 5050 5e35 2078 2050 505e 3520 746f  x PP^5 x PP^5 to
│ │ │ │ -00050c70: 2050 505e 3529 2020 2020 2020 2020 2020   PP^5)          
│ │ │ │ +00050c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6f66  -----------|.|of
│ │ │ │ +00050c60: 2050 505e 3420 7820 5050 5e35 2078 2050   PP^4 x PP^5 x P
│ │ │ │ +00050c70: 505e 3520 746f 2050 505e 3529 2020 2020  P^5 to PP^5)    
│ │ │ │  00050c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050ca0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00050ca0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00050cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050cf0: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 675f  -----+.|i12 : g_
│ │ │ │ -00050d00: 323b 2020 2020 2020 2020 2020 2020 2020  2;              
│ │ │ │ +00050cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00050d00: 3220 3a20 675f 323b 2020 2020 2020 2020  2 : g_2;        
│ │ │ │  00050d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050d40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00050d40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00050d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050d90: 2020 2020 207c 0a7c 6f31 3220 3a20 4d75       |.|o12 : Mu
│ │ │ │ -00050da0: 6c74 6968 6f6d 6f67 656e 656f 7573 5261  ltihomogeneousRa
│ │ │ │ -00050db0: 7469 6f6e 616c 4d61 7020 2864 6f6d 696e  tionalMap (domin
│ │ │ │ -00050dc0: 616e 7420 7261 7469 6f6e 616c 206d 6170  ant rational map
│ │ │ │ -00050dd0: 2066 726f 6d20 342d 6469 6d65 6e73 696f   from 4-dimensio
│ │ │ │ -00050de0: 6e61 6c20 207c 0a7c 2d2d 2d2d 2d2d 2d2d  nal  |.|--------
│ │ │ │ +00050d90: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00050da0: 3220 3a20 4d75 6c74 6968 6f6d 6f67 656e  2 : Multihomogen
│ │ │ │ +00050db0: 656f 7573 5261 7469 6f6e 616c 4d61 7020  eousRationalMap 
│ │ │ │ +00050dc0: 2864 6f6d 696e 616e 7420 7261 7469 6f6e  (dominant ration
│ │ │ │ +00050dd0: 616c 206d 6170 2066 726f 6d20 342d 6469  al map from 4-di
│ │ │ │ +00050de0: 6d65 6e73 696f 6e61 6c20 207c 0a7c 2d2d  mensional  |.|--
│ │ │ │  00050df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050e30: 2d2d 2d2d 2d7c 0a7c 7375 6276 6172 6965  -----|.|subvarie
│ │ │ │ -00050e40: 7479 206f 6620 5050 5e34 2078 2050 505e  ty of PP^4 x PP^
│ │ │ │ -00050e50: 3520 7820 5050 5e35 2074 6f20 6879 7065  5 x PP^5 to hype
│ │ │ │ -00050e60: 7273 7572 6661 6365 2069 6e20 5050 5e35  rsurface in PP^5
│ │ │ │ -00050e70: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00050e80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00050e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7375  -----------|.|su
│ │ │ │ +00050e40: 6276 6172 6965 7479 206f 6620 5050 5e34  bvariety of PP^4
│ │ │ │ +00050e50: 2078 2050 505e 3520 7820 5050 5e35 2074   x PP^5 x PP^5 t
│ │ │ │ +00050e60: 6f20 6879 7065 7273 7572 6661 6365 2069  o hypersurface i
│ │ │ │ +00050e70: 6e20 5050 5e35 2920 2020 2020 2020 2020  n PP^5)         
│ │ │ │ +00050e80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00050e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050ed0: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6465  -----+.|i13 : de
│ │ │ │ -00050ee0: 7363 7269 6265 2067 5f30 2020 2020 2020  scribe g_0      
│ │ │ │ +00050ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00050ee0: 3320 3a20 6465 7363 7269 6265 2067 5f30  3 : describe g_0
│ │ │ │  00050ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050f20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00050f20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00050f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050f70: 2020 2020 207c 0a7c 6f31 3320 3d20 7261       |.|o13 = ra
│ │ │ │ -00050f80: 7469 6f6e 616c 206d 6170 2064 6566 696e  tional map defin
│ │ │ │ -00050f90: 6564 2062 7920 6d75 6c74 6966 6f72 6d73  ed by multiforms
│ │ │ │ -00050fa0: 206f 6620 6465 6772 6565 207b 312c 2030   of degree {1, 0
│ │ │ │ -00050fb0: 2c20 307d 2020 2020 2020 2020 2020 2020  , 0}            
│ │ │ │ -00050fc0: 2020 2020 207c 0a7c 2020 2020 2020 736f       |.|      so
│ │ │ │ -00050fd0: 7572 6365 2076 6172 6965 7479 3a20 342d  urce variety: 4-
│ │ │ │ -00050fe0: 6469 6d65 6e73 696f 6e61 6c20 7375 6276  dimensional subv
│ │ │ │ -00050ff0: 6172 6965 7479 206f 6620 5050 5e34 2078  ariety of PP^4 x
│ │ │ │ -00051000: 2050 505e 3520 7820 5050 5e35 2063 7574   PP^5 x PP^5 cut
│ │ │ │ -00051010: 206f 7574 207c 0a7c 2020 2020 2020 7461   out |.|      ta
│ │ │ │ -00051020: 7267 6574 2076 6172 6965 7479 3a20 5050  rget variety: PP
│ │ │ │ -00051030: 5e34 2020 2020 2020 2020 2020 2020 2020  ^4              
│ │ │ │ +00050f70: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00050f80: 3320 3d20 7261 7469 6f6e 616c 206d 6170  3 = rational map
│ │ │ │ +00050f90: 2064 6566 696e 6564 2062 7920 6d75 6c74   defined by mult
│ │ │ │ +00050fa0: 6966 6f72 6d73 206f 6620 6465 6772 6565  iforms of degree
│ │ │ │ +00050fb0: 207b 312c 2030 2c20 307d 2020 2020 2020   {1, 0, 0}      
│ │ │ │ +00050fc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00050fd0: 2020 2020 736f 7572 6365 2076 6172 6965      source varie
│ │ │ │ +00050fe0: 7479 3a20 342d 6469 6d65 6e73 696f 6e61  ty: 4-dimensiona
│ │ │ │ +00050ff0: 6c20 7375 6276 6172 6965 7479 206f 6620  l subvariety of 
│ │ │ │ +00051000: 5050 5e34 2078 2050 505e 3520 7820 5050  PP^4 x PP^5 x PP
│ │ │ │ +00051010: 5e35 2063 7574 206f 7574 207c 0a7c 2020  ^5 cut out |.|  
│ │ │ │ +00051020: 2020 2020 7461 7267 6574 2076 6172 6965      target varie
│ │ │ │ +00051030: 7479 3a20 5050 5e34 2020 2020 2020 2020  ty: PP^4        
│ │ │ │  00051040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051060: 2020 2020 207c 0a7c 2020 2020 2020 636f       |.|      co
│ │ │ │ -00051070: 6566 6669 6369 656e 7420 7269 6e67 3a20  efficient ring: 
│ │ │ │ -00051080: 5a5a 2f31 3930 3138 3120 2020 2020 2020  ZZ/190181       
│ │ │ │ +00051060: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00051070: 2020 2020 636f 6566 6669 6369 656e 7420      coefficient 
│ │ │ │ +00051080: 7269 6e67 3a20 5a5a 2f31 3930 3138 3120  ring: ZZ/190181 
│ │ │ │  00051090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000510a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000510b0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000510b0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  000510c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000510d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000510e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000510f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051100: 2d2d 2d2d 2d7c 0a7c 6279 2033 3420 6879  -----|.|by 34 hy
│ │ │ │ -00051110: 7065 7273 7572 6661 6365 7320 6f66 2064  persurfaces of d
│ │ │ │ -00051120: 6567 7265 6573 2028 7b30 2c20 312c 2031  egrees ({0, 1, 1
│ │ │ │ -00051130: 7d2c 7b30 2c20 302c 2032 7d2c 7b30 2c20  },{0, 0, 2},{0, 
│ │ │ │ -00051140: 312c 2031 7d2c 7b30 2c20 312c 2031 7d2c  1, 1},{0, 1, 1},
│ │ │ │ -00051150: 7b30 2c20 207c 0a7c 2d2d 2d2d 2d2d 2d2d  {0,  |.|--------
│ │ │ │ +00051100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6279  -----------|.|by
│ │ │ │ +00051110: 2033 3420 6879 7065 7273 7572 6661 6365   34 hypersurface
│ │ │ │ +00051120: 7320 6f66 2064 6567 7265 6573 2028 7b30  s of degrees ({0
│ │ │ │ +00051130: 2c20 312c 2031 7d2c 7b30 2c20 302c 2032  , 1, 1},{0, 0, 2
│ │ │ │ +00051140: 7d2c 7b30 2c20 312c 2031 7d2c 7b30 2c20  },{0, 1, 1},{0, 
│ │ │ │ +00051150: 312c 2031 7d2c 7b30 2c20 207c 0a7c 2d2d  1, 1},{0,  |.|--
│ │ │ │  00051160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000511a0: 2d2d 2d2d 2d7c 0a7c 312c 2031 7d2c 7b30  -----|.|1, 1},{0
│ │ │ │ -000511b0: 2c20 312c 2031 7d2c 7b30 2c20 312c 2031  , 1, 1},{0, 1, 1
│ │ │ │ -000511c0: 7d2c 7b30 2c20 312c 2031 7d2c 7b31 2c20  },{0, 1, 1},{1, 
│ │ │ │ -000511d0: 302c 2031 7d2c 7b31 2c20 302c 2031 7d2c  0, 1},{1, 0, 1},
│ │ │ │ -000511e0: 7b30 2c20 312c 2031 7d2c 7b30 2c20 312c  {0, 1, 1},{0, 1,
│ │ │ │ -000511f0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000511a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 312c  -----------|.|1,
│ │ │ │ +000511b0: 2031 7d2c 7b30 2c20 312c 2031 7d2c 7b30   1},{0, 1, 1},{0
│ │ │ │ +000511c0: 2c20 312c 2031 7d2c 7b30 2c20 312c 2031  , 1, 1},{0, 1, 1
│ │ │ │ +000511d0: 7d2c 7b31 2c20 302c 2031 7d2c 7b31 2c20  },{1, 0, 1},{1, 
│ │ │ │ +000511e0: 302c 2031 7d2c 7b30 2c20 312c 2031 7d2c  0, 1},{0, 1, 1},
│ │ │ │ +000511f0: 7b30 2c20 312c 2020 2020 207c 0a7c 2d2d  {0, 1,     |.|--
│ │ │ │  00051200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051240: 2d2d 2d2d 2d7c 0a7c 317d 2c7b 302c 2031  -----|.|1},{0, 1
│ │ │ │ -00051250: 2c20 317d 2c7b 302c 2031 2c20 317d 2c7b  , 1},{0, 1, 1},{
│ │ │ │ -00051260: 312c 2030 2c20 317d 2c7b 312c 2030 2c20  1, 0, 1},{1, 0, 
│ │ │ │ -00051270: 317d 2c7b 312c 2030 2c20 317d 2c7b 302c  1},{1, 0, 1},{0,
│ │ │ │ -00051280: 2031 2c20 317d 2c7b 302c 2031 2c20 317d   1, 1},{0, 1, 1}
│ │ │ │ -00051290: 2c7b 302c 207c 0a7c 2d2d 2d2d 2d2d 2d2d  ,{0, |.|--------
│ │ │ │ +00051240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 317d  -----------|.|1}
│ │ │ │ +00051250: 2c7b 302c 2031 2c20 317d 2c7b 302c 2031  ,{0, 1, 1},{0, 1
│ │ │ │ +00051260: 2c20 317d 2c7b 312c 2030 2c20 317d 2c7b  , 1},{1, 0, 1},{
│ │ │ │ +00051270: 312c 2030 2c20 317d 2c7b 312c 2030 2c20  1, 0, 1},{1, 0, 
│ │ │ │ +00051280: 317d 2c7b 302c 2031 2c20 317d 2c7b 302c  1},{0, 1, 1},{0,
│ │ │ │ +00051290: 2031 2c20 317d 2c7b 302c 207c 0a7c 2d2d   1, 1},{0, |.|--
│ │ │ │  000512a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000512b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000512c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000512d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000512e0: 2d2d 2d2d 2d7c 0a7c 312c 2031 7d2c 7b30  -----|.|1, 1},{0
│ │ │ │ -000512f0: 2c20 312c 2031 7d2c 7b30 2c20 312c 2031  , 1, 1},{0, 1, 1
│ │ │ │ -00051300: 7d2c 7b31 2c20 302c 2031 7d2c 7b31 2c20  },{1, 0, 1},{1, 
│ │ │ │ -00051310: 302c 2031 7d2c 7b31 2c20 302c 2031 7d2c  0, 1},{1, 0, 1},
│ │ │ │ -00051320: 7b30 2c20 322c 2030 7d2c 7b31 2c20 312c  {0, 2, 0},{1, 1,
│ │ │ │ -00051330: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000512e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 312c  -----------|.|1,
│ │ │ │ +000512f0: 2031 7d2c 7b30 2c20 312c 2031 7d2c 7b30   1},{0, 1, 1},{0
│ │ │ │ +00051300: 2c20 312c 2031 7d2c 7b31 2c20 302c 2031  , 1, 1},{1, 0, 1
│ │ │ │ +00051310: 7d2c 7b31 2c20 302c 2031 7d2c 7b31 2c20  },{1, 0, 1},{1, 
│ │ │ │ +00051320: 302c 2031 7d2c 7b30 2c20 322c 2030 7d2c  0, 1},{0, 2, 0},
│ │ │ │ +00051330: 7b31 2c20 312c 2020 2020 207c 0a7c 2d2d  {1, 1,     |.|--
│ │ │ │  00051340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051380: 2d2d 2d2d 2d7c 0a7c 307d 2c7b 312c 2031  -----|.|0},{1, 1
│ │ │ │ -00051390: 2c20 307d 2c7b 312c 2031 2c20 307d 2c7b  , 0},{1, 1, 0},{
│ │ │ │ -000513a0: 312c 2031 2c20 307d 2c7b 312c 2031 2c20  1, 1, 0},{1, 1, 
│ │ │ │ -000513b0: 307d 2c7b 312c 2031 2c20 307d 2c7b 312c  0},{1, 1, 0},{1,
│ │ │ │ -000513c0: 2031 2c20 307d 2c7b 312c 2031 2c20 307d   1, 0},{1, 1, 0}
│ │ │ │ -000513d0: 2920 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d  )    |.+--------
│ │ │ │ +00051380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 307d  -----------|.|0}
│ │ │ │ +00051390: 2c7b 312c 2031 2c20 307d 2c7b 312c 2031  ,{1, 1, 0},{1, 1
│ │ │ │ +000513a0: 2c20 307d 2c7b 312c 2031 2c20 307d 2c7b  , 0},{1, 1, 0},{
│ │ │ │ +000513b0: 312c 2031 2c20 307d 2c7b 312c 2031 2c20  1, 1, 0},{1, 1, 
│ │ │ │ +000513c0: 307d 2c7b 312c 2031 2c20 307d 2c7b 312c  0},{1, 1, 0},{1,
│ │ │ │ +000513d0: 2031 2c20 307d 2920 2020 207c 0a2b 2d2d   1, 0})    |.+--
│ │ │ │  000513e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000513f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051420: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ -00051430: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00051440: 6e6f 7465 2067 7261 7068 2852 696e 674d  note graph(RingM
│ │ │ │ -00051450: 6170 293a 2067 7261 7068 5f6c 7052 696e  ap): graph_lpRin
│ │ │ │ -00051460: 674d 6170 5f72 702c 202d 2d20 636c 6f73  gMap_rp, -- clos
│ │ │ │ -00051470: 7572 6520 6f66 2074 6865 2067 7261 7068  ure of the graph
│ │ │ │ -00051480: 206f 6620 610a 2020 2020 7261 7469 6f6e   of a.    ration
│ │ │ │ -00051490: 616c 206d 6170 0a20 202a 202a 6e6f 7465  al map.  * *note
│ │ │ │ -000514a0: 2067 7261 7068 4964 6561 6c3a 2028 4d61   graphIdeal: (Ma
│ │ │ │ -000514b0: 6361 756c 6179 3244 6f63 2967 7261 7068  caulay2Doc)graph
│ │ │ │ -000514c0: 4964 6561 6c5f 6c70 5269 6e67 4d61 705f  Ideal_lpRingMap_
│ │ │ │ -000514d0: 7270 2c20 2d2d 2074 6865 2069 6465 616c  rp, -- the ideal
│ │ │ │ -000514e0: 206f 660a 2020 2020 7468 6520 6772 6170   of.    the grap
│ │ │ │ -000514f0: 6820 6f66 2074 6865 2072 6567 756c 6172  h of the regular
│ │ │ │ -00051500: 206d 6170 2063 6f72 7265 7370 6f6e 6469   map correspondi
│ │ │ │ -00051510: 6e67 2074 6f20 6120 7269 6e67 206d 6170  ng to a ring map
│ │ │ │ -00051520: 0a0a 5761 7973 2074 6f20 7573 6520 6772  ..Ways to use gr
│ │ │ │ -00051530: 6170 683a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  aph:.===========
│ │ │ │ -00051540: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6772  =======..  * "gr
│ │ │ │ -00051550: 6170 6828 5261 7469 6f6e 616c 4d61 7029  aph(RationalMap)
│ │ │ │ -00051560: 220a 2020 2a20 2a6e 6f74 6520 6772 6170  ".  * *note grap
│ │ │ │ -00051570: 6828 5269 6e67 4d61 7029 3a20 6772 6170  h(RingMap): grap
│ │ │ │ -00051580: 685f 6c70 5269 6e67 4d61 705f 7270 2c20  h_lpRingMap_rp, 
│ │ │ │ -00051590: 2d2d 2063 6c6f 7375 7265 206f 6620 7468  -- closure of th
│ │ │ │ -000515a0: 6520 6772 6170 6820 6f66 2061 0a20 2020  e graph of a.   
│ │ │ │ -000515b0: 2072 6174 696f 6e61 6c20 6d61 700a 0a46   rational map..F
│ │ │ │ -000515c0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -000515d0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -000515e0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -000515f0: 202a 6e6f 7465 2067 7261 7068 3a20 6772   *note graph: gr
│ │ │ │ -00051600: 6170 682c 2069 7320 6120 2a6e 6f74 6520  aph, is a *note 
│ │ │ │ -00051610: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ -00051620: 7769 7468 206f 7074 696f 6e73 3a0a 284d  with options:.(M
│ │ │ │ -00051630: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -00051640: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
│ │ │ │ -00051650: 7469 6f6e 732c 2e0a 1f0a 4669 6c65 3a20  tions,....File: 
│ │ │ │ -00051660: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ -00051670: 6465 3a20 6772 6170 685f 6c70 5269 6e67  de: graph_lpRing
│ │ │ │ -00051680: 4d61 705f 7270 2c20 4e65 7874 3a20 6964  Map_rp, Next: id
│ │ │ │ -00051690: 6561 6c5f 6c70 5261 7469 6f6e 616c 4d61  eal_lpRationalMa
│ │ │ │ -000516a0: 705f 7270 2c20 5072 6576 3a20 6772 6170  p_rp, Prev: grap
│ │ │ │ -000516b0: 682c 2055 703a 2054 6f70 0a0a 6772 6170  h, Up: Top..grap
│ │ │ │ -000516c0: 6828 5269 6e67 4d61 7029 202d 2d20 636c  h(RingMap) -- cl
│ │ │ │ -000516d0: 6f73 7572 6520 6f66 2074 6865 2067 7261  osure of the gra
│ │ │ │ -000516e0: 7068 206f 6620 6120 7261 7469 6f6e 616c  ph of a rational
│ │ │ │ -000516f0: 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a   map.***********
│ │ │ │ +00051420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ +00051430: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +00051440: 0a20 202a 202a 6e6f 7465 2067 7261 7068  .  * *note graph
│ │ │ │ +00051450: 2852 696e 674d 6170 293a 2067 7261 7068  (RingMap): graph
│ │ │ │ +00051460: 5f6c 7052 696e 674d 6170 5f72 702c 202d  _lpRingMap_rp, -
│ │ │ │ +00051470: 2d20 636c 6f73 7572 6520 6f66 2074 6865  - closure of the
│ │ │ │ +00051480: 2067 7261 7068 206f 6620 610a 2020 2020   graph of a.    
│ │ │ │ +00051490: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ +000514a0: 202a 6e6f 7465 2067 7261 7068 4964 6561   *note graphIdea
│ │ │ │ +000514b0: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ +000514c0: 2967 7261 7068 4964 6561 6c5f 6c70 5269  )graphIdeal_lpRi
│ │ │ │ +000514d0: 6e67 4d61 705f 7270 2c20 2d2d 2074 6865  ngMap_rp, -- the
│ │ │ │ +000514e0: 2069 6465 616c 206f 660a 2020 2020 7468   ideal of.    th
│ │ │ │ +000514f0: 6520 6772 6170 6820 6f66 2074 6865 2072  e graph of the r
│ │ │ │ +00051500: 6567 756c 6172 206d 6170 2063 6f72 7265  egular map corre
│ │ │ │ +00051510: 7370 6f6e 6469 6e67 2074 6f20 6120 7269  sponding to a ri
│ │ │ │ +00051520: 6e67 206d 6170 0a0a 5761 7973 2074 6f20  ng map..Ways to 
│ │ │ │ +00051530: 7573 6520 6772 6170 683a 0a3d 3d3d 3d3d  use graph:.=====
│ │ │ │ +00051540: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00051550: 202a 2022 6772 6170 6828 5261 7469 6f6e   * "graph(Ration
│ │ │ │ +00051560: 616c 4d61 7029 220a 2020 2a20 2a6e 6f74  alMap)".  * *not
│ │ │ │ +00051570: 6520 6772 6170 6828 5269 6e67 4d61 7029  e graph(RingMap)
│ │ │ │ +00051580: 3a20 6772 6170 685f 6c70 5269 6e67 4d61  : graph_lpRingMa
│ │ │ │ +00051590: 705f 7270 2c20 2d2d 2063 6c6f 7375 7265  p_rp, -- closure
│ │ │ │ +000515a0: 206f 6620 7468 6520 6772 6170 6820 6f66   of the graph of
│ │ │ │ +000515b0: 2061 0a20 2020 2072 6174 696f 6e61 6c20   a.    rational 
│ │ │ │ +000515c0: 6d61 700a 0a46 6f72 2074 6865 2070 726f  map..For the pro
│ │ │ │ +000515d0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +000515e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +000515f0: 6f62 6a65 6374 202a 6e6f 7465 2067 7261  object *note gra
│ │ │ │ +00051600: 7068 3a20 6772 6170 682c 2069 7320 6120  ph: graph, is a 
│ │ │ │ +00051610: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +00051620: 6374 696f 6e20 7769 7468 206f 7074 696f  ction with optio
│ │ │ │ +00051630: 6e73 3a0a 284d 6163 6175 6c61 7932 446f  ns:.(Macaulay2Do
│ │ │ │ +00051640: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00051650: 5769 7468 4f70 7469 6f6e 732c 2e0a 1f0a  WithOptions,....
│ │ │ │ +00051660: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ +00051670: 666f 2c20 4e6f 6465 3a20 6772 6170 685f  fo, Node: graph_
│ │ │ │ +00051680: 6c70 5269 6e67 4d61 705f 7270 2c20 4e65  lpRingMap_rp, Ne
│ │ │ │ +00051690: 7874 3a20 6964 6561 6c5f 6c70 5261 7469  xt: ideal_lpRati
│ │ │ │ +000516a0: 6f6e 616c 4d61 705f 7270 2c20 5072 6576  onalMap_rp, Prev
│ │ │ │ +000516b0: 3a20 6772 6170 682c 2055 703a 2054 6f70  : graph, Up: Top
│ │ │ │ +000516c0: 0a0a 6772 6170 6828 5269 6e67 4d61 7029  ..graph(RingMap)
│ │ │ │ +000516d0: 202d 2d20 636c 6f73 7572 6520 6f66 2074   -- closure of t
│ │ │ │ +000516e0: 6865 2067 7261 7068 206f 6620 6120 7261  he graph of a ra
│ │ │ │ +000516f0: 7469 6f6e 616c 206d 6170 0a2a 2a2a 2a2a  tional map.*****
│ │ │ │  00051700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00051710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00051720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ -00051730: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ -00051740: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -00051750: 2a6e 6f74 6520 6772 6170 683a 2067 7261  *note graph: gra
│ │ │ │ -00051760: 7068 2c0a 2020 2a20 5573 6167 653a 200a  ph,.  * Usage: .
│ │ │ │ -00051770: 2020 2020 2020 2020 6772 6170 6820 7068          graph ph
│ │ │ │ -00051780: 690a 2020 2a20 496e 7075 7473 3a0a 2020  i.  * Inputs:.  
│ │ │ │ -00051790: 2020 2020 2a20 7068 692c 2061 202a 6e6f      * phi, a *no
│ │ │ │ -000517a0: 7465 2072 696e 6720 6d61 703a 2028 4d61  te ring map: (Ma
│ │ │ │ -000517b0: 6361 756c 6179 3244 6f63 2952 696e 674d  caulay2Doc)RingM
│ │ │ │ -000517c0: 6170 2c2c 2072 6570 7265 7365 6e74 696e  ap,, representin
│ │ │ │ -000517d0: 6720 6120 7261 7469 6f6e 616c 0a20 2020  g a rational.   
│ │ │ │ -000517e0: 2020 2020 206d 6170 2024 5c50 6869 3a58       map $\Phi:X
│ │ │ │ -000517f0: 205c 6461 7368 7269 6768 7461 7272 6f77   \dashrightarrow
│ │ │ │ -00051800: 2059 240a 2020 2a20 2a6e 6f74 6520 4f70   Y$.  * *note Op
│ │ │ │ -00051810: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ -00051820: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ -00051830: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ -00051840: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ -00051850: 732c 3a0a 2020 2020 2020 2a20 2a6e 6f74  s,:.      * *not
│ │ │ │ -00051860: 6520 426c 6f77 5570 5374 7261 7465 6779  e BlowUpStrategy
│ │ │ │ -00051870: 3a20 426c 6f77 5570 5374 7261 7465 6779  : BlowUpStrategy
│ │ │ │ -00051880: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ -00051890: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ -000518a0: 2245 6c69 6d69 6e61 7465 222c 0a20 202a  "Eliminate",.  *
│ │ │ │ -000518b0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -000518c0: 2a20 6120 2a6e 6f74 6520 7269 6e67 206d  * a *note ring m
│ │ │ │ -000518d0: 6170 3a20 284d 6163 6175 6c61 7932 446f  ap: (Macaulay2Do
│ │ │ │ -000518e0: 6329 5269 6e67 4d61 702c 2c20 7265 7072  c)RingMap,, repr
│ │ │ │ -000518f0: 6573 656e 7469 6e67 2074 6865 2070 726f  esenting the pro
│ │ │ │ -00051900: 6a65 6374 696f 6e0a 2020 2020 2020 2020  jection.        
│ │ │ │ -00051910: 6f6e 2074 6865 2066 6972 7374 2066 6163  on the first fac
│ │ │ │ -00051920: 746f 7220 245c 6d61 7468 6266 7b47 7261  tor $\mathbf{Gra
│ │ │ │ -00051930: 7068 7d28 5c50 6869 2920 5c74 6f20 5824  ph}(\Phi) \to X$
│ │ │ │ -00051940: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ -00051950: 2072 696e 6720 6d61 703a 2028 4d61 6361   ring map: (Maca
│ │ │ │ -00051960: 756c 6179 3244 6f63 2952 696e 674d 6170  ulay2Doc)RingMap
│ │ │ │ -00051970: 2c2c 2072 6570 7265 7365 6e74 696e 6720  ,, representing 
│ │ │ │ -00051980: 7468 6520 7072 6f6a 6563 7469 6f6e 0a20  the projection. 
│ │ │ │ -00051990: 2020 2020 2020 206f 6e20 7468 6520 7365         on the se
│ │ │ │ -000519a0: 636f 6e64 2066 6163 746f 7220 245c 6d61  cond factor $\ma
│ │ │ │ -000519b0: 7468 6266 7b47 7261 7068 7d28 5c50 6869  thbf{Graph}(\Phi
│ │ │ │ -000519c0: 2920 5c74 6f20 5924 0a0a 4465 7363 7269  ) \to Y$..Descri
│ │ │ │ -000519d0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -000519e0: 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  =..+------------
│ │ │ │ +00051720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00051730: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ +00051740: 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675 6e63  ======..  * Func
│ │ │ │ +00051750: 7469 6f6e 3a20 2a6e 6f74 6520 6772 6170  tion: *note grap
│ │ │ │ +00051760: 683a 2067 7261 7068 2c0a 2020 2a20 5573  h: graph,.  * Us
│ │ │ │ +00051770: 6167 653a 200a 2020 2020 2020 2020 6772  age: .        gr
│ │ │ │ +00051780: 6170 6820 7068 690a 2020 2a20 496e 7075  aph phi.  * Inpu
│ │ │ │ +00051790: 7473 3a0a 2020 2020 2020 2a20 7068 692c  ts:.      * phi,
│ │ │ │ +000517a0: 2061 202a 6e6f 7465 2072 696e 6720 6d61   a *note ring ma
│ │ │ │ +000517b0: 703a 2028 4d61 6361 756c 6179 3244 6f63  p: (Macaulay2Doc
│ │ │ │ +000517c0: 2952 696e 674d 6170 2c2c 2072 6570 7265  )RingMap,, repre
│ │ │ │ +000517d0: 7365 6e74 696e 6720 6120 7261 7469 6f6e  senting a ration
│ │ │ │ +000517e0: 616c 0a20 2020 2020 2020 206d 6170 2024  al.        map $
│ │ │ │ +000517f0: 5c50 6869 3a58 205c 6461 7368 7269 6768  \Phi:X \dashrigh
│ │ │ │ +00051800: 7461 7272 6f77 2059 240a 2020 2a20 2a6e  tarrow Y$.  * *n
│ │ │ │ +00051810: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
│ │ │ │ +00051820: 7574 733a 2028 4d61 6361 756c 6179 3244  uts: (Macaulay2D
│ │ │ │ +00051830: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
│ │ │ │ +00051840: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ +00051850: 2069 6e70 7574 732c 3a0a 2020 2020 2020   inputs,:.      
│ │ │ │ +00051860: 2a20 2a6e 6f74 6520 426c 6f77 5570 5374  * *note BlowUpSt
│ │ │ │ +00051870: 7261 7465 6779 3a20 426c 6f77 5570 5374  rategy: BlowUpSt
│ │ │ │ +00051880: 7261 7465 6779 2c20 3d3e 202e 2e2e 2c20  rategy, => ..., 
│ │ │ │ +00051890: 6465 6661 756c 7420 7661 6c75 650a 2020  default value.  
│ │ │ │ +000518a0: 2020 2020 2020 2245 6c69 6d69 6e61 7465        "Eliminate
│ │ │ │ +000518b0: 222c 0a20 202a 204f 7574 7075 7473 3a0a  ",.  * Outputs:.
│ │ │ │ +000518c0: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ +000518d0: 7269 6e67 206d 6170 3a20 284d 6163 6175  ring map: (Macau
│ │ │ │ +000518e0: 6c61 7932 446f 6329 5269 6e67 4d61 702c  lay2Doc)RingMap,
│ │ │ │ +000518f0: 2c20 7265 7072 6573 656e 7469 6e67 2074  , representing t
│ │ │ │ +00051900: 6865 2070 726f 6a65 6374 696f 6e0a 2020  he projection.  
│ │ │ │ +00051910: 2020 2020 2020 6f6e 2074 6865 2066 6972        on the fir
│ │ │ │ +00051920: 7374 2066 6163 746f 7220 245c 6d61 7468  st factor $\math
│ │ │ │ +00051930: 6266 7b47 7261 7068 7d28 5c50 6869 2920  bf{Graph}(\Phi) 
│ │ │ │ +00051940: 5c74 6f20 5824 0a20 2020 2020 202a 2061  \to X$.      * a
│ │ │ │ +00051950: 202a 6e6f 7465 2072 696e 6720 6d61 703a   *note ring map:
│ │ │ │ +00051960: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ +00051970: 696e 674d 6170 2c2c 2072 6570 7265 7365  ingMap,, represe
│ │ │ │ +00051980: 6e74 696e 6720 7468 6520 7072 6f6a 6563  nting the projec
│ │ │ │ +00051990: 7469 6f6e 0a20 2020 2020 2020 206f 6e20  tion.        on 
│ │ │ │ +000519a0: 7468 6520 7365 636f 6e64 2066 6163 746f  the second facto
│ │ │ │ +000519b0: 7220 245c 6d61 7468 6266 7b47 7261 7068  r $\mathbf{Graph
│ │ │ │ +000519c0: 7d28 5c50 6869 2920 5c74 6f20 5924 0a0a  }(\Phi) \to Y$..
│ │ │ │ +000519d0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +000519e0: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d  =======..+------
│ │ │ │  000519f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051a30: 2d2b 0a7c 6931 203a 2070 6869 203d 206d  -+.|i1 : phi = m
│ │ │ │ -00051a40: 6170 2851 515b 785f 302e 2e78 5f33 5d2c  ap(QQ[x_0..x_3],
│ │ │ │ -00051a50: 5151 5b79 5f30 2e2e 795f 325d 2c7b 2d78  QQ[y_0..y_2],{-x
│ │ │ │ -00051a60: 5f31 5e32 2b78 5f30 2a78 5f32 2c2d 785f  _1^2+x_0*x_2,-x_
│ │ │ │ -00051a70: 312a 785f 322b 785f 302a 785f 332c 2d78  1*x_2+x_0*x_3,-x
│ │ │ │ -00051a80: 5f7c 0a7c 2020 2020 2020 2020 2020 2020  _|.|            
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│ │ │ │ -00052450: 206f 6620 7468 6520 7265 6775 6c61 7220   of the regular 
│ │ │ │ -00052460: 6d61 7020 636f 7272 6573 706f 6e64 696e  map correspondin
│ │ │ │ -00052470: 6720 746f 2061 2072 696e 6720 6d61 700a  g to a ring map.
│ │ │ │ -00052480: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ -00052490: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ +00052390: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ +000523a0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +000523b0: 202a 6e6f 7465 2067 7261 7068 2852 6174   *note graph(Rat
│ │ │ │ +000523c0: 696f 6e61 6c4d 6170 293a 2067 7261 7068  ionalMap): graph
│ │ │ │ +000523d0: 2c20 2d2d 2063 6c6f 7375 7265 206f 6620  , -- closure of 
│ │ │ │ +000523e0: 7468 6520 6772 6170 6820 6f66 2061 2072  the graph of a r
│ │ │ │ +000523f0: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ +00052400: 2a6e 6f74 6520 6772 6170 6849 6465 616c  *note graphIdeal
│ │ │ │ +00052410: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00052420: 6772 6170 6849 6465 616c 5f6c 7052 696e  graphIdeal_lpRin
│ │ │ │ +00052430: 674d 6170 5f72 702c 202d 2d20 7468 6520  gMap_rp, -- the 
│ │ │ │ +00052440: 6964 6561 6c20 6f66 0a20 2020 2074 6865  ideal of.    the
│ │ │ │ +00052450: 2067 7261 7068 206f 6620 7468 6520 7265   graph of the re
│ │ │ │ +00052460: 6775 6c61 7220 6d61 7020 636f 7272 6573  gular map corres
│ │ │ │ +00052470: 706f 6e64 696e 6720 746f 2061 2072 696e  ponding to a rin
│ │ │ │ +00052480: 6720 6d61 700a 0a57 6179 7320 746f 2075  g map..Ways to u
│ │ │ │ +00052490: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │  000524a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000524b0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6772  ==..  * *note gr
│ │ │ │ -000524c0: 6170 6828 5269 6e67 4d61 7029 3a20 6772  aph(RingMap): gr
│ │ │ │ -000524d0: 6170 685f 6c70 5269 6e67 4d61 705f 7270  aph_lpRingMap_rp
│ │ │ │ -000524e0: 2c20 2d2d 2063 6c6f 7375 7265 206f 6620  , -- closure of 
│ │ │ │ -000524f0: 7468 6520 6772 6170 6820 6f66 2061 0a20  the graph of a. 
│ │ │ │ -00052500: 2020 2072 6174 696f 6e61 6c20 6d61 700a     rational map.
│ │ │ │ -00052510: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -00052520: 696e 666f 2c20 4e6f 6465 3a20 6964 6561  info, Node: idea
│ │ │ │ -00052530: 6c5f 6c70 5261 7469 6f6e 616c 4d61 705f  l_lpRationalMap_
│ │ │ │ -00052540: 7270 2c20 4e65 7874 3a20 696d 6167 655f  rp, Next: image_
│ │ │ │ -00052550: 6c70 5261 7469 6f6e 616c 4d61 705f 636d  lpRationalMap_cm
│ │ │ │ -00052560: 5374 7269 6e67 5f72 702c 2050 7265 763a  String_rp, Prev:
│ │ │ │ -00052570: 2067 7261 7068 5f6c 7052 696e 674d 6170   graph_lpRingMap
│ │ │ │ -00052580: 5f72 702c 2055 703a 2054 6f70 0a0a 6964  _rp, Up: Top..id
│ │ │ │ -00052590: 6561 6c28 5261 7469 6f6e 616c 4d61 7029  eal(RationalMap)
│ │ │ │ -000525a0: 202d 2d20 6261 7365 206c 6f63 7573 206f   -- base locus o
│ │ │ │ -000525b0: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ -000525c0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +000524b0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +000524c0: 6f74 6520 6772 6170 6828 5269 6e67 4d61  ote graph(RingMa
│ │ │ │ +000524d0: 7029 3a20 6772 6170 685f 6c70 5269 6e67  p): graph_lpRing
│ │ │ │ +000524e0: 4d61 705f 7270 2c20 2d2d 2063 6c6f 7375  Map_rp, -- closu
│ │ │ │ +000524f0: 7265 206f 6620 7468 6520 6772 6170 6820  re of the graph 
│ │ │ │ +00052500: 6f66 2061 0a20 2020 2072 6174 696f 6e61  of a.    rationa
│ │ │ │ +00052510: 6c20 6d61 700a 1f0a 4669 6c65 3a20 4372  l map...File: Cr
│ │ │ │ +00052520: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ +00052530: 3a20 6964 6561 6c5f 6c70 5261 7469 6f6e  : ideal_lpRation
│ │ │ │ +00052540: 616c 4d61 705f 7270 2c20 4e65 7874 3a20  alMap_rp, Next: 
│ │ │ │ +00052550: 696d 6167 655f 6c70 5261 7469 6f6e 616c  image_lpRational
│ │ │ │ +00052560: 4d61 705f 636d 5374 7269 6e67 5f72 702c  Map_cmString_rp,
│ │ │ │ +00052570: 2050 7265 763a 2067 7261 7068 5f6c 7052   Prev: graph_lpR
│ │ │ │ +00052580: 696e 674d 6170 5f72 702c 2055 703a 2054  ingMap_rp, Up: T
│ │ │ │ +00052590: 6f70 0a0a 6964 6561 6c28 5261 7469 6f6e  op..ideal(Ration
│ │ │ │ +000525a0: 616c 4d61 7029 202d 2d20 6261 7365 206c  alMap) -- base l
│ │ │ │ +000525b0: 6f63 7573 206f 6620 6120 7261 7469 6f6e  ocus of a ration
│ │ │ │ +000525c0: 616c 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a  al map.*********
│ │ │ │  000525d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000525e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000525f0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ -00052600: 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675 6e63  ======..  * Func
│ │ │ │ -00052610: 7469 6f6e 3a20 2a6e 6f74 6520 6964 6561  tion: *note idea
│ │ │ │ -00052620: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ -00052630: 2969 6465 616c 2c0a 2020 2a20 5573 6167  )ideal,.  * Usag
│ │ │ │ -00052640: 653a 200a 2020 2020 2020 2020 6964 6561  e: .        idea
│ │ │ │ -00052650: 6c20 7068 690a 2020 2a20 496e 7075 7473  l phi.  * Inputs
│ │ │ │ -00052660: 3a0a 2020 2020 2020 2a20 7068 692c 2061  :.      * phi, a
│ │ │ │ -00052670: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ -00052680: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ -00052690: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ -000526a0: 2020 2020 202a 2061 6e20 2a6e 6f74 6520       * an *note 
│ │ │ │ -000526b0: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ -000526c0: 3244 6f63 2949 6465 616c 2c2c 2074 6865  2Doc)Ideal,, the
│ │ │ │ -000526d0: 2069 6465 616c 206f 6620 7468 6520 6261   ideal of the ba
│ │ │ │ -000526e0: 7365 206c 6f63 7573 206f 660a 2020 2020  se locus of.    
│ │ │ │ -000526f0: 2020 2020 7068 690a 0a44 6573 6372 6970      phi..Descrip
│ │ │ │ -00052700: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00052710: 0a0a 5468 6973 2069 7320 6765 6e65 7261  ..This is genera
│ │ │ │ -00052720: 6c6c 7920 6469 6666 6963 756c 742c 2062  lly difficult, b
│ │ │ │ -00052730: 7574 2069 6e20 736f 6d65 2063 6173 6573  ut in some cases
│ │ │ │ -00052740: 2069 7420 6973 2065 7175 6976 616c 656e   it is equivalen
│ │ │ │ -00052750: 7420 746f 2069 6465 616c 206d 6174 7269  t to ideal matri
│ │ │ │ -00052760: 780a 7068 692c 2077 6869 6368 2064 6f65  x.phi, which doe
│ │ │ │ -00052770: 7320 6e6f 7420 7065 7266 6f72 6d20 616e  s not perform an
│ │ │ │ -00052780: 7920 636f 6d70 7574 6174 696f 6e2e 0a0a  y computation...
│ │ │ │ -00052790: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000525f0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ +00052600: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ +00052610: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74  * Function: *not
│ │ │ │ +00052620: 6520 6964 6561 6c3a 2028 4d61 6361 756c  e ideal: (Macaul
│ │ │ │ +00052630: 6179 3244 6f63 2969 6465 616c 2c0a 2020  ay2Doc)ideal,.  
│ │ │ │ +00052640: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00052650: 2020 6964 6561 6c20 7068 690a 2020 2a20    ideal phi.  * 
│ │ │ │ +00052660: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00052670: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ +00052680: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +00052690: 6e61 6c4d 6170 2c0a 2020 2a20 4f75 7470  nalMap,.  * Outp
│ │ │ │ +000526a0: 7574 733a 0a20 2020 2020 202a 2061 6e20  uts:.      * an 
│ │ │ │ +000526b0: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ +000526c0: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ +000526d0: 2c2c 2074 6865 2069 6465 616c 206f 6620  ,, the ideal of 
│ │ │ │ +000526e0: 7468 6520 6261 7365 206c 6f63 7573 206f  the base locus o
│ │ │ │ +000526f0: 660a 2020 2020 2020 2020 7068 690a 0a44  f.        phi..D
│ │ │ │ +00052700: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00052710: 3d3d 3d3d 3d3d 0a0a 5468 6973 2069 7320  ======..This is 
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│ │ │ │ +00052730: 756c 742c 2062 7574 2069 6e20 736f 6d65  ult, but in some
│ │ │ │ +00052740: 2063 6173 6573 2069 7420 6973 2065 7175   cases it is equ
│ │ │ │ +00052750: 6976 616c 656e 7420 746f 2069 6465 616c  ivalent to ideal
│ │ │ │ +00052760: 206d 6174 7269 780a 7068 692c 2077 6869   matrix.phi, whi
│ │ │ │ +00052770: 6368 2064 6f65 7320 6e6f 7420 7065 7266  ch does not perf
│ │ │ │ +00052780: 6f72 6d20 616e 7920 636f 6d70 7574 6174  orm any computat
│ │ │ │ +00052790: 696f 6e2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ion...+---------
│ │ │ │  000527a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000527b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000527c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000527d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000527e0: 7c69 3120 3a20 7920 3d20 6765 6e73 2851  |i1 : y = gens(Q
│ │ │ │ -000527f0: 515b 785f 302e 2e78 5f35 5d2f 2878 5f32  Q[x_0..x_5]/(x_2
│ │ │ │ -00052800: 5e32 2d78 5f32 2a78 5f33 2b78 5f31 2a78  ^2-x_2*x_3+x_1*x
│ │ │ │ -00052810: 5f34 2d78 5f30 2a78 5f35 2929 3b20 2020  _4-x_0*x_5));   
│ │ │ │ -00052820: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052830: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000527d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000527e0: 2d2d 2d2d 2b0a 7c69 3120 3a20 7920 3d20  ----+.|i1 : y = 
│ │ │ │ +000527f0: 6765 6e73 2851 515b 785f 302e 2e78 5f35  gens(QQ[x_0..x_5
│ │ │ │ +00052800: 5d2f 2878 5f32 5e32 2d78 5f32 2a78 5f33  ]/(x_2^2-x_2*x_3
│ │ │ │ +00052810: 2b78 5f31 2a78 5f34 2d78 5f30 2a78 5f35  +x_1*x_4-x_0*x_5
│ │ │ │ +00052820: 2929 3b20 2020 2020 2020 2020 2020 2020  ));             
│ │ │ │ +00052830: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00052840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00052880: 7c69 3220 3a20 7068 6920 3d20 7261 7469  |i2 : phi = rati
│ │ │ │ -00052890: 6f6e 616c 4d61 7020 7b79 5f34 5e32 2d79  onalMap {y_4^2-y
│ │ │ │ -000528a0: 5f33 2a79 5f35 2c2d 795f 322a 795f 342b  _3*y_5,-y_2*y_4+
│ │ │ │ -000528b0: 795f 332a 795f 342d 795f 312a 795f 352c  y_3*y_4-y_1*y_5,
│ │ │ │ -000528c0: 202d 795f 322a 795f 332b 795f 335e 7c0a   -y_2*y_3+y_3^|.
│ │ │ │ -000528d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00052880: 2d2d 2d2d 2b0a 7c69 3220 3a20 7068 6920  ----+.|i2 : phi 
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│ │ │ │ +000528a0: 5f34 5e32 2d79 5f33 2a79 5f35 2c2d 795f  _4^2-y_3*y_5,-y_
│ │ │ │ +000528b0: 322a 795f 342b 795f 332a 795f 342d 795f  2*y_4+y_3*y_4-y_
│ │ │ │ +000528c0: 312a 795f 352c 202d 795f 322a 795f 332b  1*y_5, -y_2*y_3+
│ │ │ │ +000528d0: 795f 335e 7c0a 7c20 2020 2020 2020 2020  y_3^|.|         
│ │ │ │  000528e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000528f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052910: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052920: 7c6f 3220 3d20 2d2d 2072 6174 696f 6e61  |o2 = -- rationa
│ │ │ │ -00052930: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │ +00052910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052920: 2020 2020 7c0a 7c6f 3220 3d20 2d2d 2072      |.|o2 = -- r
│ │ │ │ +00052930: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │  00052940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052960: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052970: 7c20 2020 2020 736f 7572 6365 3a20 7375  |     source: su
│ │ │ │ -00052980: 6276 6172 6965 7479 206f 6620 5072 6f6a  bvariety of Proj
│ │ │ │ -00052990: 2851 515b 7820 2c20 7820 2c20 7820 2c20  (QQ[x , x , x , 
│ │ │ │ -000529a0: 7820 2c20 7820 2c20 7820 5d29 2064 6566  x , x , x ]) def
│ │ │ │ -000529b0: 696e 6564 2062 7920 2020 2020 2020 7c0a  ined by       |.
│ │ │ │ -000529c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052970: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ +00052980: 6365 3a20 7375 6276 6172 6965 7479 206f  ce: subvariety o
│ │ │ │ +00052990: 6620 5072 6f6a 2851 515b 7820 2c20 7820  f Proj(QQ[x , x 
│ │ │ │ +000529a0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +000529b0: 5d29 2064 6566 696e 6564 2062 7920 2020  ]) defined by   
│ │ │ │ +000529c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000529d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000529e0: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ -000529f0: 2033 2020 2034 2020 2035 2020 2020 2020   3   4   5      
│ │ │ │ -00052a00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052a10: 7c20 2020 2020 2020 2020 2020 2020 7b20  |             { 
│ │ │ │ -00052a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000529e0: 2020 2020 2020 2020 2020 2030 2020 2031             0   1
│ │ │ │ +000529f0: 2020 2032 2020 2033 2020 2034 2020 2035     2   3   4   5
│ │ │ │ +00052a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052a10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052a20: 2020 2020 7b20 2020 2020 2020 2020 2020      {           
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│ │ │ │  00052a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052a50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052a60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00052a70: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00052a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052a60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052a70: 2020 2020 2020 3220 2020 2020 2020 2020        2         
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│ │ │ │ -00052ab0: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ -00052ac0: 2020 2d20 7820 7820 202b 2078 2078 2020    - x x  + x x  
│ │ │ │ -00052ad0: 2d20 7820 7820 2020 2020 2020 2020 2020  - x x           
│ │ │ │ +00052aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052ab0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052ac0: 2020 2020 2078 2020 2d20 7820 7820 202b       x  - x x  +
│ │ │ │ +00052ad0: 2078 2078 2020 2d20 7820 7820 2020 2020   x x  - x x     
│ │ │ │  00052ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052af0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052b00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00052b10: 3220 2020 2032 2033 2020 2020 3120 3420  2    2 3    1 4 
│ │ │ │ -00052b20: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ +00052af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052b00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052b10: 2020 2020 2020 3220 2020 2032 2033 2020        2    2 3  
│ │ │ │ +00052b20: 2020 3120 3420 2020 2030 2035 2020 2020    1 4    0 5    
│ │ │ │  00052b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052b40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052b50: 7c20 2020 2020 2020 2020 2020 2020 7d20  |             } 
│ │ │ │ -00052b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00052b60: 2020 2020 7d20 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 7c0a                |.
│ │ │ │ -00052ba0: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ -00052bb0: 6f6a 2851 515b 7920 2c20 7920 2c20 7920  oj(QQ[y , y , y 
│ │ │ │ -00052bc0: 2c20 7920 2c20 7920 5d29 2020 2020 2020  , y , y ])      
│ │ │ │ +00052b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052ba0: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ +00052bb0: 6574 3a20 5072 6f6a 2851 515b 7920 2c20  et: Proj(QQ[y , 
│ │ │ │ +00052bc0: 7920 2c20 7920 2c20 7920 2c20 7920 5d29  y , y , y , y ])
│ │ │ │  00052bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052be0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052bf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00052c00: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ -00052c10: 2020 2033 2020 2034 2020 2020 2020 2020     3   4        
│ │ │ │ +00052be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052bf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052c00: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +00052c10: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │  00052c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052c30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052c40: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ -00052c50: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │ +00052c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052c40: 2020 2020 7c0a 7c20 2020 2020 6465 6669      |.|     defi
│ │ │ │ +00052c50: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │  00052c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052c80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052c90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00052ca0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00052c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052c90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052ca0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  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 7c0a                |.
│ │ │ │ -00052ce0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00052cf0: 2020 2020 2020 2078 2020 2d20 7820 7820         x  - x x 
│ │ │ │ -00052d00: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00052cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052ce0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052cf0: 2020 2020 2020 2020 2020 2020 2078 2020               x  
│ │ │ │ +00052d00: 2d20 7820 7820 2c20 2020 2020 2020 2020  - x x ,         
│ │ │ │  00052d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052d30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00052d40: 2020 2020 2020 2020 3420 2020 2033 2035          4    3 5
│ │ │ │ -00052d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052d30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052d40: 2020 2020 2020 2020 2020 2020 2020 3420                4 
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│ │ │ │  00052d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052d80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052d80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  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 7c0a                |.
│ │ │ │ -00052dd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ -00052df0: 7820 7820 202d 2078 2078 202c 2020 2020  x x  - x x ,    
│ │ │ │ -00052e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052e10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052e20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00052e30: 2020 2020 2020 2020 2020 3220 3420 2020            2 4   
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│ │ │ │ -00052e60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052e70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052dd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052de0: 2020 2020 2020 2020 2020 2020 202d 2078               - x
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│ │ │ │ +00052e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052e20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00052e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  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 7c0a                |.
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│ │ │ │ -00052f50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -00052fa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00052fb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052fb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  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 7c0a                |.
│ │ │ │ -00053000: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00053010: 2020 2020 2020 202d 2078 2078 2020 2b20         - x x  + 
│ │ │ │ -00053020: 7820 7820 202d 2078 2078 202c 2020 2020  x x  - x x ,    
│ │ │ │ -00053030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053050: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00053060: 2020 2020 2020 2020 2020 3120 3220 2020            1 2   
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│ │ │ │ -00053080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053090: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000530a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053000: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053010: 2020 2020 2020 2020 2020 2020 202d 2078               - x
│ │ │ │ +00053020: 2078 2020 2b20 7820 7820 202d 2078 2078   x  + x x  - x x
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│ │ │ │ +00053040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053050: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053070: 3120 3220 2020 2031 2033 2020 2020 3020  1 2    1 3    0 
│ │ │ │ +00053080: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +00053090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000530a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  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 7c0a                |.
│ │ │ │ -000530f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00053100: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000530e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000530f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053100: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  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 7c0a                |.
│ │ │ │ -00053140: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00053150: 2020 2020 2020 2078 2020 2d20 7820 7820         x  - x x 
│ │ │ │ -00053160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053140: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053150: 2020 2020 2020 2020 2020 2020 2078 2020               x  
│ │ │ │ +00053160: 2d20 7820 7820 2020 2020 2020 2020 2020  - x x           
│ │ │ │  00053170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053190: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000531a0: 2020 2020 2020 2020 3120 2020 2030 2033          1    0 3
│ │ │ │ -000531b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053190: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000531a0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +000531b0: 2020 2030 2033 2020 2020 2020 2020 2020     0 3          
│ │ │ │  000531c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000531d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000531e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000531f0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +000531d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000531e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  00053200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053220: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053230: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053230: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00053240: 2020 2020 2020 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 7c0a                |.
│ │ │ │ -00053280: 7c6f 3220 3a20 5261 7469 6f6e 616c 4d61  |o2 : RationalMa
│ │ │ │ -00053290: 7020 2871 7561 6472 6174 6963 2072 6174  p (quadratic rat
│ │ │ │ -000532a0: 696f 6e61 6c20 6d61 7020 6672 6f6d 2068  ional map from h
│ │ │ │ -000532b0: 7970 6572 7375 7266 6163 6520 696e 2050  ypersurface in P
│ │ │ │ -000532c0: 505e 3520 746f 2050 505e 3429 2020 7c0a  P^5 to PP^4)  |.
│ │ │ │ -000532d0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00053270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053280: 2020 2020 7c0a 7c6f 3220 3a20 5261 7469      |.|o2 : Rati
│ │ │ │ +00053290: 6f6e 616c 4d61 7020 2871 7561 6472 6174  onalMap (quadrat
│ │ │ │ +000532a0: 6963 2072 6174 696f 6e61 6c20 6d61 7020  ic rational map 
│ │ │ │ +000532b0: 6672 6f6d 2068 7970 6572 7375 7266 6163  from hypersurfac
│ │ │ │ +000532c0: 6520 696e 2050 505e 3520 746f 2050 505e  e in PP^5 to PP^
│ │ │ │ +000532d0: 3429 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  4)  |.|---------
│ │ │ │  000532e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000532f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00053320: 7c32 2d79 5f31 2a79 5f34 2c20 2d79 5f31  |2-y_1*y_4, -y_1
│ │ │ │ -00053330: 2a79 5f32 2b79 5f31 2a79 5f33 2d79 5f30  *y_2+y_1*y_3-y_0
│ │ │ │ -00053340: 2a79 5f34 2c20 795f 315e 322d 795f 302a  *y_4, y_1^2-y_0*
│ │ │ │ -00053350: 795f 337d 2020 2020 2020 2020 2020 2020  y_3}            
│ │ │ │ -00053360: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053370: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00053310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053320: 2d2d 2d2d 7c0a 7c32 2d79 5f31 2a79 5f34  ----|.|2-y_1*y_4
│ │ │ │ +00053330: 2c20 2d79 5f31 2a79 5f32 2b79 5f31 2a79  , -y_1*y_2+y_1*y
│ │ │ │ +00053340: 5f33 2d79 5f30 2a79 5f34 2c20 795f 315e  _3-y_0*y_4, y_1^
│ │ │ │ +00053350: 322d 795f 302a 795f 337d 2020 2020 2020  2-y_0*y_3}      
│ │ │ │ +00053360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053370: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00053380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000533a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000533b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000533c0: 7c69 3320 3a20 7469 6d65 2069 6465 616c  |i3 : time ideal
│ │ │ │ -000533d0: 2070 6869 2020 2020 2020 2020 2020 2020   phi            
│ │ │ │ +000533b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000533c0: 2d2d 2d2d 2b0a 7c69 3320 3a20 7469 6d65  ----+.|i3 : time
│ │ │ │ +000533d0: 2069 6465 616c 2070 6869 2020 2020 2020   ideal phi      
│ │ │ │  000533e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000533f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053410: 7c20 2d2d 2075 7365 6420 302e 3030 3339  | -- used 0.0039
│ │ │ │ -00053420: 3938 3139 7320 2863 7075 293b 2030 2e30  9819s (cpu); 0.0
│ │ │ │ -00053430: 3033 3234 3932 3473 2028 7468 7265 6164  0324924s (thread
│ │ │ │ -00053440: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -00053450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053460: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053410: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00053420: 302e 3030 3339 3837 3332 7320 2863 7075  0.00398732s (cpu
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│ │ │ │ +00053440: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +00053450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053460: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00053470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000534a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000534b0: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ -000534c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000534a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000534b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000534c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  000534d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000534e0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000534f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053500: 7c6f 3320 3d20 6964 6561 6c20 2878 2020  |o3 = ideal (x  
│ │ │ │ -00053510: 2d20 7820 7820 2c20 7820 7820 202d 2078  - x x , x x  - x
│ │ │ │ -00053520: 2078 2020 2b20 7820 7820 2c20 7820 7820   x  + x x , x x 
│ │ │ │ -00053530: 202d 2078 2020 2b20 7820 7820 2c20 7820   - x  + x x , x 
│ │ │ │ -00053540: 7820 202d 2078 2078 2020 2b20 2020 7c0a  x  - x x  +   |.
│ │ │ │ -00053550: 7c20 2020 2020 2020 2020 2020 2020 3420  |             4 
│ │ │ │ -00053560: 2020 2033 2035 2020 2032 2034 2020 2020     3 5   2 4    
│ │ │ │ -00053570: 3320 3420 2020 2031 2035 2020 2032 2033  3 4    1 5   2 3
│ │ │ │ -00053580: 2020 2020 3320 2020 2031 2034 2020 2031      3    1 4   1
│ │ │ │ -00053590: 2032 2020 2020 3120 3320 2020 2020 7c0a   2    1 3     |.
│ │ │ │ -000535a0: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +000534e0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +000534f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053500: 2020 2020 7c0a 7c6f 3320 3d20 6964 6561      |.|o3 = idea
│ │ │ │ +00053510: 6c20 2878 2020 2d20 7820 7820 2c20 7820  l (x  - x x , x 
│ │ │ │ +00053520: 7820 202d 2078 2078 2020 2b20 7820 7820  x  - x x  + x x 
│ │ │ │ +00053530: 2c20 7820 7820 202d 2078 2020 2b20 7820  , x x  - x  + x 
│ │ │ │ +00053540: 7820 2c20 7820 7820 202d 2078 2078 2020  x , x x  - x x  
│ │ │ │ +00053550: 2b20 2020 7c0a 7c20 2020 2020 2020 2020  +   |.|         
│ │ │ │ +00053560: 2020 2020 3420 2020 2033 2035 2020 2032      4    3 5   2
│ │ │ │ +00053570: 2034 2020 2020 3320 3420 2020 2031 2035   4    3 4    1 5
│ │ │ │ +00053580: 2020 2032 2033 2020 2020 3320 2020 2031     2 3    3    1
│ │ │ │ +00053590: 2034 2020 2031 2032 2020 2020 3120 3320   4   1 2    1 3 
│ │ │ │ +000535a0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  000535b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000535c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000535d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000535e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -000535f0: 7c20 2020 2020 2020 2020 2020 2032 2020  |            2  
│ │ │ │ -00053600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000535e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000535f0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ +00053600: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  00053610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053630: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053640: 7c20 2020 2020 7820 7820 2c20 7820 202d  |     x x , x  -
│ │ │ │ -00053650: 2078 2078 2029 2020 2020 2020 2020 2020   x x )          
│ │ │ │ +00053630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053640: 2020 2020 7c0a 7c20 2020 2020 7820 7820      |.|     x x 
│ │ │ │ +00053650: 2c20 7820 202d 2078 2078 2029 2020 2020  , x  - x x )    
│ │ │ │  00053660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053680: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053690: 7c20 2020 2020 2030 2034 2020 2031 2020  |      0 4   1  
│ │ │ │ -000536a0: 2020 3020 3320 2020 2020 2020 2020 2020    0 3           
│ │ │ │ +00053680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053690: 2020 2020 7c0a 7c20 2020 2020 2030 2034      |.|      0 4
│ │ │ │ +000536a0: 2020 2031 2020 2020 3020 3320 2020 2020     1    0 3     
│ │ │ │  000536b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000536c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000536d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000536e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000536d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000536e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000536f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053720: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053730: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00053740: 2020 2020 2020 5151 5b78 202e 2e78 205d        QQ[x ..x ]
│ │ │ │ -00053750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053730: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053740: 2020 2020 2020 2020 2020 2020 5151 5b78              QQ[x
│ │ │ │ +00053750: 202e 2e78 205d 2020 2020 2020 2020 2020   ..x ]          
│ │ │ │  00053760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053780: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00053790: 2020 2020 2020 2020 2020 3020 2020 3520            0   5 
│ │ │ │ -000537a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053780: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000537a0: 3020 2020 3520 2020 2020 2020 2020 2020  0   5           
│ │ │ │  000537b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000537c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000537d0: 7c6f 3320 3a20 4964 6561 6c20 6f66 202d  |o3 : Ideal of -
│ │ │ │ -000537e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000537f0: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │ +000537c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000537d0: 2020 2020 7c0a 7c6f 3320 3a20 4964 6561      |.|o3 : Idea
│ │ │ │ +000537e0: 6c20 6f66 202d 2d2d 2d2d 2d2d 2d2d 2d2d  l of -----------
│ │ │ │ +000537f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020  ------------    
│ │ │ │  00053800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053810: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053820: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00053830: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00053810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053820: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053830: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00053840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053860: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053870: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ -00053880: 2020 2d20 7820 7820 202b 2078 2078 2020    - x x  + x x  
│ │ │ │ -00053890: 2d20 7820 7820 2020 2020 2020 2020 2020  - x x           
│ │ │ │ +00053860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053870: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053880: 2020 2020 2078 2020 2d20 7820 7820 202b       x  - x x  +
│ │ │ │ +00053890: 2078 2078 2020 2d20 7820 7820 2020 2020   x x  - x x     
│ │ │ │  000538a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000538b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000538c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000538d0: 3220 2020 2032 2033 2020 2020 3120 3420  2    2 3    1 4 
│ │ │ │ -000538e0: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ +000538b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000538c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000538d0: 2020 2020 2020 3220 2020 2032 2033 2020        2    2 3  
│ │ │ │ +000538e0: 2020 3120 3420 2020 2030 2035 2020 2020    1 4    0 5    
│ │ │ │  000538f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053900: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053910: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00053900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053910: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00053920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00053960: 7c69 3420 3a20 6173 7365 7274 2869 6465  |i4 : assert(ide
│ │ │ │ -00053970: 616c 2070 6869 203d 3d20 6964 6561 6c20  al phi == ideal 
│ │ │ │ -00053980: 6d61 7472 6978 2070 6869 2920 2020 2020  matrix phi)     
│ │ │ │ -00053990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000539a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000539b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00053950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053960: 2d2d 2d2d 2b0a 7c69 3420 3a20 6173 7365  ----+.|i4 : asse
│ │ │ │ +00053970: 7274 2869 6465 616c 2070 6869 203d 3d20  rt(ideal phi == 
│ │ │ │ +00053980: 6964 6561 6c20 6d61 7472 6978 2070 6869  ideal matrix phi
│ │ │ │ +00053990: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +000539a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000539b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000539c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000539d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000539e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000539f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00053a00: 7c69 3520 3a20 7068 6927 203d 206c 6173  |i5 : phi' = las
│ │ │ │ -00053a10: 7420 6772 6170 6820 7068 6920 2020 2020  t graph phi     
│ │ │ │ -00053a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000539f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053a00: 2d2d 2d2d 2b0a 7c69 3520 3a20 7068 6927  ----+.|i5 : phi'
│ │ │ │ +00053a10: 203d 206c 6173 7420 6772 6170 6820 7068   = last graph ph
│ │ │ │ +00053a20: 6920 2020 2020 2020 2020 2020 2020 2020  i               
│ │ │ │  00053a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053a40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053a50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053a50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00053a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053a90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053aa0: 7c6f 3520 3d20 2d2d 2072 6174 696f 6e61  |o5 = -- rationa
│ │ │ │ -00053ab0: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │ +00053a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053aa0: 2020 2020 7c0a 7c6f 3520 3d20 2d2d 2072      |.|o5 = -- r
│ │ │ │ +00053ab0: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │  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 2020 7c0a                |.
│ │ │ │ -00053af0: 7c20 2020 2020 736f 7572 6365 3a20 7375  |     source: su
│ │ │ │ -00053b00: 6276 6172 6965 7479 206f 6620 5072 6f6a  bvariety of Proj
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│ │ │ │ -00053b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053b40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053af0: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ +00053b00: 6365 3a20 7375 6276 6172 6965 7479 206f  ce: subvariety o
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│ │ │ │ +00053b20: 2c20 7820 2c20 2020 2020 2020 2020 2020  , x ,           
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│ │ │ │ +00053b40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00053b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b60: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ -00053b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053b90: 7c20 2020 2020 2020 2020 2020 2020 7b20  |             { 
│ │ │ │ -00053ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00053bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053bd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053be0: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ -00053bf0: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
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│ │ │ │ +00053bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053be0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053bf0: 2020 2020 2078 2079 2020 2d20 7820 7920       x y  - x y 
│ │ │ │ +00053c00: 202b 2078 2079 202c 2020 2020 2020 2020   + x y ,        
│ │ │ │  00053c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053c30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00053c40: 3120 3220 2020 2033 2033 2020 2020 3420  1 2    3 3    4 
│ │ │ │ -00053c50: 3420 2020 2020 2020 2020 2020 2020 2020  4               
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│ │ │ │  00053ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00053cd0: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
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│ │ │ │ +00053cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053cd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053ce0: 2020 2020 2078 2079 2020 2d20 7820 7920       x y  - x y 
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│ │ │ │  00053d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -00053d30: 3020 3220 2020 2031 2033 2020 2020 3220  0 2    1 3    2 
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│ │ │ │ +00053d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00053d30: 2020 2020 2020 3020 3220 2020 2031 2033        0 2    1 3
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│ │ │ │  00053d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053d70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053d70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00053d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053db0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053dc0: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ -00053dd0: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ -00053de0: 2020 2d20 7820 7920 2c20 2020 2020 2020    - x y ,       
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│ │ │ │ +00053dc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ +00053de0: 202b 2078 2079 2020 2d20 7820 7920 2c20   + x y  - x y , 
│ │ │ │  00053df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ +00053e20: 2020 2020 2020 3220 3120 2020 2033 2031        2 1    3 1
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│ │ │ │  00053e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053e60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053e60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00053e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ea0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053eb0: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
│ │ │ │ -00053ec0: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ -00053ed0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +00053ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053eb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053ec0: 2020 2020 2078 2079 2020 2d20 7820 7920       x y  - x y 
│ │ │ │ +00053ed0: 202b 2078 2079 202c 2020 2020 2020 2020   + x y ,        
│ │ │ │  00053ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053f00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00053f10: 3120 3120 2020 2032 2032 2020 2020 3520  1 1    2 2    5 
│ │ │ │ -00053f20: 3420 2020 2020 2020 2020 2020 2020 2020  4               
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│ │ │ │ +00053f00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053f10: 2020 2020 2020 3120 3120 2020 2032 2032        1 1    2 2
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│ │ │ │  00053f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053f50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00053f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053f50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00053f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053fa0: 7c20 2020 2020 2020 2020 2020 2020 2078  |              x
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│ │ │ │ +00053f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053fa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00053fb0: 2020 2020 2078 2079 2020 2d20 7820 7920       x y  - x y 
│ │ │ │ +00053fc0: 202b 2078 2079 202c 2020 2020 2020 2020   + x y ,        
│ │ │ │  00053fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053fe0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00053ff0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00054000: 3020 3120 2020 2032 2033 2020 2020 3420  0 1    2 3    4 
│ │ │ │ -00054010: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +00053fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053ff0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  00055ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055ac0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00055ad0: 7c66 726f 6d20 342d 6469 6d65 6e73 696f  |from 4-dimensio
│ │ │ │ -00055ae0: 6e61 6c20 7375 6276 6172 6965 7479 206f  nal subvariety o
│ │ │ │ -00055af0: 6620 5050 5e35 2078 2050 505e 3420 746f  f PP^5 x PP^4 to
│ │ │ │ -00055b00: 2050 505e 3429 2020 2020 2020 2020 2020   PP^4)          
│ │ │ │ -00055b10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00055b20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00055ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055ad0: 2020 2020 7c0a 7c66 726f 6d20 342d 6469      |.|from 4-di
│ │ │ │ +00055ae0: 6d65 6e73 696f 6e61 6c20 7375 6276 6172  mensional subvar
│ │ │ │ +00055af0: 6965 7479 206f 6620 5050 5e35 2078 2050  iety of PP^5 x P
│ │ │ │ +00055b00: 505e 3420 746f 2050 505e 3429 2020 2020  P^4 to PP^4)    
│ │ │ │ +00055b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055b20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00055b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00055b70: 7c69 3620 3a20 7469 6d65 2069 6465 616c  |i6 : time ideal
│ │ │ │ -00055b80: 2070 6869 2720 2020 2020 2020 2020 2020   phi'           
│ │ │ │ +00055b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055b70: 2d2d 2d2d 2b0a 7c69 3620 3a20 7469 6d65  ----+.|i6 : time
│ │ │ │ +00055b80: 2069 6465 616c 2070 6869 2720 2020 2020   ideal phi'     
│ │ │ │  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 7c0a                |.
│ │ │ │ -00055bc0: 7c20 2d2d 2075 7365 6420 302e 3039 3030  | -- used 0.0900
│ │ │ │ -00055bd0: 3935 3573 2028 6370 7529 3b20 302e 3038  955s (cpu); 0.08
│ │ │ │ -00055be0: 3731 3837 7320 2874 6872 6561 6429 3b20  7187s (thread); 
│ │ │ │ -00055bf0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ -00055c00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00055c10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00055bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055bc0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00055bd0: 302e 3139 3736 3532 7320 2863 7075 293b  0.197652s (cpu);
│ │ │ │ +00055be0: 2030 2e31 3232 3135 7320 2874 6872 6561   0.12215s (threa
│ │ │ │ +00055bf0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00055c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055c10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  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 7c0a                |.
│ │ │ │ -00055c60: 7c6f 3620 3d20 6964 6561 6c20 3120 2020  |o6 = ideal 1   
│ │ │ │ -00055c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055c60: 2020 2020 7c0a 7c6f 3620 3d20 6964 6561      |.|o6 = idea
│ │ │ │ +00055c70: 6c20 3120 2020 2020 2020 2020 2020 2020  l 1             
│ │ │ │  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 7c0a                |.
│ │ │ │ -00055cb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00055ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055cb0: 2020 2020 7c0a 7c20 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 7c0a                |.
│ │ │ │ -00055d00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00055cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055d00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00055d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055d40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00055d50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00055d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055d50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00055d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055d90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00055da0: 7c6f 3620 3a20 4964 6561 6c20 6f66 202d  |o6 : Ideal of -
│ │ │ │ -00055db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055da0: 2020 2020 7c0a 7c6f 3620 3a20 4964 6561      |.|o6 : Idea
│ │ │ │ +00055db0: 6c20 6f66 202d 2d2d 2d2d 2d2d 2d2d 2d2d  l of -----------
│ │ │ │  00055dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00055df0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00055de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055df0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │  00055e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a                |.
│ │ │ │ -00055e40: 7c20 2020 2020 2020 2020 2020 2020 2028  |              (
│ │ │ │ -00055e50: 7820 7920 202d 2078 2079 2020 2b20 7820  x y  - x y  + x 
│ │ │ │ -00055e60: 7920 2c20 7820 7920 202d 2078 2079 2020  y , x y  - x y  
│ │ │ │ -00055e70: 2d20 7820 7920 202b 2078 2079 202c 2078  - x y  + x y , x
│ │ │ │ -00055e80: 2079 2020 2d20 7820 7920 202b 2078 7c0a   y  - x y  + x|.
│ │ │ │ -00055e90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00055ea0: 2031 2032 2020 2020 3320 3320 2020 2034   1 2    3 3    4
│ │ │ │ -00055eb0: 2034 2020 2030 2032 2020 2020 3120 3320   4   0 2    1 3 
│ │ │ │ -00055ec0: 2020 2032 2034 2020 2020 3320 3420 2020     2 4    3 4   
│ │ │ │ -00055ed0: 3220 3120 2020 2033 2031 2020 2020 7c0a  2 1    3 1    |.
│ │ │ │ -00055ee0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00055e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055e40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00055e50: 2020 2020 2028 7820 7920 202d 2078 2079       (x y  - x y
│ │ │ │ +00055e60: 2020 2b20 7820 7920 2c20 7820 7920 202d    + x y , x y  -
│ │ │ │ +00055e70: 2078 2079 2020 2d20 7820 7920 202b 2078   x y  - x y  + x
│ │ │ │ +00055e80: 2079 202c 2078 2079 2020 2d20 7820 7920   y , x y  - x y 
│ │ │ │ +00055e90: 202b 2078 7c0a 7c20 2020 2020 2020 2020   + x|.|         
│ │ │ │ +00055ea0: 2020 2020 2020 2031 2032 2020 2020 3320         1 2    3 
│ │ │ │ +00055eb0: 3320 2020 2034 2034 2020 2030 2032 2020  3    4 4   0 2  
│ │ │ │ +00055ec0: 2020 3120 3320 2020 2032 2034 2020 2020    1 3    2 4    
│ │ │ │ +00055ed0: 3320 3420 2020 3220 3120 2020 2033 2031  3 4   2 1    3 1
│ │ │ │ +00055ee0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │  00055ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00055f30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00055f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055f30: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │  00055f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055f50: 5151 5b78 202e 2e78 202c 2079 202e 2e79  QQ[x ..x , y ..y
│ │ │ │ -00055f60: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ -00055f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00055f80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00055f50: 2020 2020 2020 5151 5b78 202e 2e78 202c        QQ[x ..x ,
│ │ │ │ +00055f60: 2079 202e 2e79 205d 2020 2020 2020 2020   y ..y ]        
│ │ │ │ +00055f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055f80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00055f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055fa0: 2020 2020 3020 2020 3520 2020 3020 2020      0   5   0   
│ │ │ │ -00055fb0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00055fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00055fd0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00055fa0: 2020 2020 2020 2020 2020 3020 2020 3520            0   5 
│ │ │ │ +00055fb0: 2020 3020 2020 3420 2020 2020 2020 2020    0   4         
│ │ │ │ +00055fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00055fd0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │  00055fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00056020: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00056010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056020: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │  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 7c0a                |.
│ │ │ │ -00056070: 7c20 7920 202d 2078 2079 202c 2078 2079  | y  - x y , x y
│ │ │ │ -00056080: 2020 2d20 7820 7920 202b 2078 2079 202c    - x y  + x y ,
│ │ │ │ -00056090: 2078 2079 2020 2d20 7820 7920 202b 2078   x y  - x y  + x
│ │ │ │ -000560a0: 2079 202c 2078 2079 2020 2d20 7820 7920   y , x y  - x y 
│ │ │ │ -000560b0: 202b 2078 2079 2020 2d20 7820 7920 7c0a   + x y  - x y |.
│ │ │ │ -000560c0: 7c34 2032 2020 2020 3520 3320 2020 3120  |4 2    5 3   1 
│ │ │ │ -000560d0: 3120 2020 2032 2032 2020 2020 3520 3420  1    2 2    5 4 
│ │ │ │ -000560e0: 2020 3020 3120 2020 2032 2033 2020 2020    0 1    2 3    
│ │ │ │ -000560f0: 3420 3420 2020 3220 3020 2020 2033 2030  4 4   2 0    3 0
│ │ │ │ -00056100: 2020 2020 3420 3120 2020 2035 2032 7c0a      4 1    5 2|.
│ │ │ │ -00056110: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00056060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056070: 2020 2020 7c0a 7c20 7920 202d 2078 2079      |.| y  - x y
│ │ │ │ +00056080: 202c 2078 2079 2020 2d20 7820 7920 202b   , x y  - x y  +
│ │ │ │ +00056090: 2078 2079 202c 2078 2079 2020 2d20 7820   x y , x y  - x 
│ │ │ │ +000560a0: 7920 202b 2078 2079 202c 2078 2079 2020  y  + x y , x y  
│ │ │ │ +000560b0: 2d20 7820 7920 202b 2078 2079 2020 2d20  - x y  + x y  - 
│ │ │ │ +000560c0: 7820 7920 7c0a 7c34 2032 2020 2020 3520  x y |.|4 2    5 
│ │ │ │ +000560d0: 3320 2020 3120 3120 2020 2032 2032 2020  3   1 1    2 2  
│ │ │ │ +000560e0: 2020 3520 3420 2020 3020 3120 2020 2032    5 4   0 1    2
│ │ │ │ +000560f0: 2033 2020 2020 3420 3420 2020 3220 3020   3    4 4   2 0 
│ │ │ │ +00056100: 2020 2033 2030 2020 2020 3420 3120 2020     3 0    4 1   
│ │ │ │ +00056110: 2035 2032 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d   5 2|.|---------
│ │ │ │  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 7c0a  --------------|.
│ │ │ │ -00056160: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00056150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056160: 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  ----|.|---------
│ │ │ │  00056170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000561a0: 2d2d 2d20 2020 2020 2020 2020 2020 7c0a  ---           |.
│ │ │ │ -000561b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000561a0: 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020 2020  ---------       
│ │ │ │ +000561b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000561c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000561d0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000561e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000561f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00056200: 7c2c 2078 2079 2020 2d20 7820 7920 202b  |, x y  - x y  +
│ │ │ │ -00056210: 2078 2079 202c 2078 2079 2020 2d20 7820   x y , x y  - x 
│ │ │ │ -00056220: 7920 202b 2078 2079 202c 2078 2020 2d20  y  + x y , x  - 
│ │ │ │ -00056230: 7820 7820 202b 2078 2078 2020 2d20 7820  x x  + x x  - x 
│ │ │ │ -00056240: 7820 2920 2020 2020 2020 2020 2020 7c0a  x )           |.
│ │ │ │ -00056250: 7c20 2020 3120 3020 2020 2033 2031 2020  |   1 0    3 1  
│ │ │ │ -00056260: 2020 3420 3220 2020 3020 3020 2020 2032    4 2   0 0    2
│ │ │ │ -00056270: 2032 2020 2020 3420 3320 2020 3220 2020   2    4 3   2   
│ │ │ │ -00056280: 2032 2033 2020 2020 3120 3420 2020 2030   2 3    1 4    0
│ │ │ │ -00056290: 2035 2020 2020 2020 2020 2020 2020 7c0a   5            |.
│ │ │ │ -000562a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000561d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000561e0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000561f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056200: 2020 2020 7c0a 7c2c 2078 2079 2020 2d20      |.|, x y  - 
│ │ │ │ +00056210: 7820 7920 202b 2078 2079 202c 2078 2079  x y  + x y , x y
│ │ │ │ +00056220: 2020 2d20 7820 7920 202b 2078 2079 202c    - x y  + x y ,
│ │ │ │ +00056230: 2078 2020 2d20 7820 7820 202b 2078 2078   x  - x x  + x x
│ │ │ │ +00056240: 2020 2d20 7820 7820 2920 2020 2020 2020    - x x )       
│ │ │ │ +00056250: 2020 2020 7c0a 7c20 2020 3120 3020 2020      |.|   1 0   
│ │ │ │ +00056260: 2033 2031 2020 2020 3420 3220 2020 3020   3 1    4 2   0 
│ │ │ │ +00056270: 3020 2020 2032 2032 2020 2020 3420 3320  0    2 2    4 3 
│ │ │ │ +00056280: 2020 3220 2020 2032 2033 2020 2020 3120    2    2 3    1 
│ │ │ │ +00056290: 3420 2020 2030 2035 2020 2020 2020 2020  4    0 5        
│ │ │ │ +000562a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000562b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000562c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000562d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000562e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000562f0: 7c69 3720 3a20 6173 7365 7274 2869 6465  |i7 : assert(ide
│ │ │ │ -00056300: 616c 2070 6869 2720 213d 2069 6465 616c  al phi' != ideal
│ │ │ │ -00056310: 206d 6174 7269 7820 7068 6927 2920 2020   matrix phi')   
│ │ │ │ -00056320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056330: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00056340: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000562e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000562f0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6173 7365  ----+.|i7 : asse
│ │ │ │ +00056300: 7274 2869 6465 616c 2070 6869 2720 213d  rt(ideal phi' !=
│ │ │ │ +00056310: 2069 6465 616c 206d 6174 7269 7820 7068   ideal matrix ph
│ │ │ │ +00056320: 6927 2920 2020 2020 2020 2020 2020 2020  i')             
│ │ │ │ +00056330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056340: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00056350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00056390: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -000563a0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6973  ==..  * *note is
│ │ │ │ -000563b0: 4d6f 7270 6869 736d 3a20 6973 4d6f 7270  Morphism: isMorp
│ │ │ │ -000563c0: 6869 736d 2c20 2d2d 2077 6865 7468 6572  hism, -- whether
│ │ │ │ -000563d0: 2061 2072 6174 696f 6e61 6c20 6d61 7020   a rational map 
│ │ │ │ -000563e0: 6973 2061 206d 6f72 7068 6973 6d0a 2020  is a morphism.  
│ │ │ │ -000563f0: 2a20 2a6e 6f74 6520 6d61 7472 6978 2852  * *note matrix(R
│ │ │ │ -00056400: 6174 696f 6e61 6c4d 6170 293a 206d 6174  ationalMap): mat
│ │ │ │ -00056410: 7269 785f 6c70 5261 7469 6f6e 616c 4d61  rix_lpRationalMa
│ │ │ │ -00056420: 705f 7270 2c20 2d2d 2074 6865 206d 6174  p_rp, -- the mat
│ │ │ │ -00056430: 7269 780a 2020 2020 6173 736f 6369 6174  rix.    associat
│ │ │ │ -00056440: 6564 2074 6f20 6120 7261 7469 6f6e 616c  ed to a rational
│ │ │ │ -00056450: 206d 6170 0a0a 5761 7973 2074 6f20 7573   map..Ways to us
│ │ │ │ -00056460: 6520 7468 6973 206d 6574 686f 643a 0a3d  e this method:.=
│ │ │ │ -00056470: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00056480: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ -00056490: 7465 2069 6465 616c 2852 6174 696f 6e61  te ideal(Rationa
│ │ │ │ -000564a0: 6c4d 6170 293a 2069 6465 616c 5f6c 7052  lMap): ideal_lpR
│ │ │ │ -000564b0: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ -000564c0: 2d20 6261 7365 206c 6f63 7573 206f 6620  - base locus of 
│ │ │ │ -000564d0: 610a 2020 2020 7261 7469 6f6e 616c 206d  a.    rational m
│ │ │ │ -000564e0: 6170 0a1f 0a46 696c 653a 2043 7265 6d6f  ap...File: Cremo
│ │ │ │ -000564f0: 6e61 2e69 6e66 6f2c 204e 6f64 653a 2069  na.info, Node: i
│ │ │ │ -00056500: 6d61 6765 5f6c 7052 6174 696f 6e61 6c4d  mage_lpRationalM
│ │ │ │ -00056510: 6170 5f63 6d53 7472 696e 675f 7270 2c20  ap_cmString_rp, 
│ │ │ │ -00056520: 4e65 7874 3a20 696d 6167 655f 6c70 5261  Next: image_lpRa
│ │ │ │ -00056530: 7469 6f6e 616c 4d61 705f 636d 5a5a 5f72  tionalMap_cmZZ_r
│ │ │ │ -00056540: 702c 2050 7265 763a 2069 6465 616c 5f6c  p, Prev: ideal_l
│ │ │ │ -00056550: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -00056560: 2055 703a 2054 6f70 0a0a 696d 6167 6528   Up: Top..image(
│ │ │ │ -00056570: 5261 7469 6f6e 616c 4d61 702c 5374 7269  RationalMap,Stri
│ │ │ │ -00056580: 6e67 2920 2d2d 2063 6c6f 7375 7265 206f  ng) -- closure o
│ │ │ │ -00056590: 6620 7468 6520 696d 6167 6520 6f66 2061  f the image of a
│ │ │ │ -000565a0: 2072 6174 696f 6e61 6c20 6d61 7020 7573   rational map us
│ │ │ │ -000565b0: 696e 6720 7468 6520 4634 2061 6c67 6f72  ing the F4 algor
│ │ │ │ -000565c0: 6974 686d 2028 6578 7065 7269 6d65 6e74  ithm (experiment
│ │ │ │ -000565d0: 616c 290a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  al).************
│ │ │ │ +00056380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056390: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ +000563a0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +000563b0: 6f74 6520 6973 4d6f 7270 6869 736d 3a20  ote isMorphism: 
│ │ │ │ +000563c0: 6973 4d6f 7270 6869 736d 2c20 2d2d 2077  isMorphism, -- w
│ │ │ │ +000563d0: 6865 7468 6572 2061 2072 6174 696f 6e61  hether a rationa
│ │ │ │ +000563e0: 6c20 6d61 7020 6973 2061 206d 6f72 7068  l map is a morph
│ │ │ │ +000563f0: 6973 6d0a 2020 2a20 2a6e 6f74 6520 6d61  ism.  * *note ma
│ │ │ │ +00056400: 7472 6978 2852 6174 696f 6e61 6c4d 6170  trix(RationalMap
│ │ │ │ +00056410: 293a 206d 6174 7269 785f 6c70 5261 7469  ): matrix_lpRati
│ │ │ │ +00056420: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2074  onalMap_rp, -- t
│ │ │ │ +00056430: 6865 206d 6174 7269 780a 2020 2020 6173  he matrix.    as
│ │ │ │ +00056440: 736f 6369 6174 6564 2074 6f20 6120 7261  sociated to a ra
│ │ │ │ +00056450: 7469 6f6e 616c 206d 6170 0a0a 5761 7973  tional map..Ways
│ │ │ │ +00056460: 2074 6f20 7573 6520 7468 6973 206d 6574   to use this met
│ │ │ │ +00056470: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  hod:.===========
│ │ │ │ +00056480: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00056490: 202a 202a 6e6f 7465 2069 6465 616c 2852   * *note ideal(R
│ │ │ │ +000564a0: 6174 696f 6e61 6c4d 6170 293a 2069 6465  ationalMap): ide
│ │ │ │ +000564b0: 616c 5f6c 7052 6174 696f 6e61 6c4d 6170  al_lpRationalMap
│ │ │ │ +000564c0: 5f72 702c 202d 2d20 6261 7365 206c 6f63  _rp, -- base loc
│ │ │ │ +000564d0: 7573 206f 6620 610a 2020 2020 7261 7469  us of a.    rati
│ │ │ │ +000564e0: 6f6e 616c 206d 6170 0a1f 0a46 696c 653a  onal map...File:
│ │ │ │ +000564f0: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ +00056500: 6f64 653a 2069 6d61 6765 5f6c 7052 6174  ode: image_lpRat
│ │ │ │ +00056510: 696f 6e61 6c4d 6170 5f63 6d53 7472 696e  ionalMap_cmStrin
│ │ │ │ +00056520: 675f 7270 2c20 4e65 7874 3a20 696d 6167  g_rp, Next: imag
│ │ │ │ +00056530: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +00056540: 636d 5a5a 5f72 702c 2050 7265 763a 2069  cmZZ_rp, Prev: i
│ │ │ │ +00056550: 6465 616c 5f6c 7052 6174 696f 6e61 6c4d  deal_lpRationalM
│ │ │ │ +00056560: 6170 5f72 702c 2055 703a 2054 6f70 0a0a  ap_rp, Up: Top..
│ │ │ │ +00056570: 696d 6167 6528 5261 7469 6f6e 616c 4d61  image(RationalMa
│ │ │ │ +00056580: 702c 5374 7269 6e67 2920 2d2d 2063 6c6f  p,String) -- clo
│ │ │ │ +00056590: 7375 7265 206f 6620 7468 6520 696d 6167  sure of the imag
│ │ │ │ +000565a0: 6520 6f66 2061 2072 6174 696f 6e61 6c20  e of a rational 
│ │ │ │ +000565b0: 6d61 7020 7573 696e 6720 7468 6520 4634  map using the F4
│ │ │ │ +000565c0: 2061 6c67 6f72 6974 686d 2028 6578 7065   algorithm (expe
│ │ │ │ +000565d0: 7269 6d65 6e74 616c 290a 2a2a 2a2a 2a2a  rimental).******
│ │ │ │  000565e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000565f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00056600: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00056610: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00056620: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00056630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ -00056640: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ -00056650: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -00056660: 2a6e 6f74 6520 696d 6167 653a 2028 4d61  *note image: (Ma
│ │ │ │ -00056670: 6361 756c 6179 3244 6f63 2969 6d61 6765  caulay2Doc)image
│ │ │ │ -00056680: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ -00056690: 2020 2020 2020 696d 6167 6528 5068 692c        image(Phi,
│ │ │ │ -000566a0: 2246 3422 290a 2020 2020 2020 2020 696d  "F4").        im
│ │ │ │ -000566b0: 6167 6528 5068 692c 224d 4742 2229 0a20  age(Phi,"MGB"). 
│ │ │ │ -000566c0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -000566d0: 202a 2050 6869 2c20 6120 2a6e 6f74 6520   * Phi, a *note 
│ │ │ │ -000566e0: 7261 7469 6f6e 616c 206d 6170 3a20 5261  rational map: Ra
│ │ │ │ -000566f0: 7469 6f6e 616c 4d61 702c 0a20 2020 2020  tionalMap,.     
│ │ │ │ -00056700: 202a 2022 4634 2220 6f72 2022 4d47 4222   * "F4" or "MGB"
│ │ │ │ -00056710: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00056720: 2020 2020 2a20 7468 6520 2a6e 6f74 6520      * the *note 
│ │ │ │ -00056730: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ -00056740: 3244 6f63 2949 6465 616c 2c20 6465 6669  2Doc)Ideal, defi
│ │ │ │ -00056750: 6e69 6e67 2074 6865 2063 6c6f 7375 7265  ning the closure
│ │ │ │ -00056760: 206f 6620 7468 6520 696d 6167 650a 2020   of the image.  
│ │ │ │ -00056770: 2020 2020 2020 6f66 2050 6869 3b20 7468        of Phi; th
│ │ │ │ -00056780: 6520 6361 6c63 756c 6174 696f 6e20 7061  e calculation pa
│ │ │ │ -00056790: 7373 6573 2074 6872 6f75 6768 202a 6e6f  sses through *no
│ │ │ │ -000567a0: 7465 2067 726f 6562 6e65 7242 6173 6973  te groebnerBasis
│ │ │ │ -000567b0: 3a0a 2020 2020 2020 2020 284d 6163 6175  :.        (Macau
│ │ │ │ -000567c0: 6c61 7932 446f 6329 6772 6f65 626e 6572  lay2Doc)groebner
│ │ │ │ -000567d0: 4261 7369 732c 282e 2e2e 2c53 7472 6174  Basis,(...,Strat
│ │ │ │ -000567e0: 6567 793d 3e22 4634 2229 206f 7220 2a6e  egy=>"F4") or *n
│ │ │ │ -000567f0: 6f74 650a 2020 2020 2020 2020 6772 6f65  ote.        groe
│ │ │ │ -00056800: 626e 6572 4261 7369 733a 2028 4d61 6361  bnerBasis: (Maca
│ │ │ │ -00056810: 756c 6179 3244 6f63 2967 726f 6562 6e65  ulay2Doc)groebne
│ │ │ │ -00056820: 7242 6173 6973 2c28 2e2e 2e2c 5374 7261  rBasis,(...,Stra
│ │ │ │ -00056830: 7465 6779 3d3e 224d 4742 2229 2e0a 0a53  tegy=>"MGB")...S
│ │ │ │ -00056840: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -00056850: 0a0a 2020 2a20 2a6e 6f74 6520 696d 6167  ..  * *note imag
│ │ │ │ -00056860: 6528 5261 7469 6f6e 616c 4d61 7029 3a20  e(RationalMap): 
│ │ │ │ -00056870: 696d 6167 655f 6c70 5261 7469 6f6e 616c  image_lpRational
│ │ │ │ -00056880: 4d61 705f 636d 5a5a 5f72 702c 202d 2d20  Map_cmZZ_rp, -- 
│ │ │ │ -00056890: 636c 6f73 7572 6520 6f66 2074 6865 0a20  closure of the. 
│ │ │ │ -000568a0: 2020 2069 6d61 6765 206f 6620 6120 7261     image of a ra
│ │ │ │ -000568b0: 7469 6f6e 616c 206d 6170 0a20 202a 202a  tional map.  * *
│ │ │ │ -000568c0: 6e6f 7465 2069 6d61 6765 2852 6174 696f  note image(Ratio
│ │ │ │ -000568d0: 6e61 6c4d 6170 2c5a 5a29 3a20 696d 6167  nalMap,ZZ): imag
│ │ │ │ -000568e0: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ -000568f0: 636d 5a5a 5f72 702c 202d 2d20 636c 6f73  cmZZ_rp, -- clos
│ │ │ │ -00056900: 7572 6520 6f66 2074 6865 0a20 2020 2069  ure of the.    i
│ │ │ │ -00056910: 6d61 6765 206f 6620 6120 7261 7469 6f6e  mage of a ration
│ │ │ │ -00056920: 616c 206d 6170 0a20 202a 202a 6e6f 7465  al map.  * *note
│ │ │ │ -00056930: 2067 726f 6562 6e65 7242 6173 6973 3a20   groebnerBasis: 
│ │ │ │ -00056940: 284d 6163 6175 6c61 7932 446f 6329 6772  (Macaulay2Doc)gr
│ │ │ │ -00056950: 6f65 626e 6572 4261 7369 732c 202d 2d20  oebnerBasis, -- 
│ │ │ │ -00056960: 4772 c3b6 626e 6572 2062 6173 6973 2c20  Gr..bner basis, 
│ │ │ │ -00056970: 6173 2061 0a20 2020 206d 6174 7269 780a  as a.    matrix.
│ │ │ │ -00056980: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ -00056990: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ +00056630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00056640: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ +00056650: 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675 6e63  ======..  * Func
│ │ │ │ +00056660: 7469 6f6e 3a20 2a6e 6f74 6520 696d 6167  tion: *note imag
│ │ │ │ +00056670: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +00056680: 2969 6d61 6765 2c0a 2020 2a20 5573 6167  )image,.  * Usag
│ │ │ │ +00056690: 653a 200a 2020 2020 2020 2020 696d 6167  e: .        imag
│ │ │ │ +000566a0: 6528 5068 692c 2246 3422 290a 2020 2020  e(Phi,"F4").    
│ │ │ │ +000566b0: 2020 2020 696d 6167 6528 5068 692c 224d      image(Phi,"M
│ │ │ │ +000566c0: 4742 2229 0a20 202a 2049 6e70 7574 733a  GB").  * Inputs:
│ │ │ │ +000566d0: 0a20 2020 2020 202a 2050 6869 2c20 6120  .      * Phi, a 
│ │ │ │ +000566e0: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +000566f0: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ +00056700: 0a20 2020 2020 202a 2022 4634 2220 6f72  .      * "F4" or
│ │ │ │ +00056710: 2022 4d47 4222 0a20 202a 204f 7574 7075   "MGB".  * Outpu
│ │ │ │ +00056720: 7473 3a0a 2020 2020 2020 2a20 7468 6520  ts:.      * the 
│ │ │ │ +00056730: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ +00056740: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ +00056750: 2c20 6465 6669 6e69 6e67 2074 6865 2063  , defining the c
│ │ │ │ +00056760: 6c6f 7375 7265 206f 6620 7468 6520 696d  losure of the im
│ │ │ │ +00056770: 6167 650a 2020 2020 2020 2020 6f66 2050  age.        of P
│ │ │ │ +00056780: 6869 3b20 7468 6520 6361 6c63 756c 6174  hi; the calculat
│ │ │ │ +00056790: 696f 6e20 7061 7373 6573 2074 6872 6f75  ion passes throu
│ │ │ │ +000567a0: 6768 202a 6e6f 7465 2067 726f 6562 6e65  gh *note groebne
│ │ │ │ +000567b0: 7242 6173 6973 3a0a 2020 2020 2020 2020  rBasis:.        
│ │ │ │ +000567c0: 284d 6163 6175 6c61 7932 446f 6329 6772  (Macaulay2Doc)gr
│ │ │ │ +000567d0: 6f65 626e 6572 4261 7369 732c 282e 2e2e  oebnerBasis,(...
│ │ │ │ +000567e0: 2c53 7472 6174 6567 793d 3e22 4634 2229  ,Strategy=>"F4")
│ │ │ │ +000567f0: 206f 7220 2a6e 6f74 650a 2020 2020 2020   or *note.      
│ │ │ │ +00056800: 2020 6772 6f65 626e 6572 4261 7369 733a    groebnerBasis:
│ │ │ │ +00056810: 2028 4d61 6361 756c 6179 3244 6f63 2967   (Macaulay2Doc)g
│ │ │ │ +00056820: 726f 6562 6e65 7242 6173 6973 2c28 2e2e  roebnerBasis,(..
│ │ │ │ +00056830: 2e2c 5374 7261 7465 6779 3d3e 224d 4742  .,Strategy=>"MGB
│ │ │ │ +00056840: 2229 2e0a 0a53 6565 2061 6c73 6f0a 3d3d  ")...See also.==
│ │ │ │ +00056850: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +00056860: 6520 696d 6167 6528 5261 7469 6f6e 616c  e image(Rational
│ │ │ │ +00056870: 4d61 7029 3a20 696d 6167 655f 6c70 5261  Map): image_lpRa
│ │ │ │ +00056880: 7469 6f6e 616c 4d61 705f 636d 5a5a 5f72  tionalMap_cmZZ_r
│ │ │ │ +00056890: 702c 202d 2d20 636c 6f73 7572 6520 6f66  p, -- closure of
│ │ │ │ +000568a0: 2074 6865 0a20 2020 2069 6d61 6765 206f   the.    image o
│ │ │ │ +000568b0: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ +000568c0: 0a20 202a 202a 6e6f 7465 2069 6d61 6765  .  * *note image
│ │ │ │ +000568d0: 2852 6174 696f 6e61 6c4d 6170 2c5a 5a29  (RationalMap,ZZ)
│ │ │ │ +000568e0: 3a20 696d 6167 655f 6c70 5261 7469 6f6e  : image_lpRation
│ │ │ │ +000568f0: 616c 4d61 705f 636d 5a5a 5f72 702c 202d  alMap_cmZZ_rp, -
│ │ │ │ +00056900: 2d20 636c 6f73 7572 6520 6f66 2074 6865  - closure of the
│ │ │ │ +00056910: 0a20 2020 2069 6d61 6765 206f 6620 6120  .    image of a 
│ │ │ │ +00056920: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ +00056930: 202a 6e6f 7465 2067 726f 6562 6e65 7242   *note groebnerB
│ │ │ │ +00056940: 6173 6973 3a20 284d 6163 6175 6c61 7932  asis: (Macaulay2
│ │ │ │ +00056950: 446f 6329 6772 6f65 626e 6572 4261 7369  Doc)groebnerBasi
│ │ │ │ +00056960: 732c 202d 2d20 4772 c3b6 626e 6572 2062  s, -- Gr..bner b
│ │ │ │ +00056970: 6173 6973 2c20 6173 2061 0a20 2020 206d  asis, as a.    m
│ │ │ │ +00056980: 6174 7269 780a 0a57 6179 7320 746f 2075  atrix..Ways to u
│ │ │ │ +00056990: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │  000569a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000569b0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 696d  ==..  * *note im
│ │ │ │ -000569c0: 6167 6528 5261 7469 6f6e 616c 4d61 702c  age(RationalMap,
│ │ │ │ -000569d0: 5374 7269 6e67 293a 2069 6d61 6765 5f6c  String): image_l
│ │ │ │ -000569e0: 7052 6174 696f 6e61 6c4d 6170 5f63 6d53  pRationalMap_cmS
│ │ │ │ -000569f0: 7472 696e 675f 7270 2c20 2d2d 0a20 2020  tring_rp, --.   
│ │ │ │ -00056a00: 2063 6c6f 7375 7265 206f 6620 7468 6520   closure of the 
│ │ │ │ -00056a10: 696d 6167 6520 6f66 2061 2072 6174 696f  image of a ratio
│ │ │ │ -00056a20: 6e61 6c20 6d61 7020 7573 696e 6720 7468  nal map using th
│ │ │ │ -00056a30: 6520 4634 2061 6c67 6f72 6974 686d 0a20  e F4 algorithm. 
│ │ │ │ -00056a40: 2020 2028 6578 7065 7269 6d65 6e74 616c     (experimental
│ │ │ │ -00056a50: 290a 1f0a 4669 6c65 3a20 4372 656d 6f6e  )...File: Cremon
│ │ │ │ -00056a60: 612e 696e 666f 2c20 4e6f 6465 3a20 696d  a.info, Node: im
│ │ │ │ -00056a70: 6167 655f 6c70 5261 7469 6f6e 616c 4d61  age_lpRationalMa
│ │ │ │ -00056a80: 705f 636d 5a5a 5f72 702c 204e 6578 743a  p_cmZZ_rp, Next:
│ │ │ │ -00056a90: 2069 6e76 6572 7365 5f6c 7052 6174 696f   inverse_lpRatio
│ │ │ │ -00056aa0: 6e61 6c4d 6170 5f72 702c 2050 7265 763a  nalMap_rp, Prev:
│ │ │ │ -00056ab0: 2069 6d61 6765 5f6c 7052 6174 696f 6e61   image_lpRationa
│ │ │ │ -00056ac0: 6c4d 6170 5f63 6d53 7472 696e 675f 7270  lMap_cmString_rp
│ │ │ │ -00056ad0: 2c20 5570 3a20 546f 700a 0a69 6d61 6765  , Up: Top..image
│ │ │ │ -00056ae0: 2852 6174 696f 6e61 6c4d 6170 2c5a 5a29  (RationalMap,ZZ)
│ │ │ │ -00056af0: 202d 2d20 636c 6f73 7572 6520 6f66 2074   -- closure of t
│ │ │ │ -00056b00: 6865 2069 6d61 6765 206f 6620 6120 7261  he image of a ra
│ │ │ │ -00056b10: 7469 6f6e 616c 206d 6170 0a2a 2a2a 2a2a  tional map.*****
│ │ │ │ -00056b20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000569b0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +000569c0: 6f74 6520 696d 6167 6528 5261 7469 6f6e  ote image(Ration
│ │ │ │ +000569d0: 616c 4d61 702c 5374 7269 6e67 293a 2069  alMap,String): i
│ │ │ │ +000569e0: 6d61 6765 5f6c 7052 6174 696f 6e61 6c4d  mage_lpRationalM
│ │ │ │ +000569f0: 6170 5f63 6d53 7472 696e 675f 7270 2c20  ap_cmString_rp, 
│ │ │ │ +00056a00: 2d2d 0a20 2020 2063 6c6f 7375 7265 206f  --.    closure o
│ │ │ │ +00056a10: 6620 7468 6520 696d 6167 6520 6f66 2061  f the image of a
│ │ │ │ +00056a20: 2072 6174 696f 6e61 6c20 6d61 7020 7573   rational map us
│ │ │ │ +00056a30: 696e 6720 7468 6520 4634 2061 6c67 6f72  ing the F4 algor
│ │ │ │ +00056a40: 6974 686d 0a20 2020 2028 6578 7065 7269  ithm.    (experi
│ │ │ │ +00056a50: 6d65 6e74 616c 290a 1f0a 4669 6c65 3a20  mental)...File: 
│ │ │ │ +00056a60: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ +00056a70: 6465 3a20 696d 6167 655f 6c70 5261 7469  de: image_lpRati
│ │ │ │ +00056a80: 6f6e 616c 4d61 705f 636d 5a5a 5f72 702c  onalMap_cmZZ_rp,
│ │ │ │ +00056a90: 204e 6578 743a 2069 6e76 6572 7365 5f6c   Next: inverse_l
│ │ │ │ +00056aa0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +00056ab0: 2050 7265 763a 2069 6d61 6765 5f6c 7052   Prev: image_lpR
│ │ │ │ +00056ac0: 6174 696f 6e61 6c4d 6170 5f63 6d53 7472  ationalMap_cmStr
│ │ │ │ +00056ad0: 696e 675f 7270 2c20 5570 3a20 546f 700a  ing_rp, Up: Top.
│ │ │ │ +00056ae0: 0a69 6d61 6765 2852 6174 696f 6e61 6c4d  .image(RationalM
│ │ │ │ +00056af0: 6170 2c5a 5a29 202d 2d20 636c 6f73 7572  ap,ZZ) -- closur
│ │ │ │ +00056b00: 6520 6f66 2074 6865 2069 6d61 6765 206f  e of the image o
│ │ │ │ +00056b10: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ +00056b20: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  00056b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00056b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00056b50: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ -00056b60: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ -00056b70: 202a 2046 756e 6374 696f 6e3a 202a 6e6f   * Function: *no
│ │ │ │ -00056b80: 7465 2069 6d61 6765 3a20 284d 6163 6175  te image: (Macau
│ │ │ │ -00056b90: 6c61 7932 446f 6329 696d 6167 652c 0a20  lay2Doc)image,. 
│ │ │ │ -00056ba0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -00056bb0: 2020 2069 6d61 6765 2050 6869 200a 2020     image Phi .  
│ │ │ │ -00056bc0: 2020 2020 2020 696d 6167 6528 5068 692c        image(Phi,
│ │ │ │ -00056bd0: 6429 0a20 202a 2049 6e70 7574 733a 0a20  d).  * Inputs:. 
│ │ │ │ -00056be0: 2020 2020 202a 2050 6869 2c20 6120 2a6e       * Phi, a *n
│ │ │ │ -00056bf0: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ -00056c00: 3a20 5261 7469 6f6e 616c 4d61 702c 0a20  : RationalMap,. 
│ │ │ │ -00056c10: 2020 2020 202a 2064 2c20 616e 202a 6e6f       * d, an *no
│ │ │ │ -00056c20: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ -00056c30: 6175 6c61 7932 446f 6329 5a5a 2c0a 2020  aulay2Doc)ZZ,.  
│ │ │ │ -00056c40: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -00056c50: 202a 2061 6e20 2a6e 6f74 6520 6964 6561   * an *note idea
│ │ │ │ -00056c60: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ -00056c70: 2949 6465 616c 2c2c 2074 6865 2069 6465  )Ideal,, the ide
│ │ │ │ -00056c80: 616c 2064 6566 696e 696e 6720 7468 6520  al defining the 
│ │ │ │ -00056c90: 636c 6f73 7572 6520 6f66 0a20 2020 2020  closure of.     
│ │ │ │ -00056ca0: 2020 2074 6865 2069 6d61 6765 206f 6620     the image of 
│ │ │ │ -00056cb0: 5068 692c 206f 7220 6974 7320 6465 6772  Phi, or its degr
│ │ │ │ -00056cc0: 6565 2064 2068 6f6d 6f67 656e 656f 7573  ee d homogeneous
│ │ │ │ -00056cd0: 2063 6f6d 706f 6e65 6e74 2069 6620 6420   component if d 
│ │ │ │ -00056ce0: 6973 2070 6173 7365 640a 0a44 6573 6372  is passed..Descr
│ │ │ │ -00056cf0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -00056d00: 3d3d 0a0a 5468 6973 2063 6f6d 7075 7461  ==..This computa
│ │ │ │ -00056d10: 7469 6f6e 2069 7320 646f 6e65 2074 6872  tion is done thr
│ │ │ │ -00056d20: 6f75 6768 2074 6865 206b 6572 6e65 6c20  ough the kernel 
│ │ │ │ -00056d30: 6f66 2061 2072 696e 6720 6d61 7020 7265  of a ring map re
│ │ │ │ -00056d40: 7072 6573 656e 7469 6e67 2074 6865 0a72  presenting the.r
│ │ │ │ -00056d50: 6174 696f 6e61 6c20 6d61 702e 2053 6565  ational map. See
│ │ │ │ -00056d60: 202a 6e6f 7465 206b 6572 6e65 6c28 5269   *note kernel(Ri
│ │ │ │ -00056d70: 6e67 4d61 7029 3a20 284d 6163 6175 6c61  ngMap): (Macaula
│ │ │ │ -00056d80: 7932 446f 6329 6b65 726e 656c 5f6c 7052  y2Doc)kernel_lpR
│ │ │ │ -00056d90: 696e 674d 6170 5f72 702c 2061 6e64 0a2a  ingMap_rp, and.*
│ │ │ │ -00056da0: 6e6f 7465 206b 6572 6e65 6c28 5269 6e67  note kernel(Ring
│ │ │ │ -00056db0: 4d61 702c 5a5a 293a 206b 6572 6e65 6c5f  Map,ZZ): kernel_
│ │ │ │ -00056dc0: 6c70 5269 6e67 4d61 705f 636d 5a5a 5f72  lpRingMap_cmZZ_r
│ │ │ │ -00056dd0: 702c 2066 6f72 206d 6f72 6520 6465 7461  p, for more deta
│ │ │ │ -00056de0: 696c 732e 0a0a 5365 6520 616c 736f 0a3d  ils...See also.=
│ │ │ │ -00056df0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 206b 6572  =======..  * ker
│ │ │ │ -00056e00: 6e65 6c28 5269 6e67 4d61 702c 5a5a 2920  nel(RingMap,ZZ) 
│ │ │ │ -00056e10: 286d 6973 7369 6e67 2064 6f63 756d 656e  (missing documen
│ │ │ │ -00056e20: 7461 7469 6f6e 290a 2020 2a20 2a6e 6f74  tation).  * *not
│ │ │ │ -00056e30: 6520 6b65 726e 656c 2852 696e 674d 6170  e kernel(RingMap
│ │ │ │ -00056e40: 293a 2028 4d61 6361 756c 6179 3244 6f63  ): (Macaulay2Doc
│ │ │ │ -00056e50: 296b 6572 6e65 6c5f 6c70 5269 6e67 4d61  )kernel_lpRingMa
│ │ │ │ -00056e60: 705f 7270 2c20 2d2d 206b 6572 6e65 6c20  p_rp, -- kernel 
│ │ │ │ -00056e70: 6f66 2061 0a20 2020 2072 696e 676d 6170  of a.    ringmap
│ │ │ │ -00056e80: 0a20 202a 202a 6e6f 7465 2069 6d61 6765  .  * *note image
│ │ │ │ -00056e90: 2852 6174 696f 6e61 6c4d 6170 2c53 7472  (RationalMap,Str
│ │ │ │ -00056ea0: 696e 6729 3a20 696d 6167 655f 6c70 5261  ing): image_lpRa
│ │ │ │ -00056eb0: 7469 6f6e 616c 4d61 705f 636d 5374 7269  tionalMap_cmStri
│ │ │ │ -00056ec0: 6e67 5f72 702c 202d 2d0a 2020 2020 636c  ng_rp, --.    cl
│ │ │ │ -00056ed0: 6f73 7572 6520 6f66 2074 6865 2069 6d61  osure of the ima
│ │ │ │ -00056ee0: 6765 206f 6620 6120 7261 7469 6f6e 616c  ge of a rational
│ │ │ │ -00056ef0: 206d 6170 2075 7369 6e67 2074 6865 2046   map using the F
│ │ │ │ -00056f00: 3420 616c 676f 7269 7468 6d0a 2020 2020  4 algorithm.    
│ │ │ │ -00056f10: 2865 7870 6572 696d 656e 7461 6c29 0a0a  (experimental)..
│ │ │ │ -00056f20: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
│ │ │ │ -00056f30: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   method:.=======
│ │ │ │ +00056b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00056b60: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ +00056b70: 3d3d 3d0a 0a20 202a 2046 756e 6374 696f  ===..  * Functio
│ │ │ │ +00056b80: 6e3a 202a 6e6f 7465 2069 6d61 6765 3a20  n: *note image: 
│ │ │ │ +00056b90: 284d 6163 6175 6c61 7932 446f 6329 696d  (Macaulay2Doc)im
│ │ │ │ +00056ba0: 6167 652c 0a20 202a 2055 7361 6765 3a20  age,.  * Usage: 
│ │ │ │ +00056bb0: 0a20 2020 2020 2020 2069 6d61 6765 2050  .        image P
│ │ │ │ +00056bc0: 6869 200a 2020 2020 2020 2020 696d 6167  hi .        imag
│ │ │ │ +00056bd0: 6528 5068 692c 6429 0a20 202a 2049 6e70  e(Phi,d).  * Inp
│ │ │ │ +00056be0: 7574 733a 0a20 2020 2020 202a 2050 6869  uts:.      * Phi
│ │ │ │ +00056bf0: 2c20 6120 2a6e 6f74 6520 7261 7469 6f6e  , a *note ration
│ │ │ │ +00056c00: 616c 206d 6170 3a20 5261 7469 6f6e 616c  al map: Rational
│ │ │ │ +00056c10: 4d61 702c 0a20 2020 2020 202a 2064 2c20  Map,.      * d, 
│ │ │ │ +00056c20: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ +00056c30: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00056c40: 5a5a 2c0a 2020 2a20 4f75 7470 7574 733a  ZZ,.  * Outputs:
│ │ │ │ +00056c50: 0a20 2020 2020 202a 2061 6e20 2a6e 6f74  .      * an *not
│ │ │ │ +00056c60: 6520 6964 6561 6c3a 2028 4d61 6361 756c  e ideal: (Macaul
│ │ │ │ +00056c70: 6179 3244 6f63 2949 6465 616c 2c2c 2074  ay2Doc)Ideal,, t
│ │ │ │ +00056c80: 6865 2069 6465 616c 2064 6566 696e 696e  he ideal definin
│ │ │ │ +00056c90: 6720 7468 6520 636c 6f73 7572 6520 6f66  g the closure of
│ │ │ │ +00056ca0: 0a20 2020 2020 2020 2074 6865 2069 6d61  .        the ima
│ │ │ │ +00056cb0: 6765 206f 6620 5068 692c 206f 7220 6974  ge of Phi, or it
│ │ │ │ +00056cc0: 7320 6465 6772 6565 2064 2068 6f6d 6f67  s degree d homog
│ │ │ │ +00056cd0: 656e 656f 7573 2063 6f6d 706f 6e65 6e74  eneous component
│ │ │ │ +00056ce0: 2069 6620 6420 6973 2070 6173 7365 640a   if d is passed.
│ │ │ │ +00056cf0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +00056d00: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2063  ========..This c
│ │ │ │ +00056d10: 6f6d 7075 7461 7469 6f6e 2069 7320 646f  omputation is do
│ │ │ │ +00056d20: 6e65 2074 6872 6f75 6768 2074 6865 206b  ne through the k
│ │ │ │ +00056d30: 6572 6e65 6c20 6f66 2061 2072 696e 6720  ernel of a ring 
│ │ │ │ +00056d40: 6d61 7020 7265 7072 6573 656e 7469 6e67  map representing
│ │ │ │ +00056d50: 2074 6865 0a72 6174 696f 6e61 6c20 6d61   the.rational ma
│ │ │ │ +00056d60: 702e 2053 6565 202a 6e6f 7465 206b 6572  p. See *note ker
│ │ │ │ +00056d70: 6e65 6c28 5269 6e67 4d61 7029 3a20 284d  nel(RingMap): (M
│ │ │ │ +00056d80: 6163 6175 6c61 7932 446f 6329 6b65 726e  acaulay2Doc)kern
│ │ │ │ +00056d90: 656c 5f6c 7052 696e 674d 6170 5f72 702c  el_lpRingMap_rp,
│ │ │ │ +00056da0: 2061 6e64 0a2a 6e6f 7465 206b 6572 6e65   and.*note kerne
│ │ │ │ +00056db0: 6c28 5269 6e67 4d61 702c 5a5a 293a 206b  l(RingMap,ZZ): k
│ │ │ │ +00056dc0: 6572 6e65 6c5f 6c70 5269 6e67 4d61 705f  ernel_lpRingMap_
│ │ │ │ +00056dd0: 636d 5a5a 5f72 702c 2066 6f72 206d 6f72  cmZZ_rp, for mor
│ │ │ │ +00056de0: 6520 6465 7461 696c 732e 0a0a 5365 6520  e details...See 
│ │ │ │ +00056df0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ +00056e00: 202a 206b 6572 6e65 6c28 5269 6e67 4d61   * kernel(RingMa
│ │ │ │ +00056e10: 702c 5a5a 2920 286d 6973 7369 6e67 2064  p,ZZ) (missing d
│ │ │ │ +00056e20: 6f63 756d 656e 7461 7469 6f6e 290a 2020  ocumentation).  
│ │ │ │ +00056e30: 2a20 2a6e 6f74 6520 6b65 726e 656c 2852  * *note kernel(R
│ │ │ │ +00056e40: 696e 674d 6170 293a 2028 4d61 6361 756c  ingMap): (Macaul
│ │ │ │ +00056e50: 6179 3244 6f63 296b 6572 6e65 6c5f 6c70  ay2Doc)kernel_lp
│ │ │ │ +00056e60: 5269 6e67 4d61 705f 7270 2c20 2d2d 206b  RingMap_rp, -- k
│ │ │ │ +00056e70: 6572 6e65 6c20 6f66 2061 0a20 2020 2072  ernel of a.    r
│ │ │ │ +00056e80: 696e 676d 6170 0a20 202a 202a 6e6f 7465  ingmap.  * *note
│ │ │ │ +00056e90: 2069 6d61 6765 2852 6174 696f 6e61 6c4d   image(RationalM
│ │ │ │ +00056ea0: 6170 2c53 7472 696e 6729 3a20 696d 6167  ap,String): imag
│ │ │ │ +00056eb0: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +00056ec0: 636d 5374 7269 6e67 5f72 702c 202d 2d0a  cmString_rp, --.
│ │ │ │ +00056ed0: 2020 2020 636c 6f73 7572 6520 6f66 2074      closure of t
│ │ │ │ +00056ee0: 6865 2069 6d61 6765 206f 6620 6120 7261  he image of a ra
│ │ │ │ +00056ef0: 7469 6f6e 616c 206d 6170 2075 7369 6e67  tional map using
│ │ │ │ +00056f00: 2074 6865 2046 3420 616c 676f 7269 7468   the F4 algorith
│ │ │ │ +00056f10: 6d0a 2020 2020 2865 7870 6572 696d 656e  m.    (experimen
│ │ │ │ +00056f20: 7461 6c29 0a0a 5761 7973 2074 6f20 7573  tal)..Ways to us
│ │ │ │ +00056f30: 6520 7468 6973 206d 6574 686f 643a 0a3d  e this method:.=
│ │ │ │  00056f40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00056f50: 3d0a 0a20 202a 2022 696d 6167 6528 5261  =..  * "image(Ra
│ │ │ │ -00056f60: 7469 6f6e 616c 4d61 7029 220a 2020 2a20  tionalMap)".  * 
│ │ │ │ -00056f70: 2a6e 6f74 6520 696d 6167 6528 5261 7469  *note image(Rati
│ │ │ │ -00056f80: 6f6e 616c 4d61 702c 5a5a 293a 2069 6d61  onalMap,ZZ): ima
│ │ │ │ -00056f90: 6765 5f6c 7052 6174 696f 6e61 6c4d 6170  ge_lpRationalMap
│ │ │ │ -00056fa0: 5f63 6d5a 5a5f 7270 2c20 2d2d 2063 6c6f  _cmZZ_rp, -- clo
│ │ │ │ -00056fb0: 7375 7265 206f 6620 7468 650a 2020 2020  sure of the.    
│ │ │ │ -00056fc0: 696d 6167 6520 6f66 2061 2072 6174 696f  image of a ratio
│ │ │ │ -00056fd0: 6e61 6c20 6d61 700a 1f0a 4669 6c65 3a20  nal map...File: 
│ │ │ │ -00056fe0: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ -00056ff0: 6465 3a20 696e 7665 7273 655f 6c70 5261  de: inverse_lpRa
│ │ │ │ -00057000: 7469 6f6e 616c 4d61 705f 7270 2c20 4e65  tionalMap_rp, Ne
│ │ │ │ -00057010: 7874 3a20 696e 7665 7273 654d 6170 2c20  xt: inverseMap, 
│ │ │ │ -00057020: 5072 6576 3a20 696d 6167 655f 6c70 5261  Prev: image_lpRa
│ │ │ │ -00057030: 7469 6f6e 616c 4d61 705f 636d 5a5a 5f72  tionalMap_cmZZ_r
│ │ │ │ -00057040: 702c 2055 703a 2054 6f70 0a0a 696e 7665  p, Up: Top..inve
│ │ │ │ -00057050: 7273 6528 5261 7469 6f6e 616c 4d61 7029  rse(RationalMap)
│ │ │ │ -00057060: 202d 2d20 696e 7665 7273 6520 6f66 2061   -- inverse of a
│ │ │ │ -00057070: 2062 6972 6174 696f 6e61 6c20 6d61 700a   birational map.
│ │ │ │ -00057080: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00056f50: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 696d  =======..  * "im
│ │ │ │ +00056f60: 6167 6528 5261 7469 6f6e 616c 4d61 7029  age(RationalMap)
│ │ │ │ +00056f70: 220a 2020 2a20 2a6e 6f74 6520 696d 6167  ".  * *note imag
│ │ │ │ +00056f80: 6528 5261 7469 6f6e 616c 4d61 702c 5a5a  e(RationalMap,ZZ
│ │ │ │ +00056f90: 293a 2069 6d61 6765 5f6c 7052 6174 696f  ): image_lpRatio
│ │ │ │ +00056fa0: 6e61 6c4d 6170 5f63 6d5a 5a5f 7270 2c20  nalMap_cmZZ_rp, 
│ │ │ │ +00056fb0: 2d2d 2063 6c6f 7375 7265 206f 6620 7468  -- closure of th
│ │ │ │ +00056fc0: 650a 2020 2020 696d 6167 6520 6f66 2061  e.    image of a
│ │ │ │ +00056fd0: 2072 6174 696f 6e61 6c20 6d61 700a 1f0a   rational map...
│ │ │ │ +00056fe0: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ +00056ff0: 666f 2c20 4e6f 6465 3a20 696e 7665 7273  fo, Node: invers
│ │ │ │ +00057000: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +00057010: 7270 2c20 4e65 7874 3a20 696e 7665 7273  rp, Next: invers
│ │ │ │ +00057020: 654d 6170 2c20 5072 6576 3a20 696d 6167  eMap, Prev: imag
│ │ │ │ +00057030: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +00057040: 636d 5a5a 5f72 702c 2055 703a 2054 6f70  cmZZ_rp, Up: Top
│ │ │ │ +00057050: 0a0a 696e 7665 7273 6528 5261 7469 6f6e  ..inverse(Ration
│ │ │ │ +00057060: 616c 4d61 7029 202d 2d20 696e 7665 7273  alMap) -- invers
│ │ │ │ +00057070: 6520 6f66 2061 2062 6972 6174 696f 6e61  e of a birationa
│ │ │ │ +00057080: 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a  l map.**********
│ │ │ │  00057090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000570a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000570b0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ -000570c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675 6e63  ======..  * Func
│ │ │ │ -000570d0: 7469 6f6e 3a20 2a6e 6f74 6520 696e 7665  tion: *note inve
│ │ │ │ -000570e0: 7273 653a 2028 4d61 6361 756c 6179 3244  rse: (Macaulay2D
│ │ │ │ -000570f0: 6f63 2969 6e76 6572 7365 2c0a 2020 2a20  oc)inverse,.  * 
│ │ │ │ -00057100: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00057110: 696e 7665 7273 6520 7068 690a 2020 2020  inverse phi.    
│ │ │ │ -00057120: 2020 2020 696e 7665 7273 6528 7068 692c      inverse(phi,
│ │ │ │ -00057130: 4365 7274 6966 793d 3e62 290a 2020 2a20  Certify=>b).  * 
│ │ │ │ -00057140: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00057150: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ -00057160: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00057170: 6e61 6c4d 6170 2c2c 2077 6869 6368 2068  nalMap,, which h
│ │ │ │ -00057180: 6173 2074 6f20 6265 2062 6972 6174 696f  as to be biratio
│ │ │ │ -00057190: 6e61 6c2c 0a20 2020 2020 2020 2061 6e64  nal,.        and
│ │ │ │ -000571a0: 2062 2069 7320 6120 2a6e 6f74 6520 626f   b is a *note bo
│ │ │ │ -000571b0: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61  olean value: (Ma
│ │ │ │ -000571c0: 6361 756c 6179 3244 6f63 2942 6f6f 6c65  caulay2Doc)Boole
│ │ │ │ -000571d0: 616e 2c2c 2074 6861 7420 6973 2c20 7472  an,, that is, tr
│ │ │ │ -000571e0: 7565 0a20 2020 2020 2020 206f 7220 6661  ue.        or fa
│ │ │ │ -000571f0: 6c73 6520 2874 6865 2064 6566 6175 6c74  lse (the default
│ │ │ │ -00057200: 2076 616c 7565 2069 7320 6661 6c73 6529   value is false)
│ │ │ │ -00057210: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00057220: 2020 2020 2a20 6120 2a6e 6f74 6520 7261      * a *note ra
│ │ │ │ -00057230: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ -00057240: 6f6e 616c 4d61 702c 2c20 7468 6520 696e  onalMap,, the in
│ │ │ │ -00057250: 7665 7273 6520 6d61 7020 6f66 2070 6869  verse map of phi
│ │ │ │ -00057260: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -00057270: 3d3d 3d3d 3d3d 3d3d 3d0a 0a69 6e76 6572  =========..inver
│ │ │ │ -00057280: 7365 2870 6869 2c43 6572 7469 6679 3d3e  se(phi,Certify=>
│ │ │ │ -00057290: 7472 7565 2920 6973 2074 6865 2073 616d  true) is the sam
│ │ │ │ -000572a0: 6520 6173 202a 6e6f 7465 2069 6e76 6572  e as *note inver
│ │ │ │ -000572b0: 7365 4d61 703a 0a69 6e76 6572 7365 4d61  seMap:.inverseMa
│ │ │ │ -000572c0: 702c 2870 6869 2c43 6572 7469 6679 3d3e  p,(phi,Certify=>
│ │ │ │ -000572d0: 7472 7565 2c56 6572 626f 7365 3d3e 6661  true,Verbose=>fa
│ │ │ │ -000572e0: 6c73 6529 2c20 7768 696c 650a 696e 7665  lse), while.inve
│ │ │ │ -000572f0: 7273 6528 7068 692c 4365 7274 6966 793d  rse(phi,Certify=
│ │ │ │ -00057300: 3e66 616c 7365 2920 6170 706c 6965 7320  >false) applies 
│ │ │ │ -00057310: 6120 6d69 6464 6c65 2067 726f 756e 6420  a middle ground 
│ │ │ │ -00057320: 6170 7072 6f61 6368 2062 6574 7765 656e  approach between
│ │ │ │ -00057330: 202a 6e6f 7465 0a69 6e76 6572 7365 4d61   *note.inverseMa
│ │ │ │ -00057340: 703a 2069 6e76 6572 7365 4d61 702c 2870  p: inverseMap,(p
│ │ │ │ -00057350: 6869 2c43 6572 7469 6679 3d3e 7472 7565  hi,Certify=>true
│ │ │ │ -00057360: 2920 616e 6420 2a6e 6f74 6520 696e 7665  ) and *note inve
│ │ │ │ -00057370: 7273 654d 6170 3a0a 696e 7665 7273 654d  rseMap:.inverseM
│ │ │ │ -00057380: 6170 2c28 7068 692c 4365 7274 6966 793d  ap,(phi,Certify=
│ │ │ │ -00057390: 3e66 616c 7365 292e 2054 6865 2070 726f  >false). The pro
│ │ │ │ -000573a0: 6365 6475 7265 2066 6f72 2074 6865 206c  cedure for the l
│ │ │ │ -000573b0: 6174 7465 7220 6973 2061 7320 666f 6c6c  atter is as foll
│ │ │ │ -000573c0: 6f77 733a 2049 740a 6669 7273 7420 636f  ows: It.first co
│ │ │ │ -000573d0: 6d70 7574 6573 2074 6865 2069 6e76 6572  mputes the inver
│ │ │ │ -000573e0: 7365 206d 6170 206f 6620 7068 6920 7573  se map of phi us
│ │ │ │ -000573f0: 696e 6720 7073 6920 3d20 2a6e 6f74 6520  ing psi = *note 
│ │ │ │ -00057400: 696e 7665 7273 654d 6170 3a20 696e 7665  inverseMap: inve
│ │ │ │ -00057410: 7273 654d 6170 2c0a 7068 692e 2054 6865  rseMap,.phi. The
│ │ │ │ -00057420: 6e20 6974 2069 7320 6368 6563 6b65 6420  n it is checked 
│ │ │ │ -00057430: 7468 6174 202a 6e6f 7465 2070 7369 2070  that *note psi p
│ │ │ │ -00057440: 6869 2070 3a20 5261 7469 6f6e 616c 4d61  hi p: RationalMa
│ │ │ │ -00057450: 7020 5f75 735f 7374 2c20 3d3d 2070 2061  p _us_st, == p a
│ │ │ │ -00057460: 6e64 0a2a 6e6f 7465 2070 6869 2070 7369  nd.*note phi psi
│ │ │ │ -00057470: 2071 3a20 5261 7469 6f6e 616c 4d61 7020   q: RationalMap 
│ │ │ │ -00057480: 5f75 735f 7374 2c20 3d3d 2071 2c20 7768  _us_st, == q, wh
│ │ │ │ -00057490: 6572 6520 702c 7120 6172 652c 2072 6573  ere p,q are, res
│ │ │ │ -000574a0: 7065 6374 6976 656c 792c 2061 202a 6e6f  pectively, a *no
│ │ │ │ -000574b0: 7465 0a72 616e 646f 6d20 706f 696e 743a  te.random point:
│ │ │ │ -000574c0: 2070 6f69 6e74 2c20 6f6e 2074 6865 202a   point, on the *
│ │ │ │ -000574d0: 6e6f 7465 2073 6f75 7263 653a 2073 6f75  note source: sou
│ │ │ │ -000574e0: 7263 655f 6c70 5261 7469 6f6e 616c 4d61  rce_lpRationalMa
│ │ │ │ -000574f0: 705f 7270 2c20 616e 6420 7468 650a 2a6e  p_rp, and the.*n
│ │ │ │ -00057500: 6f74 6520 7461 7267 6574 3a20 7461 7267  ote target: targ
│ │ │ │ -00057510: 6574 5f6c 7052 6174 696f 6e61 6c4d 6170  et_lpRationalMap
│ │ │ │ -00057520: 5f72 702c 206f 6620 7068 692e 2046 696e  _rp, of phi. Fin
│ │ │ │ -00057530: 616c 6c79 2c20 6966 2074 6865 2074 6573  ally, if the tes
│ │ │ │ -00057540: 7473 2070 6173 732c 2074 6865 0a63 6f6d  ts pass, the.com
│ │ │ │ -00057550: 6d61 6e64 202a 6e6f 7465 2066 6f72 6365  mand *note force
│ │ │ │ -00057560: 496e 7665 7273 654d 6170 3a20 666f 7263  InverseMap: forc
│ │ │ │ -00057570: 6549 6e76 6572 7365 4d61 702c 2870 6869  eInverseMap,(phi
│ │ │ │ -00057580: 2c70 7369 2920 6973 2069 6e76 6f6b 6564  ,psi) is invoked
│ │ │ │ -00057590: 2061 6e64 2070 7369 2069 730a 7265 7475   and psi is.retu
│ │ │ │ -000575a0: 726e 6564 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  rned...+--------
│ │ │ │ +000570b0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70  *********..Synop
│ │ │ │ +000570c0: 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  sis.========..  
│ │ │ │ +000570d0: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74  * Function: *not
│ │ │ │ +000570e0: 6520 696e 7665 7273 653a 2028 4d61 6361  e inverse: (Maca
│ │ │ │ +000570f0: 756c 6179 3244 6f63 2969 6e76 6572 7365  ulay2Doc)inverse
│ │ │ │ +00057100: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ +00057110: 2020 2020 2020 696e 7665 7273 6520 7068        inverse ph
│ │ │ │ +00057120: 690a 2020 2020 2020 2020 696e 7665 7273  i.        invers
│ │ │ │ +00057130: 6528 7068 692c 4365 7274 6966 793d 3e62  e(phi,Certify=>b
│ │ │ │ +00057140: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ +00057150: 2020 2020 2a20 7068 692c 2061 202a 6e6f      * phi, a *no
│ │ │ │ +00057160: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ +00057170: 2052 6174 696f 6e61 6c4d 6170 2c2c 2077   RationalMap,, w
│ │ │ │ +00057180: 6869 6368 2068 6173 2074 6f20 6265 2062  hich has to be b
│ │ │ │ +00057190: 6972 6174 696f 6e61 6c2c 0a20 2020 2020  irational,.     
│ │ │ │ +000571a0: 2020 2061 6e64 2062 2069 7320 6120 2a6e     and b is a *n
│ │ │ │ +000571b0: 6f74 6520 626f 6f6c 6561 6e20 7661 6c75  ote boolean valu
│ │ │ │ +000571c0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +000571d0: 2942 6f6f 6c65 616e 2c2c 2074 6861 7420  )Boolean,, that 
│ │ │ │ +000571e0: 6973 2c20 7472 7565 0a20 2020 2020 2020  is, true.       
│ │ │ │ +000571f0: 206f 7220 6661 6c73 6520 2874 6865 2064   or false (the d
│ │ │ │ +00057200: 6566 6175 6c74 2076 616c 7565 2069 7320  efault value is 
│ │ │ │ +00057210: 6661 6c73 6529 0a20 202a 204f 7574 7075  false).  * Outpu
│ │ │ │ +00057220: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ +00057230: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ +00057240: 3a20 5261 7469 6f6e 616c 4d61 702c 2c20  : RationalMap,, 
│ │ │ │ +00057250: 7468 6520 696e 7665 7273 6520 6d61 7020  the inverse map 
│ │ │ │ +00057260: 6f66 2070 6869 0a0a 4465 7363 7269 7074  of phi..Descript
│ │ │ │ +00057270: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +00057280: 0a69 6e76 6572 7365 2870 6869 2c43 6572  .inverse(phi,Cer
│ │ │ │ +00057290: 7469 6679 3d3e 7472 7565 2920 6973 2074  tify=>true) is t
│ │ │ │ +000572a0: 6865 2073 616d 6520 6173 202a 6e6f 7465  he same as *note
│ │ │ │ +000572b0: 2069 6e76 6572 7365 4d61 703a 0a69 6e76   inverseMap:.inv
│ │ │ │ +000572c0: 6572 7365 4d61 702c 2870 6869 2c43 6572  erseMap,(phi,Cer
│ │ │ │ +000572d0: 7469 6679 3d3e 7472 7565 2c56 6572 626f  tify=>true,Verbo
│ │ │ │ +000572e0: 7365 3d3e 6661 6c73 6529 2c20 7768 696c  se=>false), whil
│ │ │ │ +000572f0: 650a 696e 7665 7273 6528 7068 692c 4365  e.inverse(phi,Ce
│ │ │ │ +00057300: 7274 6966 793d 3e66 616c 7365 2920 6170  rtify=>false) ap
│ │ │ │ +00057310: 706c 6965 7320 6120 6d69 6464 6c65 2067  plies a middle g
│ │ │ │ +00057320: 726f 756e 6420 6170 7072 6f61 6368 2062  round approach b
│ │ │ │ +00057330: 6574 7765 656e 202a 6e6f 7465 0a69 6e76  etween *note.inv
│ │ │ │ +00057340: 6572 7365 4d61 703a 2069 6e76 6572 7365  erseMap: inverse
│ │ │ │ +00057350: 4d61 702c 2870 6869 2c43 6572 7469 6679  Map,(phi,Certify
│ │ │ │ +00057360: 3d3e 7472 7565 2920 616e 6420 2a6e 6f74  =>true) and *not
│ │ │ │ +00057370: 6520 696e 7665 7273 654d 6170 3a0a 696e  e inverseMap:.in
│ │ │ │ +00057380: 7665 7273 654d 6170 2c28 7068 692c 4365  verseMap,(phi,Ce
│ │ │ │ +00057390: 7274 6966 793d 3e66 616c 7365 292e 2054  rtify=>false). T
│ │ │ │ +000573a0: 6865 2070 726f 6365 6475 7265 2066 6f72  he procedure for
│ │ │ │ +000573b0: 2074 6865 206c 6174 7465 7220 6973 2061   the latter is a
│ │ │ │ +000573c0: 7320 666f 6c6c 6f77 733a 2049 740a 6669  s follows: It.fi
│ │ │ │ +000573d0: 7273 7420 636f 6d70 7574 6573 2074 6865  rst computes the
│ │ │ │ +000573e0: 2069 6e76 6572 7365 206d 6170 206f 6620   inverse map of 
│ │ │ │ +000573f0: 7068 6920 7573 696e 6720 7073 6920 3d20  phi using psi = 
│ │ │ │ +00057400: 2a6e 6f74 6520 696e 7665 7273 654d 6170  *note inverseMap
│ │ │ │ +00057410: 3a20 696e 7665 7273 654d 6170 2c0a 7068  : inverseMap,.ph
│ │ │ │ +00057420: 692e 2054 6865 6e20 6974 2069 7320 6368  i. Then it is ch
│ │ │ │ +00057430: 6563 6b65 6420 7468 6174 202a 6e6f 7465  ecked that *note
│ │ │ │ +00057440: 2070 7369 2070 6869 2070 3a20 5261 7469   psi phi p: Rati
│ │ │ │ +00057450: 6f6e 616c 4d61 7020 5f75 735f 7374 2c20  onalMap _us_st, 
│ │ │ │ +00057460: 3d3d 2070 2061 6e64 0a2a 6e6f 7465 2070  == p and.*note p
│ │ │ │ +00057470: 6869 2070 7369 2071 3a20 5261 7469 6f6e  hi psi q: Ration
│ │ │ │ +00057480: 616c 4d61 7020 5f75 735f 7374 2c20 3d3d  alMap _us_st, ==
│ │ │ │ +00057490: 2071 2c20 7768 6572 6520 702c 7120 6172   q, where p,q ar
│ │ │ │ +000574a0: 652c 2072 6573 7065 6374 6976 656c 792c  e, respectively,
│ │ │ │ +000574b0: 2061 202a 6e6f 7465 0a72 616e 646f 6d20   a *note.random 
│ │ │ │ +000574c0: 706f 696e 743a 2070 6f69 6e74 2c20 6f6e  point: point, on
│ │ │ │ +000574d0: 2074 6865 202a 6e6f 7465 2073 6f75 7263   the *note sourc
│ │ │ │ +000574e0: 653a 2073 6f75 7263 655f 6c70 5261 7469  e: source_lpRati
│ │ │ │ +000574f0: 6f6e 616c 4d61 705f 7270 2c20 616e 6420  onalMap_rp, and 
│ │ │ │ +00057500: 7468 650a 2a6e 6f74 6520 7461 7267 6574  the.*note target
│ │ │ │ +00057510: 3a20 7461 7267 6574 5f6c 7052 6174 696f  : target_lpRatio
│ │ │ │ +00057520: 6e61 6c4d 6170 5f72 702c 206f 6620 7068  nalMap_rp, of ph
│ │ │ │ +00057530: 692e 2046 696e 616c 6c79 2c20 6966 2074  i. Finally, if t
│ │ │ │ +00057540: 6865 2074 6573 7473 2070 6173 732c 2074  he tests pass, t
│ │ │ │ +00057550: 6865 0a63 6f6d 6d61 6e64 202a 6e6f 7465  he.command *note
│ │ │ │ +00057560: 2066 6f72 6365 496e 7665 7273 654d 6170   forceInverseMap
│ │ │ │ +00057570: 3a20 666f 7263 6549 6e76 6572 7365 4d61  : forceInverseMa
│ │ │ │ +00057580: 702c 2870 6869 2c70 7369 2920 6973 2069  p,(phi,psi) is i
│ │ │ │ +00057590: 6e76 6f6b 6564 2061 6e64 2070 7369 2069  nvoked and psi i
│ │ │ │ +000575a0: 730a 7265 7475 726e 6564 2e0a 0a2b 2d2d  s.returned...+--
│ │ │ │  000575b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000575c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000575d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000575e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000575f0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ -00057600: 2051 515b 785f 302e 2e78 5f34 5d3b 2070   QQ[x_0..x_4]; p
│ │ │ │ -00057610: 6869 203d 2072 6174 696f 6e61 6c4d 6170  hi = rationalMap
│ │ │ │ -00057620: 206d 696e 6f72 7328 342c 7261 6e64 6f6d   minors(4,random
│ │ │ │ -00057630: 2852 5e7b 343a 317d 2c20 2020 2020 2020  (R^{4:1},       
│ │ │ │ -00057640: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000575f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00057600: 203a 2052 203d 2051 515b 785f 302e 2e78   : R = QQ[x_0..x
│ │ │ │ +00057610: 5f34 5d3b 2070 6869 203d 2072 6174 696f  _4]; phi = ratio
│ │ │ │ +00057620: 6e61 6c4d 6170 206d 696e 6f72 7328 342c  nalMap minors(4,
│ │ │ │ +00057630: 7261 6e64 6f6d 2852 5e7b 343a 317d 2c20  random(R^{4:1}, 
│ │ │ │ +00057640: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057690: 2020 2020 207c 0a7c 6f32 203d 202d 2d20       |.|o2 = -- 
│ │ │ │ -000576a0: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ -000576b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057690: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +000576a0: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +000576b0: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  000576c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000576d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000576e0: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -000576f0: 7263 653a 2050 726f 6a28 5151 5b78 202c  rce: Proj(QQ[x ,
│ │ │ │ -00057700: 2078 202c 2078 202c 2078 202c 2078 205d   x , x , x , x ]
│ │ │ │ -00057710: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +000576e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000576f0: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ +00057700: 5151 5b78 202c 2078 202c 2078 202c 2078  QQ[x , x , x , x
│ │ │ │ +00057710: 202c 2078 205d 2920 2020 2020 2020 2020   , x ])         
│ │ │ │  00057720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057730: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057740: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00057750: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ -00057760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057730: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057750: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +00057760: 3320 2020 3420 2020 2020 2020 2020 2020  3   4           
│ │ │ │  00057770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057780: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -00057790: 6765 743a 2050 726f 6a28 5151 5b78 202c  get: Proj(QQ[x ,
│ │ │ │ -000577a0: 2078 202c 2078 202c 2078 202c 2078 205d   x , x , x , x ]
│ │ │ │ -000577b0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00057780: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057790: 2020 2074 6172 6765 743a 2050 726f 6a28     target: Proj(
│ │ │ │ +000577a0: 5151 5b78 202c 2078 202c 2078 202c 2078  QQ[x , x , x , x
│ │ │ │ +000577b0: 202c 2078 205d 2920 2020 2020 2020 2020   , x ])         
│ │ │ │  000577c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000577d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000577e0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000577f0: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ -00057800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000577d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000577e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000577f0: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +00057800: 3320 2020 3420 2020 2020 2020 2020 2020  3   4           
│ │ │ │  00057810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057820: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -00057830: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ -00057840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057820: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057830: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
│ │ │ │ +00057840: 733a 207b 2020 2020 2020 2020 2020 2020  s: {            
│ │ │ │  00057850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057870: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057870: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057890: 3837 3731 3320 3420 2020 3137 3139 3430  87713 4   171940
│ │ │ │ -000578a0: 3636 3920 3320 2020 2020 3738 3636 3933  669 3     786693
│ │ │ │ -000578b0: 3231 3720 3220 3220 2020 2020 2020 2020  217 2 2         
│ │ │ │ -000578c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000578d0: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -000578e0: 2d2d 2d2d 2d78 2020 2b20 2d2d 2d2d 2d2d  -----x  + ------
│ │ │ │ -000578f0: 2d2d 2d78 2078 2020 2d20 2d2d 2d2d 2d2d  ---x x  - ------
│ │ │ │ -00057900: 2d2d 2d78 2078 2020 2d20 2020 2020 2020  ---x x  -       
│ │ │ │ -00057910: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057890: 2020 2020 2020 3837 3731 3320 3420 2020        87713 4   
│ │ │ │ +000578a0: 3137 3139 3430 3636 3920 3320 2020 2020  171940669 3     
│ │ │ │ +000578b0: 3738 3636 3933 3231 3720 3220 3220 2020  786693217 2 2   
│ │ │ │ +000578c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000578d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000578e0: 2020 2020 2d20 2d2d 2d2d 2d78 2020 2b20      - -----x  + 
│ │ │ │ +000578f0: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020 2d20  ---------x x  - 
│ │ │ │ +00057900: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020 2d20  ---------x x  - 
│ │ │ │ +00057910: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057930: 2037 3834 3020 3020 2020 2033 3632 3838   7840 0    36288
│ │ │ │ -00057940: 3030 2020 3020 3120 2020 2031 3639 3334  00  0 1    16934
│ │ │ │ -00057950: 3430 3020 3020 3120 2020 2020 2020 2020  400 0 1         
│ │ │ │ -00057960: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057930: 2020 2020 2020 2037 3834 3020 3020 2020         7840 0   
│ │ │ │ +00057940: 2033 3632 3838 3030 2020 3020 3120 2020   3628800  0 1   
│ │ │ │ +00057950: 2031 3639 3334 3430 3020 3020 3120 2020   16934400 0 1   
│ │ │ │ +00057960: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 3737                77
│ │ │ │ -000579d0: 3638 3720 3420 2020 3138 3933 3337 3531  687 4   18933751
│ │ │ │ -000579e0: 3633 2033 2020 2020 2037 3435 3939 3030  63 3     7459900
│ │ │ │ -000579f0: 3831 2032 2032 2020 2020 2020 2020 2020  81 2 2          
│ │ │ │ -00057a00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057a10: 2020 2020 2020 2020 2020 2020 2020 2d2d                --
│ │ │ │ -00057a20: 2d2d 2d78 2020 2d20 2d2d 2d2d 2d2d 2d2d  ---x  - --------
│ │ │ │ -00057a30: 2d2d 7820 7820 202d 202d 2d2d 2d2d 2d2d  --x x  - -------
│ │ │ │ -00057a40: 2d2d 7820 7820 202b 2020 2020 2020 2020  --x x  +        
│ │ │ │ -00057a50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057a60: 2020 2020 2020 2020 2020 2020 2020 3434                44
│ │ │ │ -00057a70: 3130 3020 3020 2020 2037 3430 3838 3030  100 0    7408800
│ │ │ │ -00057a80: 3020 2030 2031 2020 2020 3337 3034 3430  0  0 1    370440
│ │ │ │ -00057a90: 3020 2030 2031 2020 2020 2020 2020 2020  0  0 1          
│ │ │ │ -00057aa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000579b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000579c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000579d0: 2020 2020 3737 3638 3720 3420 2020 3138      77687 4   18
│ │ │ │ +000579e0: 3933 3337 3531 3633 2033 2020 2020 2037  93375163 3     7
│ │ │ │ +000579f0: 3435 3939 3030 3831 2032 2032 2020 2020  45990081 2 2    
│ │ │ │ +00057a00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057a20: 2020 2020 2d2d 2d2d 2d78 2020 2d20 2d2d      -----x  - --
│ │ │ │ +00057a30: 2d2d 2d2d 2d2d 2d2d 7820 7820 202d 202d  --------x x  - -
│ │ │ │ +00057a40: 2d2d 2d2d 2d2d 2d2d 7820 7820 202b 2020  --------x x  +  
│ │ │ │ +00057a50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057a70: 2020 2020 3434 3130 3020 3020 2020 2037      44100 0    7
│ │ │ │ +00057a80: 3430 3838 3030 3020 2030 2031 2020 2020  4088000  0 1    
│ │ │ │ +00057a90: 3337 3034 3430 3020 2030 2031 2020 2020  3704400  0 1    
│ │ │ │ +00057aa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057af0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057b00: 2020 2020 2020 2020 2020 2020 2020 3230                20
│ │ │ │ -00057b10: 3437 3032 3137 2034 2020 2036 3139 3732  470217 4   61972
│ │ │ │ -00057b20: 3930 3231 3320 3320 2020 2020 3134 3234  90213 3     1424
│ │ │ │ -00057b30: 3237 3839 3135 3637 2020 2020 2020 2020  27891567        
│ │ │ │ -00057b40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057b50: 2020 2020 2020 2020 2020 2020 2020 2d2d                --
│ │ │ │ -00057b60: 2d2d 2d2d 2d2d 7820 202d 202d 2d2d 2d2d  ------x  - -----
│ │ │ │ -00057b70: 2d2d 2d2d 2d78 2078 2020 2d20 2d2d 2d2d  -----x x  - ----
│ │ │ │ -00057b80: 2d2d 2d2d 2d2d 2d2d 7820 2020 2020 2020  --------x       
│ │ │ │ -00057b90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057ba0: 2020 2020 2020 2020 2020 2020 2020 2038                 8
│ │ │ │ -00057bb0: 3832 3030 3020 2030 2020 2020 3432 3333  82000  0    4233
│ │ │ │ -00057bc0: 3630 3030 2020 3020 3120 2020 2020 3136  6000  0 1     16
│ │ │ │ -00057bd0: 3933 3434 3030 3020 2020 2020 2020 2020  9344000         
│ │ │ │ -00057be0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057af0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057b10: 2020 2020 3230 3437 3032 3137 2034 2020      20470217 4  
│ │ │ │ +00057b20: 2036 3139 3732 3930 3231 3320 3320 2020   6197290213 3   
│ │ │ │ +00057b30: 2020 3134 3234 3237 3839 3135 3637 2020    142427891567  
│ │ │ │ +00057b40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057b60: 2020 2020 2d2d 2d2d 2d2d 2d2d 7820 202d      --------x  -
│ │ │ │ +00057b70: 202d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020   ----------x x  
│ │ │ │ +00057b80: 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820  - ------------x 
│ │ │ │ +00057b90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057bb0: 2020 2020 2038 3832 3030 3020 2030 2020       882000  0  
│ │ │ │ +00057bc0: 2020 3432 3333 3630 3030 2020 3020 3120    42336000  0 1 
│ │ │ │ +00057bd0: 2020 2020 3136 3933 3434 3030 3020 2020      169344000   
│ │ │ │ +00057be0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057c30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057c30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057c50: 3135 3030 3339 3120 3420 2020 3832 3735  1500391 4   8275
│ │ │ │ -00057c60: 3735 3235 3633 2033 2020 2020 2032 3536  752563 3     256
│ │ │ │ -00057c70: 3334 3130 3839 2032 2020 2020 2020 2020  341089 2        
│ │ │ │ -00057c80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057c90: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -00057ca0: 2d2d 2d2d 2d2d 2d78 2020 2d20 2d2d 2d2d  -------x  - ----
│ │ │ │ -00057cb0: 2d2d 2d2d 2d2d 7820 7820 202b 202d 2d2d  ------x x  + ---
│ │ │ │ -00057cc0: 2d2d 2d2d 2d2d 7820 7820 2020 2020 2020  ------x x       
│ │ │ │ -00057cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057c50: 2020 2020 2020 3135 3030 3339 3120 3420        1500391 4 
│ │ │ │ +00057c60: 2020 3832 3735 3735 3235 3633 2033 2020    8275752563 3  
│ │ │ │ +00057c70: 2020 2032 3536 3334 3130 3839 2032 2020     256341089 2  
│ │ │ │ +00057c80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057ca0: 2020 2020 2d20 2d2d 2d2d 2d2d 2d78 2020      - -------x  
│ │ │ │ +00057cb0: 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  - ----------x x 
│ │ │ │ +00057cc0: 202b 202d 2d2d 2d2d 2d2d 2d2d 7820 7820   + ---------x x 
│ │ │ │ +00057cd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057cf0: 2032 3131 3638 3020 3020 2020 2038 3839   211680 0    889
│ │ │ │ -00057d00: 3035 3630 3020 2030 2031 2020 2020 3337  05600  0 1    37
│ │ │ │ -00057d10: 3034 3430 3020 2030 2020 2020 2020 2020  04400  0        
│ │ │ │ -00057d20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057cf0: 2020 2020 2020 2032 3131 3638 3020 3020         211680 0 
│ │ │ │ +00057d00: 2020 2038 3839 3035 3630 3020 2030 2031     88905600  0 1
│ │ │ │ +00057d10: 2020 2020 3337 3034 3430 3020 2030 2020      3704400  0  
│ │ │ │ +00057d20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057d70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057d70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057d90: 3235 3139 3720 3420 2020 3134 3036 3136  25197 4   140616
│ │ │ │ -00057da0: 3730 3920 3320 2020 2020 3133 3532 3033  709 3     135203
│ │ │ │ -00057db0: 3932 3839 3120 3220 3220 2020 2020 2020  92891 2 2       
│ │ │ │ -00057dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057dd0: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -00057de0: 2d2d 2d2d 2d78 2020 2d20 2d2d 2d2d 2d2d  -----x  - ------
│ │ │ │ -00057df0: 2d2d 2d78 2078 2020 2b20 2d2d 2d2d 2d2d  ---x x  + ------
│ │ │ │ -00057e00: 2d2d 2d2d 2d78 2078 2020 2020 2020 2020  -----x x        
│ │ │ │ -00057e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057d90: 2020 2020 2020 3235 3139 3720 3420 2020        25197 4   
│ │ │ │ +00057da0: 3134 3036 3136 3730 3920 3320 2020 2020  140616709 3     
│ │ │ │ +00057db0: 3133 3532 3033 3932 3839 3120 3220 3220  13520392891 2 2 
│ │ │ │ +00057dc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057de0: 2020 2020 2d20 2d2d 2d2d 2d78 2020 2d20      - -----x  - 
│ │ │ │ +00057df0: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020 2b20  ---------x x  + 
│ │ │ │ +00057e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  -----------x x  
│ │ │ │ +00057e10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e30: 2032 3830 3020 3020 2020 2036 3335 3034   2800 0    63504
│ │ │ │ -00057e40: 3030 2020 3020 3120 2020 2020 3530 3830  00  0 1     5080
│ │ │ │ -00057e50: 3332 3030 2020 3020 3120 2020 2020 2020  3200  0 1       
│ │ │ │ -00057e60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00057e70: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ -00057e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057e30: 2020 2020 2020 2032 3830 3020 3020 2020         2800 0   
│ │ │ │ +00057e40: 2036 3335 3034 3030 2020 3020 3120 2020   6350400  0 1   
│ │ │ │ +00057e50: 2020 3530 3830 3332 3030 2020 3020 3120    50803200  0 1 
│ │ │ │ +00057e60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00057e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057e80: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  00057e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057eb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057eb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00057ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f00: 2020 2020 207c 0a7c 6f32 203a 2052 6174       |.|o2 : Rat
│ │ │ │ -00057f10: 696f 6e61 6c4d 6170 2028 7261 7469 6f6e  ionalMap (ration
│ │ │ │ -00057f20: 616c 206d 6170 2066 726f 6d20 5050 5e34  al map from PP^4
│ │ │ │ -00057f30: 2074 6f20 5050 5e34 2920 2020 2020 2020   to PP^4)       
│ │ │ │ +00057f00: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00057f10: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +00057f20: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ +00057f30: 6d20 5050 5e34 2074 6f20 5050 5e34 2920  m PP^4 to PP^4) 
│ │ │ │  00057f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f50: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00057f50: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  00057f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057fa0: 2d2d 2d2d 2d7c 0a7c 525e 3529 2920 2d2d  -----|.|R^5)) --
│ │ │ │ -00057fb0: 2073 7065 6369 616c 2043 7265 6d6f 6e61   special Cremona
│ │ │ │ -00057fc0: 2074 7261 6e73 666f 726d 6174 696f 6e20   transformation 
│ │ │ │ -00057fd0: 6f66 2050 5e34 206f 6620 7479 7065 2028  of P^4 of type (
│ │ │ │ -00057fe0: 342c 3429 2020 2020 2020 2020 2020 2020  4,4)            
│ │ │ │ -00057ff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00057fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 525e  -----------|.|R^
│ │ │ │ +00057fb0: 3529 2920 2d2d 2073 7065 6369 616c 2043  5)) -- special C
│ │ │ │ +00057fc0: 7265 6d6f 6e61 2074 7261 6e73 666f 726d  remona transform
│ │ │ │ +00057fd0: 6174 696f 6e20 6f66 2050 5e34 206f 6620  ation of P^4 of 
│ │ │ │ +00057fe0: 7479 7065 2028 342c 3429 2020 2020 2020  type (4,4)      
│ │ │ │ +00057ff0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00058000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058040: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058040: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00058050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058090: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000580a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000580b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000580c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000580d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000580e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000580e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000580f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058130: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058130: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00058140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058180: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058180: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00058190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000581a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000581b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000581c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000581d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000581d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000581e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000581f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058220: 2020 2020 207c 0a7c 2031 3137 3632 3631       |.| 1176261
│ │ │ │ -00058230: 2020 2033 2020 2031 3734 3537 3520 3420     3   174575 4 
│ │ │ │ -00058240: 2020 3435 3632 3136 3439 3920 3320 2020    456216499 3   
│ │ │ │ -00058250: 2020 3532 3335 3931 3934 3932 3720 3220    52359194927 2 
│ │ │ │ -00058260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058270: 2020 2020 207c 0a7c 202d 2d2d 2d2d 2d2d       |.| -------
│ │ │ │ -00058280: 7820 7820 202b 202d 2d2d 2d2d 2d78 2020  x x  + ------x  
│ │ │ │ -00058290: 2d20 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  - ---------x x  
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│ │ │ │ +0005a220: 2032 2033 2020 2020 3331 3735 3230 3020   2 3    3175200 
│ │ │ │ +0005a230: 2030 2033 2020 2020 3334 3239 3231 3630   0 3    34292160
│ │ │ │ +0005a240: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +0005a250: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  0005a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0005a2b0: 3320 2020 2020 3136 3736 3939 3730 3437  3     1676997047
│ │ │ │ -0005a2c0: 2032 2032 2020 2031 3631 3438 3732 3539   2 2   161487259
│ │ │ │ -0005a2d0: 3931 2020 2020 2032 2020 2020 2020 2020  91     2        
│ │ │ │ +0005a2a0: 2020 2020 2020 2020 2020 207c 0a7c 2039             |.| 9
│ │ │ │ +0005a2b0: 3830 3035 3420 3320 2020 2020 3136 3736  80054 3     1676
│ │ │ │ +0005a2c0: 3939 3730 3437 2032 2032 2020 2031 3631  997047 2 2   161
│ │ │ │ +0005a2d0: 3438 3732 3539 3931 2020 2020 2032 2020  48725991     2  
│ │ │ │  0005a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a2f0: 2020 2020 207c 0a7c 202d 2d2d 2d2d 2d78       |.| ------x
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│ │ │ │  0005a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a340: 2020 2020 207c 0a7c 2032 3331 3532 3520       |.| 231525 
│ │ │ │ -0005a350: 3220 3320 2020 2032 3038 3337 3235 3020  2 3    20837250 
│ │ │ │ -0005a360: 2030 2033 2020 2020 3139 3035 3132 3030   0 3    19051200
│ │ │ │ -0005a370: 3020 2030 2031 2033 2020 2020 2020 2020  0  0 1 3        
│ │ │ │ +0005a340: 2020 2020 2020 2020 2020 207c 0a7c 2032             |.| 2
│ │ │ │ +0005a350: 3331 3532 3520 3220 3320 2020 2032 3038  31525 2 3    208
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│ │ │ │ +0005a370: 3035 3132 3030 3020 2030 2031 2033 2020  0512000  0 1 3  
│ │ │ │  0005a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a390: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005a390: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -0005a3e0: 2020 2020 207c 0a7c 3220 2020 2020 3330       |.|2     30
│ │ │ │ -0005a3f0: 3437 3535 3737 2033 2020 2020 2033 3537  475577 3     357
│ │ │ │ -0005a400: 3736 3831 3920 3220 3220 2020 3135 3534  76819 2 2   1554
│ │ │ │ -0005a410: 3233 3037 3720 2020 2020 3220 2020 2020  23077     2     
│ │ │ │ -0005a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a430: 2020 2020 207c 0a7c 2078 2020 2d20 2d2d       |.| x  - --
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│ │ │ │ +0005a3e0: 2020 2020 2020 2020 2020 207c 0a7c 3220             |.|2 
│ │ │ │ +0005a3f0: 2020 2020 3330 3437 3535 3737 2033 2020      30475577 3  
│ │ │ │ +0005a400: 2020 2033 3537 3736 3831 3920 3220 3220     35776819 2 2 
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│ │ │ │ +0005a420: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │  0005a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a480: 2020 2020 207c 0a7c 3220 3320 2020 2031       |.|2 3    1
│ │ │ │ -0005a490: 3531 3230 3030 2032 2033 2020 2020 3730  512000 2 3    70
│ │ │ │ -0005a4a0: 3536 3030 2020 3020 3320 2020 2020 3334  5600  0 3     34
│ │ │ │ -0005a4b0: 3536 3030 2020 3020 3120 3320 2020 2020  5600  0 1 3     
│ │ │ │ -0005a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0005a480: 2020 2020 2020 2020 2020 207c 0a7c 3220             |.|2 
│ │ │ │ +0005a490: 3320 2020 2031 3531 3230 3030 2032 2033  3    1512000 2 3
│ │ │ │ +0005a4a0: 2020 2020 3730 3536 3030 2020 3020 3320      705600  0 3 
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│ │ │ │ +0005a4c0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0005a4d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0005a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0005a530: 3735 3934 3920 3320 2020 2020 3135 3134  75949 3     1514
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│ │ │ │ -0005a550: 3131 3836 3920 2020 2020 3220 2020 2020  11869     2     
│ │ │ │ -0005a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0005a520: 2020 2020 2020 2020 2020 207c 0a7c 3220             |.|2 
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│ │ │ │ +0005a5a0: 2b20 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078  + ---------x x x
│ │ │ │  0005a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a5c0: 2020 2020 207c 0a7c 3220 3320 2020 3335       |.|2 3   35
│ │ │ │ -0005a5d0: 3238 3030 3020 3220 3320 2020 2031 3034  28000 2 3    104
│ │ │ │ -0005a5e0: 3138 3632 3520 3020 3320 2020 2036 3335  18625 0 3    635
│ │ │ │ -0005a5f0: 3034 3030 3020 3020 3120 3320 2020 2020  04000 0 1 3     
│ │ │ │ -0005a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005a5c0: 2020 2020 2020 2020 2020 207c 0a7c 3220             |.|2 
│ │ │ │ +0005a5d0: 3320 2020 3335 3238 3030 3020 3220 3320  3   3528000 2 3 
│ │ │ │ +0005a5e0: 2020 2031 3034 3138 3632 3520 3020 3320     10418625 0 3 
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│ │ │ │ +0005a600: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0005a610: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -0005a660: 2020 2020 207c 0a7c 2020 3220 2020 2020       |.|  2     
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│ │ │ │ -0005a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0005a660: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  0005a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0005a7c0: 3334 3639 3133 3631 3037 3920 2020 2020  34691361079     
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│ │ │ │ +0005a7d0: 3920 2020 2020 3220 2020 3130 3436 3735  9     2   104675
│ │ │ │ +0005a7e0: 3332 3338 3720 2020 2020 3220 2020 3538  32387     2   58
│ │ │ │ +0005a7f0: 3832 3438 3936 3633 3320 327c 0a7c 7820  824896633 2|.|x 
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│ │ │ │ +0005a890: 3333 3335 3834 3030 3020 327c 0a7c 2020  333584000 2|.|  
│ │ │ │  0005a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0005a980: 2d2d 2d2d 2d7c 0a7c 2031 3038 3836 3430  -----|.| 1088640
│ │ │ │ -0005a990: 3020 3120 3320 2020 2036 3636 3739 3230  0 1 3    6667920
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│ │ │ │ -0005a9b0: 3838 3430 3030 2031 2032 2033 2020 2020  884000 1 2 3    
│ │ │ │ -0005a9c0: 3636 3637 3932 3030 2020 3220 3320 2020  66679200  2 3   
│ │ │ │ -0005a9d0: 2034 3736 327c 0a7c 2020 2020 2020 2020   4762|.|        
│ │ │ │ +0005a8e0: 2020 2020 2020 2020 2020 207c 0a7c 3534             |.|54
│ │ │ │ +0005a8f0: 3132 3637 3738 3120 3220 3220 2020 3335  1267781 2 2   35
│ │ │ │ +0005a900: 3731 3134 3333 3831 3720 2020 2020 3220  711433817     2 
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│ │ │ │ +0005a990: 3038 3836 3430 3020 3120 3320 2020 2036  0886400 1 3    6
│ │ │ │ +0005a9a0: 3636 3739 3230 3030 2020 3020 3220 3320  66792000  0 2 3 
│ │ │ │ +0005a9b0: 2020 2031 3432 3838 3430 3030 2031 2032     142884000 1 2
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│ │ │ │  0005aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0005aa50: 3036 3938 3236 3337 3720 2020 2020 3220  069826377     2 
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│ │ │ │ -0005aac0: 202d 2d2d 2d7c 0a7c 2020 2020 2036 3232   ----|.|     622
│ │ │ │ -0005aad0: 3038 3020 2031 2033 2020 2020 3132 3730  080  1 3    1270
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│ │ │ │ -0005aaf0: 3330 3633 3638 3030 2020 3120 3220 3320  30636800  1 2 3 
│ │ │ │ -0005ab00: 2020 2031 3538 3736 3030 2032 2033 2020     1587600 2 3  
│ │ │ │ -0005ab10: 2020 3135 317c 0a7c 2020 2020 2020 2020    151|.|        
│ │ │ │ +0005aa20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0005b280: 3030 2033 2020 2020 3935 3235 3630 3030  00 3    95256000
│ │ │ │ +0005b290: 2030 2034 2020 2020 3838 397c 0a7c 2020   0 4    889|.|  
│ │ │ │  0005b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b2e0: 2020 2020 207c 0a7c 3220 2020 3133 3536       |.|2   1356
│ │ │ │ -0005b2f0: 3239 3320 2020 3320 2020 3930 3338 3530  293   3   903850
│ │ │ │ -0005b300: 3231 2020 2033 2020 2031 3038 3031 3730  21   3   1080170
│ │ │ │ -0005b310: 3936 3920 2020 3320 2020 3234 3034 3339  969   3   240439
│ │ │ │ -0005b320: 2034 2020 2037 3038 3431 3733 3433 2033   4   708417343 3
│ │ │ │ -0005b330: 2020 2020 207c 0a7c 2020 2b20 2d2d 2d2d       |.|  + ----
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│ │ │ │ -0005b380: 7820 202d 207c 0a7c 3320 2020 2033 3532  x  - |.|3    352
│ │ │ │ -0005b390: 3830 3020 3020 3320 2020 2036 3034 3830  800 0 3    60480
│ │ │ │ -0005b3a0: 3020 2031 2033 2020 2020 3331 3735 3230  0  1 3    317520
│ │ │ │ -0005b3b0: 3030 2020 3220 3320 2020 2031 3531 3230  00  2 3    15120
│ │ │ │ -0005b3c0: 2033 2020 2020 3437 3632 3830 3030 2030   3    47628000 0
│ │ │ │ -0005b3d0: 2034 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d   4   |.|--------
│ │ │ │ +0005b2e0: 2020 2020 2020 2020 2020 207c 0a7c 3220             |.|2 
│ │ │ │ +0005b2f0: 2020 3133 3536 3239 3320 2020 3320 2020    1356293   3   
│ │ │ │ +0005b300: 3930 3338 3530 3231 2020 2033 2020 2031  90385021   3   1
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│ │ │ │ +0005b320: 3234 3034 3339 2034 2020 2037 3038 3431  240439 4   70841
│ │ │ │ +0005b330: 3733 3433 2033 2020 2020 207c 0a7c 2020  7343 3     |.|  
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│ │ │ │ +0005b380: 2d2d 2d2d 7820 7820 202d 207c 0a7c 3320  ----x x  - |.|3 
│ │ │ │ +0005b390: 2020 2033 3532 3830 3020 3020 3320 2020     352800 0 3   
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│ │ │ │ +0005b3b0: 3331 3735 3230 3030 2020 3220 3320 2020  31752000  2 3   
│ │ │ │ +0005b3c0: 2031 3531 3230 2033 2020 2020 3437 3632   15120 3    4762
│ │ │ │ +0005b3d0: 3830 3030 2030 2034 2020 207c 0a7c 2d2d  8000 0 4   |.|--
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│ │ │ │  0005b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b420: 2d2d 2d2d 2d7c 0a7c 3830 3336 3434 3231  -----|.|80364421
│ │ │ │ -0005b430: 2032 2020 2020 2020 2031 3238 3530 3435   2       1285045
│ │ │ │ -0005b440: 3036 3333 2020 2032 2020 2020 2033 3034  0633   2     304
│ │ │ │ -0005b450: 3730 3039 3239 2033 2020 2020 2036 3030  700929 3     600
│ │ │ │ -0005b460: 3731 3338 3837 2032 2020 2020 2020 2035  713887 2       5
│ │ │ │ -0005b470: 3230 3234 307c 0a7c 2d2d 2d2d 2d2d 2d2d  20240|.|--------
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│ │ │ │ -0005b4a0: 2d2d 2d2d 2d2d 7820 7820 202d 202d 2d2d  ------x x  - ---
│ │ │ │ -0005b4b0: 2d2d 2d2d 2d2d 7820 7820 7820 202b 202d  ------x x x  + -
│ │ │ │ -0005b4c0: 2d2d 2d2d 2d7c 0a7c 3438 3139 3230 3020  -----|.|4819200 
│ │ │ │ -0005b4d0: 2030 2031 2034 2020 2020 3135 3234 3039   0 1 4    152409
│ │ │ │ -0005b4e0: 3630 3020 2030 2031 2034 2020 2020 3736  600  0 1 4    76
│ │ │ │ -0005b4f0: 3230 3438 3020 2031 2034 2020 2031 3237  20480  1 4   127
│ │ │ │ -0005b500: 3030 3830 3030 2030 2032 2034 2020 2020  008000 0 2 4    
│ │ │ │ -0005b510: 3335 3536 327c 0a7c 2020 2020 2020 2020  35562|.|        
│ │ │ │ +0005b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3830  -----------|.|80
│ │ │ │ +0005b430: 3336 3434 3231 2032 2020 2020 2020 2031  364421 2       1
│ │ │ │ +0005b440: 3238 3530 3435 3036 3333 2020 2032 2020  2850450633   2  
│ │ │ │ +0005b450: 2020 2033 3034 3730 3039 3239 2033 2020     304700929 3  
│ │ │ │ +0005b460: 2020 2036 3030 3731 3338 3837 2032 2020     600713887 2  
│ │ │ │ +0005b470: 2020 2020 2035 3230 3234 307c 0a7c 2d2d       520240|.|--
│ │ │ │ +0005b480: 2d2d 2d2d 2d2d 7820 7820 7820 202d 202d  ------x x x  - -
│ │ │ │ +0005b490: 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820 7820  ----------x x x 
│ │ │ │ +0005b4a0: 202d 202d 2d2d 2d2d 2d2d 2d2d 7820 7820   - ---------x x 
│ │ │ │ +0005b4b0: 202d 202d 2d2d 2d2d 2d2d 2d2d 7820 7820   - ---------x x 
│ │ │ │ +0005b4c0: 7820 202b 202d 2d2d 2d2d 2d7c 0a7c 3438  x  + ------|.|48
│ │ │ │ +0005b4d0: 3139 3230 3020 2030 2031 2034 2020 2020  19200  0 1 4    
│ │ │ │ +0005b4e0: 3135 3234 3039 3630 3020 2030 2031 2034  152409600  0 1 4
│ │ │ │ +0005b4f0: 2020 2020 3736 3230 3438 3020 2031 2034      7620480  1 4
│ │ │ │ +0005b500: 2020 2031 3237 3030 3830 3030 2030 2032     127008000 0 2
│ │ │ │ +0005b510: 2034 2020 2020 3335 3536 327c 0a7c 2020   4    35562|.|  
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│ │ │ │ -0005b560: 2020 2020 207c 0a7c 3739 2032 2020 2020       |.|79 2    
│ │ │ │ -0005b570: 2020 2034 3835 3439 3032 3537 3233 2020     48549025723  
│ │ │ │ -0005b580: 2032 2020 2020 2037 3334 3232 3439 3331   2     734224931
│ │ │ │ -0005b590: 2033 2020 2020 2034 3039 3036 3335 3431   3     409063541
│ │ │ │ -0005b5a0: 2032 2020 2020 2020 2031 3831 3933 3137   2       1819317
│ │ │ │ -0005b5b0: 3232 3920 207c 0a7c 2d2d 7820 7820 7820  229  |.|--x x x 
│ │ │ │ -0005b5c0: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820   + -----------x 
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│ │ │ │ -0005b5e0: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d2d  x x  + ---------
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│ │ │ │ -0005b600: 2d2d 2d78 207c 0a7c 3020 2030 2031 2034  ---x |.|0  0 1 4
│ │ │ │ -0005b610: 2020 2020 3235 3430 3136 3030 3020 2030      254016000  0
│ │ │ │ -0005b620: 2031 2034 2020 2020 3234 3139 3230 3020   1 4    2419200 
│ │ │ │ -0005b630: 2031 2034 2020 2020 3133 3630 3830 3030   1 4    13608000
│ │ │ │ -0005b640: 2030 2032 2034 2020 2020 2037 3035 3630   0 2 4     70560
│ │ │ │ -0005b650: 3030 2020 307c 0a7c 2020 2020 2020 2020  00  0|.|        
│ │ │ │ +0005b560: 2020 2020 2020 2020 2020 207c 0a7c 3739             |.|79
│ │ │ │ +0005b570: 2032 2020 2020 2020 2034 3835 3439 3032   2       4854902
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│ │ │ │ +0005b590: 3232 3439 3331 2033 2020 2020 2034 3039  224931 3     409
│ │ │ │ +0005b5a0: 3036 3335 3431 2032 2020 2020 2020 2031  063541 2       1
│ │ │ │ +0005b5b0: 3831 3933 3137 3232 3920 207c 0a7c 2d2d  819317229  |.|--
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│ │ │ │ +0005b5f0: 2d2d 2d2d 2d2d 7820 7820 7820 202b 202d  ------x x x  + -
│ │ │ │ +0005b600: 2d2d 2d2d 2d2d 2d2d 2d78 207c 0a7c 3020  ---------x |.|0 
│ │ │ │ +0005b610: 2030 2031 2034 2020 2020 3235 3430 3136   0 1 4    254016
│ │ │ │ +0005b620: 3030 3020 2030 2031 2034 2020 2020 3234  000  0 1 4    24
│ │ │ │ +0005b630: 3139 3230 3020 2031 2034 2020 2020 3133  19200  1 4    13
│ │ │ │ +0005b640: 3630 3830 3030 2030 2032 2034 2020 2020  608000 0 2 4    
│ │ │ │ +0005b650: 2037 3035 3630 3030 2020 307c 0a7c 2020   7056000  0|.|  
│ │ │ │  0005b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0005b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b6a0: 2020 2020 207c 0a7c 2020 2020 2034 3732       |.|     472
│ │ │ │ -0005b6b0: 3033 3835 3235 3131 2020 2032 2020 2020  03852511   2    
│ │ │ │ -0005b6c0: 2034 3034 3339 3137 3637 2033 2020 2020   404391767 3    
│ │ │ │ -0005b6d0: 2038 3738 3439 3837 3835 3133 2032 2020   87849878513 2  
│ │ │ │ -0005b6e0: 2020 2020 2039 3235 3633 3338 3439 3433       92563384943
│ │ │ │ -0005b6f0: 2020 2020 207c 0a7c 7820 202b 202d 2d2d       |.|x  + ---
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│ │ │ │ -0005b710: 202d 2d2d 2d2d 2d2d 2d2d 7820 7820 202b   ---------x x  +
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│ │ │ │ -0005b790: 2030 2031 207c 0a7c 2020 2020 2020 2020   0 1 |.|        
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│ │ │ │ +0005b6c0: 2032 2020 2020 2034 3034 3339 3137 3637   2     404391767
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│ │ │ │ +0005b6e0: 3133 2032 2020 2020 2020 2039 3235 3633  13 2       92563
│ │ │ │ +0005b6f0: 3338 3439 3433 2020 2020 207c 0a7c 7820  384943     |.|x 
│ │ │ │ +0005b700: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820   + -----------x 
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│ │ │ │ +0005b770: 2031 2034 2020 2020 3139 3035 3132 3030   1 4    19051200
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│ │ │ │ +0005b790: 3136 3030 3020 2030 2031 207c 0a7c 2020  16000  0 1 |.|  
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│ │ │ │ -0005b880: 202d 2d2d 2d7c 0a7c 3035 3630 3030 2020   ----|.|056000  
│ │ │ │ -0005b890: 3020 3120 3420 2020 2020 3533 3334 3333  0 1 4     533433
│ │ │ │ -0005b8a0: 3630 3030 2020 3020 3120 3420 2020 2033  6000  0 1 4    3
│ │ │ │ -0005b8b0: 3831 3032 3430 3020 2031 2034 2020 2020  8102400  1 4    
│ │ │ │ -0005b8c0: 3335 3732 3130 3030 2030 2032 2034 2020  35721000 0 2 4  
│ │ │ │ -0005b8d0: 2020 3537 317c 0a7c 2020 2020 2020 2020    571|.|        
│ │ │ │ +0005b7e0: 2020 2020 2020 2020 2020 207c 0a7c 3338             |.|38
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│ │ │ │ +0005b870: 7820 202d 202d 2d2d 2d2d 2d2d 2d2d 7820  x  - ---------x 
│ │ │ │ +0005b880: 7820 7820 202d 202d 2d2d 2d7c 0a7c 3035  x x  - ----|.|05
│ │ │ │ +0005b890: 3630 3030 2020 3020 3120 3420 2020 2020  6000  0 1 4     
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│ │ │ │ +0005b8b0: 3420 2020 2033 3831 3032 3430 3020 2031  4    38102400  1
│ │ │ │ +0005b8c0: 2034 2020 2020 3335 3732 3130 3030 2030   4    35721000 0
│ │ │ │ +0005b8d0: 2032 2034 2020 2020 3537 317c 0a7c 2020   2 4    571|.|  
│ │ │ │  0005b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0005b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0005b940: 3830 3232 3937 3431 2020 2032 2020 2020  80229741   2    
│ │ │ │ -0005b950: 2038 3233 3432 3639 3135 3720 3320 2020   8234269157 3   
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│ │ │ │ -0005b970: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ -0005b9c0: 2078 2020 2d7c 0a7c 2034 3434 3532 3830   x  -|.| 4445280
│ │ │ │ -0005b9d0: 3030 2020 3020 3120 3420 2020 2034 3236  00  0 1 4    426
│ │ │ │ -0005b9e0: 3734 3638 3830 3020 2030 2031 2034 2020  7468800  0 1 4  
│ │ │ │ -0005b9f0: 2020 3135 3234 3039 3630 3020 3120 3420    152409600 1 4 
│ │ │ │ -0005ba00: 2020 2037 3134 3432 3030 3030 2020 3020     714420000  0 
│ │ │ │ -0005ba10: 3220 3420 207c 0a7c 2d2d 2d2d 2d2d 2d2d  2 4  |.|--------
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│ │ │ │ +0005b9c0: 2d2d 2d78 2078 2078 2020 2d7c 0a7c 2034  ---x x x  -|.| 4
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│ │ │ │ +0005b9e0: 2020 2034 3236 3734 3638 3830 3020 2030     4267468800  0
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│ │ │ │ +0005ba10: 3030 2020 3020 3220 3420 207c 0a7c 2d2d  00  0 2 4  |.|--
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│ │ │ │  0005ba30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0005ba50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0005ba70: 2020 2020 2020 2020 2031 3233 3532 3137           1235217
│ │ │ │ -0005ba80: 3438 3739 2032 2020 2020 2020 2031 3932  4879 2       192
│ │ │ │ -0005ba90: 3034 3937 3531 2020 2032 2020 2020 2033  049751   2     3
│ │ │ │ -0005baa0: 3330 3238 3031 3936 3320 2020 3220 2020  302801963   2   
│ │ │ │ -0005bab0: 2020 3336 397c 0a7c 2d2d 2d2d 2d2d 7820    369|.|------x 
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│ │ │ │ -0005bb00: 2b20 2d2d 2d7c 0a7c 3234 3030 3020 2030  + ---|.|24000  0
│ │ │ │ -0005bb10: 2031 2032 2034 2020 2020 3236 3637 3136   1 2 4    266716
│ │ │ │ -0005bb20: 3830 3020 2031 2032 2034 2020 2020 3432  800  1 2 4    42
│ │ │ │ -0005bb30: 3333 3630 3030 2030 2032 2034 2020 2020  336000 0 2 4    
│ │ │ │ -0005bb40: 3834 3637 3230 3030 2020 3120 3220 3420  84672000  1 2 4 
│ │ │ │ -0005bb50: 2020 2037 307c 0a7c 2020 2020 2020 2020     70|.|        
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│ │ │ │ +0005ba80: 3233 3532 3137 3438 3739 2032 2020 2020  2352174879 2    
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│ │ │ │ +0005baa0: 2020 2020 2033 3330 3238 3031 3936 3320       3302801963 
│ │ │ │ +0005bab0: 2020 3220 2020 2020 3336 397c 0a7c 2d2d    2     369|.|--
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│ │ │ │ +0005bb00: 2078 2078 2020 2b20 2d2d 2d7c 0a7c 3234   x x  + ---|.|24
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│ │ │ │ +0005bb30: 2020 2020 3432 3333 3630 3030 2030 2032      42336000 0 2
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│ │ │ │ +0005bb50: 3120 3220 3420 2020 2037 307c 0a7c 2020  1 2 4    70|.|  
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│ │ │ │ -0005bba0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005bbb0: 2039 3837 3932 3735 3239 3633 2032 2020   98792752963 2  
│ │ │ │ -0005bbc0: 2020 2020 2032 3734 3539 3632 3831 2020       274596281  
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│ │ │ │ -0005bbe0: 3233 2020 2032 2020 2020 2031 3232 3539  23   2     12259
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│ │ │ │ -0005bc60: 2034 2020 2020 3935 3235 3630 3030 2030   4    95256000 0
│ │ │ │ -0005bc70: 2032 2034 2020 2020 3533 3334 3333 3630   2 4    53343360
│ │ │ │ -0005bc80: 3020 2031 2032 2034 2020 2020 3135 3837  0  1 2 4    1587
│ │ │ │ -0005bc90: 3630 3030 207c 0a7c 2020 2020 2020 2020  6000 |.|        
│ │ │ │ +0005bba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0005bbc0: 3633 2032 2020 2020 2020 2032 3734 3539  63 2       27459
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│ │ │ │ +0005bbf0: 2031 3232 3539 3236 3931 207c 0a7c 7820   122592691 |.|x 
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│ │ │ │ +0005bc40: 202d 2d2d 2d2d 2d2d 2d2d 787c 0a7c 2031   ---------x|.| 1
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│ │ │ │ +0005bc90: 2020 3135 3837 3630 3030 207c 0a7c 2020    15876000 |.|  
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│ │ │ │ -0005bce0: 2020 2020 207c 0a7c 2020 2020 2020 3437       |.|      47
│ │ │ │ -0005bcf0: 3632 3233 3632 3937 3233 2032 2020 2020  6223629723 2    
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│ │ │ │ -0005bd20: 2020 2032 2020 2020 2031 3630 3637 3739     2     1606779
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│ │ │ │ -0005bd50: 202d 202d 2d2d 2d2d 2d2d 2d2d 7820 7820   - ---------x x 
│ │ │ │ -0005bd60: 7820 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d  x  + -----------
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│ │ │ │ -0005bd80: 2d78 2078 207c 0a7c 3220 3420 2020 2020  -x x |.|2 4     
│ │ │ │ -0005bd90: 3530 3830 3332 3030 3020 2031 2032 2034  508032000  1 2 4
│ │ │ │ -0005bda0: 2020 2020 3736 3230 3438 3030 2030 2032      76204800 0 2
│ │ │ │ -0005bdb0: 2034 2020 2020 3132 3730 3038 3030 3020   4    127008000 
│ │ │ │ -0005bdc0: 2031 2032 2034 2020 2020 3138 3134 3430   1 2 4    181440
│ │ │ │ -0005bdd0: 3020 3220 347c 0a7c 2020 2020 2020 2020  0 2 4|.|        
│ │ │ │ +0005bce0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005bcf0: 2020 2020 3437 3632 3233 3632 3937 3233      476223629723
│ │ │ │ +0005bd00: 2032 2020 2020 2020 2032 3433 3834 3330   2       2438430
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│ │ │ │ +0005bd80: 2d2d 2d2d 2d2d 2d78 2078 207c 0a7c 3220  -------x x |.|2 
│ │ │ │ +0005bd90: 3420 2020 2020 3530 3830 3332 3030 3020  4     508032000 
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│ │ │ │ +0005bdb0: 3030 2030 2032 2034 2020 2020 3132 3730  00 0 2 4    1270
│ │ │ │ +0005bdc0: 3038 3030 3020 2031 2032 2034 2020 2020  08000  1 2 4    
│ │ │ │ +0005bdd0: 3138 3134 3430 3020 3220 347c 0a7c 2020  1814400 2 4|.|  
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│ │ │ │ -0005be20: 2020 2020 207c 0a7c 3736 3738 3937 3733       |.|76789773
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│ │ │ │ -0005be50: 3434 3132 3930 3739 2020 2032 2020 2020  44129079   2    
│ │ │ │ -0005be60: 2031 3131 3034 3034 3837 3230 3320 2020   111040487203   
│ │ │ │ -0005be70: 3220 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d  2    |.|--------
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│ │ │ │ -0005beb0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078   ------------x x
│ │ │ │ -0005bec0: 2078 2020 2d7c 0a7c 3533 3630 3030 3020   x  -|.|5360000 
│ │ │ │ -0005bed0: 2030 2031 2032 2034 2020 2020 3338 3130   0 1 2 4    3810
│ │ │ │ -0005bee0: 3234 3030 3020 3120 3220 3420 2020 2039  24000 1 2 4    9
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│ │ │ │ -0005bf00: 2020 2039 3532 3536 3030 3030 2020 3120     952560000  1 
│ │ │ │ -0005bf10: 3220 3420 207c 0a7c 2020 2020 2020 2020  2 4  |.|        
│ │ │ │ +0005be20: 2020 2020 2020 2020 2020 207c 0a7c 3736             |.|76
│ │ │ │ +0005be30: 3738 3937 3733 2020 2020 2020 2020 2020  789773          
│ │ │ │ +0005be40: 2037 3139 3635 3137 3931 3320 3220 2020   7196517913 2   
│ │ │ │ +0005be50: 2020 2020 3439 3434 3132 3930 3739 2020      4944129079  
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│ │ │ │ +0005be70: 3230 3320 2020 3220 2020 207c 0a7c 2d2d  203   2    |.|--
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│ │ │ │ +0005be90: 202d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078   ----------x x x
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│ │ │ │ +0005bec0: 2d2d 2d78 2078 2078 2020 2d7c 0a7c 3533  ---x x x  -|.|53
│ │ │ │ +0005bed0: 3630 3030 3020 2030 2031 2032 2034 2020  60000  0 1 2 4  
│ │ │ │ +0005bee0: 2020 3338 3130 3234 3030 3020 3120 3220    381024000 1 2 
│ │ │ │ +0005bef0: 3420 2020 2039 3532 3536 3030 3020 2030  4    95256000  0
│ │ │ │ +0005bf00: 2032 2034 2020 2020 2039 3532 3536 3030   2 4     9525600
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│ │ │ │ -0005bfb0: 3237 3637 207c 0a7c 202d 2d2d 2d2d 2d2d  2767 |.| -------
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│ │ │ │ -0005c000: 2d2d 2d2d 787c 0a7c 2020 3136 3030 3330  ----x|.|  160030
│ │ │ │ -0005c010: 3038 3030 3020 2030 2031 2032 2034 2020  08000  0 1 2 4  
│ │ │ │ -0005c020: 2020 3135 3234 3039 3630 3030 2020 3120    1524096000  1 
│ │ │ │ -0005c030: 3220 3420 2020 2039 3532 3536 3030 3020  2 4    95256000 
│ │ │ │ -0005c040: 2030 2032 2034 2020 2020 3533 3334 3333   0 2 4    533433
│ │ │ │ -0005c050: 3630 3020 207c 0a7c 2d2d 2d2d 2d2d 2d2d  600  |.|--------
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│ │ │ │ +0005bfb0: 3338 3336 3439 3237 3637 207c 0a7c 202d  3836492767 |.| -
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│ │ │ │ -0005c160: 3320 3420 2020 2037 3833 3832 3038 3020  3 4    78382080 
│ │ │ │ -0005c170: 2030 2031 2033 2034 2020 2020 3534 3836   0 1 3 4    5486
│ │ │ │ -0005c180: 3734 3536 2020 3120 3320 3420 2020 2035  7456  1 3 4    5
│ │ │ │ -0005c190: 3333 3433 337c 0a7c 2020 2020 2020 2020  33433|.|        
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│ │ │ │ +0005c0b0: 3636 3120 3320 2020 2020 3538 3835 3737  661 3     588577
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│ │ │ │ +0005c0d0: 3739 3635 3039 2020 2020 2020 2020 2020  796509          
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│ │ │ │ -0005c1f0: 3135 3933 3436 3720 3220 2020 2020 2020  1593467 2       
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│ │ │ │ -0005c2b0: 3120 3320 3420 2020 2031 3136 3634 3030  1 3 4    1166400
│ │ │ │ -0005c2c0: 3020 2031 2033 2034 2020 2020 3137 3134  0  1 3 4    1714
│ │ │ │ -0005c2d0: 3630 3830 307c 0a7c 2020 2020 2020 2020  60800|.|        
│ │ │ │ +0005c1e0: 2020 2020 2020 2020 2020 207c 0a7c 3320             |.|3 
│ │ │ │ +0005c1f0: 2020 2020 2036 3135 3933 3436 3720 3220       61593467 2 
│ │ │ │ +0005c200: 2020 2020 2020 3134 3330 3434 3138 3135        1430441815
│ │ │ │ +0005c210: 3231 3720 2020 2020 2020 2020 2020 3734  217           74
│ │ │ │ +0005c220: 3239 3733 3335 3931 2032 2020 2020 2020  29733591 2      
│ │ │ │ +0005c230: 2038 3330 3539 3836 3539 367c 0a7c 2078   8305986596|.| x
│ │ │ │ +0005c240: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78 2078    + ---------x x
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│ │ │ │ +0005c270: 2d2d 2d2d 2d2d 2d2d 7820 7820 7820 202b  --------x x x  +
│ │ │ │ +0005c280: 202d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3220   ----------|.|2 
│ │ │ │ +0005c290: 3420 2020 3230 3030 3337 3630 3020 3020  4   200037600 0 
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│ │ │ │ +0005c2b0: 3030 2020 3020 3120 3320 3420 2020 2031  00  0 1 3 4    1
│ │ │ │ +0005c2c0: 3136 3634 3030 3020 2031 2033 2034 2020  1664000  1 3 4  
│ │ │ │ +0005c2d0: 2020 3137 3134 3630 3830 307c 0a7c 2020    171460800|.|  
│ │ │ │  0005c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c320: 2020 2020 207c 0a7c 2020 2032 3533 3634       |.|   25364
│ │ │ │ -0005c330: 3139 3439 2032 2020 2020 2020 2031 3032  1949 2       102
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│ │ │ │ -0005c350: 2031 3330 3839 3537 3338 3320 3220 2020   1308957383 2   
│ │ │ │ -0005c360: 2020 2020 3139 3935 3936 3339 3738 3720      19959639787 
│ │ │ │ -0005c370: 2020 2020 207c 0a7c 202b 202d 2d2d 2d2d       |.| + -----
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│ │ │ │ -0005c3c0: 2078 2078 207c 0a7c 2020 2020 3339 3639   x x |.|    3969
│ │ │ │ -0005c3d0: 3030 3020 2030 2033 2034 2020 2020 3331  000  0 3 4    31
│ │ │ │ -0005c3e0: 3735 3230 3030 2030 2031 2033 2034 2020  752000 0 1 3 4  
│ │ │ │ -0005c3f0: 2020 2038 3036 3430 3020 2020 3120 3320     806400   1 3 
│ │ │ │ -0005c400: 3420 2020 2033 3831 3032 3430 3030 2020  4    381024000  
│ │ │ │ -0005c410: 3020 3220 337c 0a7c 2020 2020 2020 2020  0 2 3|.|        
│ │ │ │ +0005c320: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0005c350: 2020 2020 2020 2031 3330 3839 3537 3338         130895738
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│ │ │ │ +0005c370: 3339 3738 3720 2020 2020 207c 0a7c 202b  39787      |.| +
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│ │ │ │ +0005c3d0: 2020 3339 3639 3030 3020 2030 2033 2034    3969000  0 3 4
│ │ │ │ +0005c3e0: 2020 2020 3331 3735 3230 3030 2030 2031      31752000 0 1
│ │ │ │ +0005c3f0: 2033 2034 2020 2020 2038 3036 3430 3020   3 4     806400 
│ │ │ │ +0005c400: 2020 3120 3320 3420 2020 2033 3831 3032    1 3 4    38102
│ │ │ │ +0005c410: 3430 3030 2020 3020 3220 337c 0a7c 2020  4000  0 2 3|.|  
│ │ │ │  0005c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0005c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c460: 2020 2020 207c 0a7c 2031 3936 3037 3131       |.| 1960711
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│ │ │ │ -0005c480: 3837 3720 3220 2020 2020 2020 3239 3835  877 2       2985
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│ │ │ │ -0005c4a0: 2020 3233 3136 3135 3438 3531 3120 3220    23161548511 2 
│ │ │ │ -0005c4b0: 2020 2020 207c 0a7c 202d 2d2d 2d2d 2d2d       |.| -------
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│ │ │ │ -0005c500: 2078 2020 2b7c 0a7c 2020 3834 3030 3020   x  +|.|  84000 
│ │ │ │ -0005c510: 2032 2034 2020 2020 3130 3030 3138 3830   2 4    10001880
│ │ │ │ -0005c520: 3030 3020 3020 3320 3420 2020 2032 3238  000 0 3 4    228
│ │ │ │ -0005c530: 3631 3434 3030 2020 3020 3120 3320 3420  614400  0 1 3 4 
│ │ │ │ -0005c540: 2020 2032 3238 3631 3434 3030 2020 3120     228614400  1 
│ │ │ │ -0005c550: 3320 3420 207c 0a7c 2020 2020 2020 2020  3 4  |.|        
│ │ │ │ +0005c460: 2020 2020 2020 2020 2020 207c 0a7c 2031             |.| 1
│ │ │ │ +0005c470: 3936 3037 3131 2033 2020 2020 2037 3338  960711 3     738
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│ │ │ │ +0005c4a0: 2020 2020 2020 2020 3233 3136 3135 3438          23161548
│ │ │ │ +0005c4b0: 3531 3120 3220 2020 2020 207c 0a7c 202d  511 2      |.| -
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│ │ │ │ +0005c4e0: 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078  + -----------x x
│ │ │ │ +0005c4f0: 2078 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d   x x  - --------
│ │ │ │ +0005c500: 2d2d 2d78 2078 2078 2020 2b7c 0a7c 2020  ---x x x  +|.|  
│ │ │ │ +0005c510: 3834 3030 3020 2032 2034 2020 2020 3130  84000  2 4    10
│ │ │ │ +0005c520: 3030 3138 3830 3030 3020 3020 3320 3420  001880000 0 3 4 
│ │ │ │ +0005c530: 2020 2032 3238 3631 3434 3030 2020 3020     228614400  0 
│ │ │ │ +0005c540: 3120 3320 3420 2020 2032 3238 3631 3434  1 3 4    2286144
│ │ │ │ +0005c550: 3030 2020 3120 3320 3420 207c 0a7c 2020  00  1 3 4  |.|  
│ │ │ │  0005c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c5a0: 2020 2020 207c 0a7c 2020 3220 2020 2020       |.|  2     
│ │ │ │ -0005c5b0: 3630 3833 3231 3920 3320 2020 2020 3639  6083219 3     69
│ │ │ │ -0005c5c0: 3236 3830 3831 3730 3920 3220 2020 2020  268081709 2     
│ │ │ │ -0005c5d0: 2020 3136 3632 3934 3531 3436 3437 2020    166294514647  
│ │ │ │ -0005c5e0: 2020 2020 2020 2020 2031 3838 3230 3238           1882028
│ │ │ │ -0005c5f0: 3932 3630 317c 0a7c 2078 2078 2020 2d20  92601|.| x x  - 
│ │ │ │ -0005c600: 2d2d 2d2d 2d2d 2d78 2078 2020 2b20 2d2d  -------x x  + --
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│ │ │ │ -0005c620: 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820  + ------------x 
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│ │ │ │ -0005c640: 2d2d 2d2d 2d7c 0a7c 3120 3220 3420 2020  -----|.|1 2 4   
│ │ │ │ -0005c650: 2032 3131 3638 3020 3220 3420 2020 2031   211680 2 4    1
│ │ │ │ -0005c660: 3030 3031 3838 3030 3020 3020 3320 3420  000188000 0 3 4 
│ │ │ │ -0005c670: 2020 2032 3636 3731 3638 3030 3020 2030     2667168000  0
│ │ │ │ -0005c680: 2031 2033 2034 2020 2020 2034 3537 3232   1 3 4     45722
│ │ │ │ -0005c690: 3838 3030 207c 0a7c 2d2d 2d2d 2d2d 2d2d  8800 |.|--------
│ │ │ │ +0005c5a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005c5b0: 3220 2020 2020 3630 3833 3231 3920 3320  2     6083219 3 
│ │ │ │ +0005c5c0: 2020 2020 3639 3236 3830 3831 3730 3920      69268081709 
│ │ │ │ +0005c5d0: 3220 2020 2020 2020 3136 3632 3934 3531  2       16629451
│ │ │ │ +0005c5e0: 3436 3437 2020 2020 2020 2020 2020 2031  4647           1
│ │ │ │ +0005c5f0: 3838 3230 3238 3932 3630 317c 0a7c 2078  88202892601|.| x
│ │ │ │ +0005c600: 2078 2020 2d20 2d2d 2d2d 2d2d 2d78 2078   x  - -------x x
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│ │ │ │ +0005c630: 2d2d 2d2d 7820 7820 7820 7820 202b 202d  ----x x x x  + -
│ │ │ │ +0005c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3120  -----------|.|1 
│ │ │ │ +0005c650: 3220 3420 2020 2032 3131 3638 3020 3220  2 4    211680 2 
│ │ │ │ +0005c660: 3420 2020 2031 3030 3031 3838 3030 3020  4    1000188000 
│ │ │ │ +0005c670: 3020 3320 3420 2020 2032 3636 3731 3638  0 3 4    2667168
│ │ │ │ +0005c680: 3030 3020 2030 2031 2033 2034 2020 2020  000  0 1 3 4    
│ │ │ │ +0005c690: 2034 3537 3232 3838 3030 207c 0a7c 2d2d   457228800 |.|--
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│ │ │ │  0005c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c6e0: 2d2d 2d2d 2d7c 0a7c 3633 3633 2020 2020  -----|.|6363    
│ │ │ │ -0005c6f0: 2020 2020 2020 2031 3037 3631 3938 3433         107619843
│ │ │ │ -0005c700: 3631 2020 2020 2020 2020 2020 2031 3534  61           154
│ │ │ │ -0005c710: 3037 3533 3233 3720 3220 2020 2020 2020  0753237 2       
│ │ │ │ -0005c720: 3738 3330 3533 3230 3537 2020 2032 2020  7830532057   2  
│ │ │ │ -0005c730: 2020 2031 307c 0a7c 2d2d 2d2d 7820 7820     10|.|----x x 
│ │ │ │ -0005c740: 7820 7820 202d 202d 2d2d 2d2d 2d2d 2d2d  x x  - ---------
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│ │ │ │ -0005c780: 202b 202d 2d7c 0a7c 3630 3020 2030 2032   + --|.|600  0 2
│ │ │ │ -0005c790: 2033 2034 2020 2020 2036 3533 3138 3430   3 4     6531840
│ │ │ │ -0005c7a0: 3020 2031 2032 2033 2034 2020 2020 3135  0  1 2 3 4    15
│ │ │ │ -0005c7b0: 3837 3630 3030 2020 3220 3320 3420 2020  876000  2 3 4   
│ │ │ │ -0005c7c0: 2034 3238 3635 3230 3030 2030 2033 2034   428652000 0 3 4
│ │ │ │ -0005c7d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005c6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3633  -----------|.|63
│ │ │ │ +0005c6f0: 3633 2020 2020 2020 2020 2020 2031 3037  63           107
│ │ │ │ +0005c700: 3631 3938 3433 3631 2020 2020 2020 2020  61984361        
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│ │ │ │ +0005c790: 3020 2030 2032 2033 2034 2020 2020 2036  0  0 2 3 4     6
│ │ │ │ +0005c7a0: 3533 3138 3430 3020 2031 2032 2033 2034  5318400  1 2 3 4
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│ │ │ │ +0005c7d0: 2030 2033 2034 2020 2020 207c 0a7c 2020   0 3 4     |.|  
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│ │ │ │ -0005c8d0: 3420 2020 2033 3432 3932 3136 3030 3030  4    34292160000
│ │ │ │ -0005c8e0: 2020 3120 3220 3320 3420 2020 2034 3434    1 2 3 4    444
│ │ │ │ -0005c8f0: 3532 3830 3030 2020 3220 3320 3420 2020  528000  2 3 4   
│ │ │ │ -0005c900: 2031 3432 3838 3430 3030 3020 2030 2033   1428840000  0 3
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│ │ │ │ -0005ca10: 3433 3037 3230 3030 2020 3120 3220 3320  43072000  1 2 3 
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│ │ │ │ +0005c9d0: 2078 2078 2078 2020 2b20 2d2d 2d2d 2d2d   x x x  + ------
│ │ │ │ +0005c9e0: 2d2d 2d2d 7820 7820 7820 202b 202d 2d2d  ----x x x  + ---
│ │ │ │ +0005c9f0: 2d2d 2d2d 2d2d 2d78 2078 2078 2020 2d20  -------x x x  - 
│ │ │ │ +0005ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 787c 0a7c 2034  ----------x|.| 4
│ │ │ │ +0005ca10: 2020 2020 3131 3433 3037 3230 3030 2020      1143072000  
│ │ │ │ +0005ca20: 3120 3220 3320 3420 2020 2032 3835 3736  1 2 3 4    28576
│ │ │ │ +0005ca30: 3830 3030 2032 2033 2034 2020 2020 3437  8000 2 3 4    47
│ │ │ │ +0005ca40: 3632 3830 3030 2020 3020 3320 3420 2020  628000  0 3 4   
│ │ │ │ +0005ca50: 2020 3635 3331 3834 3020 207c 0a7c 2020    6531840  |.|  
│ │ │ │  0005ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005caa0: 2020 2020 207c 0a7c 2033 3530 3934 3833       |.| 3509483
│ │ │ │ -0005cab0: 3833 2020 2020 2020 2020 2020 2035 3030  83           500
│ │ │ │ -0005cac0: 3836 3730 3637 3730 3920 2020 2020 2020  867067709       
│ │ │ │ -0005cad0: 2020 2020 3335 3837 3835 3232 3639 2032      3587852269 2
│ │ │ │ -0005cae0: 2020 2020 2020 2033 3535 3837 3232 3230         355872220
│ │ │ │ -0005caf0: 3637 2020 207c 0a7c 202d 2d2d 2d2d 2d2d  67   |.| -------
│ │ │ │ -0005cb00: 2d2d 7820 7820 7820 7820 202b 202d 2d2d  --x x x x  + ---
│ │ │ │ -0005cb10: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2078  ---------x x x x
│ │ │ │ -0005cb20: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 7820    + ----------x 
│ │ │ │ -0005cb30: 7820 7820 202d 202d 2d2d 2d2d 2d2d 2d2d  x x  - ---------
│ │ │ │ -0005cb40: 2d2d 7820 787c 0a7c 2020 3334 3939 3230  --x x|.|  349920
│ │ │ │ -0005cb50: 3020 2030 2032 2033 2034 2020 2020 3137  0  0 2 3 4    17
│ │ │ │ -0005cb60: 3134 3630 3830 3030 2020 3120 3220 3320  14608000  1 2 3 
│ │ │ │ -0005cb70: 3420 2020 2031 3035 3834 3030 3030 2032  4    105840000 2
│ │ │ │ -0005cb80: 2033 2034 2020 2020 3830 3031 3530 3430   3 4    80015040
│ │ │ │ -0005cb90: 3030 2030 207c 0a7c 2020 2020 2020 2020  00 0 |.|        
│ │ │ │ +0005caa0: 2020 2020 2020 2020 2020 207c 0a7c 2033             |.| 3
│ │ │ │ +0005cab0: 3530 3934 3833 3833 2020 2020 2020 2020  50948383        
│ │ │ │ +0005cac0: 2020 2035 3030 3836 3730 3637 3730 3920     500867067709 
│ │ │ │ +0005cad0: 2020 2020 2020 2020 2020 3335 3837 3835            358785
│ │ │ │ +0005cae0: 3232 3639 2032 2020 2020 2020 2033 3535  2269 2       355
│ │ │ │ +0005caf0: 3837 3232 3230 3637 2020 207c 0a7c 202d  87222067   |.| -
│ │ │ │ +0005cb00: 2d2d 2d2d 2d2d 2d2d 7820 7820 7820 7820  --------x x x x 
│ │ │ │ +0005cb10: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78   + ------------x
│ │ │ │ +0005cb20: 2078 2078 2078 2020 2b20 2d2d 2d2d 2d2d   x x x  + ------
│ │ │ │ +0005cb30: 2d2d 2d2d 7820 7820 7820 202d 202d 2d2d  ----x x x  - ---
│ │ │ │ +0005cb40: 2d2d 2d2d 2d2d 2d2d 7820 787c 0a7c 2020  --------x x|.|  
│ │ │ │ +0005cb50: 3334 3939 3230 3020 2030 2032 2033 2034  3499200  0 2 3 4
│ │ │ │ +0005cb60: 2020 2020 3137 3134 3630 3830 3030 2020      1714608000  
│ │ │ │ +0005cb70: 3120 3220 3320 3420 2020 2031 3035 3834  1 2 3 4    10584
│ │ │ │ +0005cb80: 3030 3030 2032 2033 2034 2020 2020 3830  0000 2 3 4    80
│ │ │ │ +0005cb90: 3031 3530 3430 3030 2030 207c 0a7c 2020  01504000 0 |.|  
│ │ │ │  0005cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cbe0: 2020 2020 207c 0a7c 2032 2020 2020 2020       |.| 2      
│ │ │ │ -0005cbf0: 2033 3039 3239 3636 3539 3937 2020 2020   30929665997    
│ │ │ │ -0005cc00: 2020 2020 2020 2031 3536 3534 3733 3433         156547343
│ │ │ │ -0005cc10: 3931 3433 3320 2020 2020 2020 2020 2020  91433           
│ │ │ │ -0005cc20: 3538 3431 3630 3339 3720 3220 2020 2020  584160397 2     
│ │ │ │ -0005cc30: 2020 3539 307c 0a7c 7820 7820 7820 202b    590|.|x x x  +
│ │ │ │ -0005cc40: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820   -----------x x 
│ │ │ │ -0005cc50: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d2d  x x  + ---------
│ │ │ │ -0005cc60: 2d2d 2d2d 2d78 2078 2078 2078 2020 2b20  -----x x x x  + 
│ │ │ │ -0005cc70: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2020  ---------x x x  
│ │ │ │ -0005cc80: 2d20 2d2d 2d7c 0a7c 2031 2033 2034 2020  - ---|.| 1 3 4  
│ │ │ │ -0005cc90: 2020 3234 3030 3435 3132 3020 2030 2032    240045120  0 2
│ │ │ │ -0005cca0: 2033 2034 2020 2020 2034 3830 3039 3032   3 4     4800902
│ │ │ │ -0005ccb0: 3430 3030 2020 3120 3220 3320 3420 2020  4000  1 2 3 4   
│ │ │ │ -0005ccc0: 2031 3438 3137 3630 3020 3220 3320 3420   14817600 2 3 4 
│ │ │ │ -0005ccd0: 2020 2034 307c 0a7c 2d2d 2d2d 2d2d 2d2d     40|.|--------
│ │ │ │ +0005cbe0: 2020 2020 2020 2020 2020 207c 0a7c 2032             |.| 2
│ │ │ │ +0005cbf0: 2020 2020 2020 2033 3039 3239 3636 3539         309296659
│ │ │ │ +0005cc00: 3937 2020 2020 2020 2020 2020 2031 3536  97           156
│ │ │ │ +0005cc10: 3534 3733 3433 3931 3433 3320 2020 2020  54734391433     
│ │ │ │ +0005cc20: 2020 2020 2020 3538 3431 3630 3339 3720        584160397 
│ │ │ │ +0005cc30: 3220 2020 2020 2020 3539 307c 0a7c 7820  2       590|.|x 
│ │ │ │ +0005cc40: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d2d  x x  + ---------
│ │ │ │ +0005cc50: 2d2d 7820 7820 7820 7820 202b 202d 2d2d  --x x x x  + ---
│ │ │ │ +0005cc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078  -----------x x x
│ │ │ │ +0005cc70: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78   x  + ---------x
│ │ │ │ +0005cc80: 2078 2078 2020 2d20 2d2d 2d7c 0a7c 2031   x x  - ---|.| 1
│ │ │ │ +0005cc90: 2033 2034 2020 2020 3234 3030 3435 3132   3 4    24004512
│ │ │ │ +0005cca0: 3020 2030 2032 2033 2034 2020 2020 2034  0  0 2 3 4     4
│ │ │ │ +0005ccb0: 3830 3039 3032 3430 3030 2020 3120 3220  8009024000  1 2 
│ │ │ │ +0005ccc0: 3320 3420 2020 2031 3438 3137 3630 3020  3 4    14817600 
│ │ │ │ +0005ccd0: 3220 3320 3420 2020 2034 307c 0a7c 2d2d  2 3 4    40|.|--
│ │ │ │  0005cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cd20: 2d2d 2d2d 2d7c 0a7c 3336 3037 3339 3831  -----|.|36073981
│ │ │ │ -0005cd30: 3920 2020 3220 2020 2020 3132 3638 3535  9   2     126855
│ │ │ │ -0005cd40: 3039 3733 3739 2020 2032 2020 2020 2039  097379   2     9
│ │ │ │ -0005cd50: 3738 3136 3636 3920 3320 2020 2020 2020  7816669 3       
│ │ │ │ +0005cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3336  -----------|.|36
│ │ │ │ +0005cd30: 3037 3339 3831 3920 2020 3220 2020 2020  0739819   2     
│ │ │ │ +0005cd40: 3132 3638 3535 3039 3733 3739 2020 2032  126855097379   2
│ │ │ │ +0005cd50: 2020 2020 2039 3738 3136 3636 3920 3320       97816669 3 
│ │ │ │  0005cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cd70: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005cd80: 2d78 2078 2078 2020 2b20 2d2d 2d2d 2d2d  -x x x  + ------
│ │ │ │ -0005cd90: 2d2d 2d2d 2d2d 7820 7820 7820 202b 202d  ------x x x  + -
│ │ │ │ -0005cda0: 2d2d 2d2d 2d2d 2d78 2078 2020 2d20 2020  -------x x  -   
│ │ │ │ -0005cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cdc0: 2020 2020 207c 0a7c 3334 3239 3231 3630       |.|34292160
│ │ │ │ -0005cdd0: 2020 3120 3320 3420 2020 2031 3333 3335    1 3 4    13335
│ │ │ │ -0005cde0: 3834 3030 3020 2032 2033 2034 2020 2020  84000  2 3 4    
│ │ │ │ -0005cdf0: 3633 3530 3430 3020 3320 3420 2020 2020  6350400 3 4     
│ │ │ │ -0005ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ce10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005cd70: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +0005cd80: 2d2d 2d2d 2d2d 2d78 2078 2078 2020 2b20  -------x x x  + 
│ │ │ │ +0005cd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  ------------x x 
│ │ │ │ +0005cda0: 7820 202b 202d 2d2d 2d2d 2d2d 2d78 2078  x  + --------x x
│ │ │ │ +0005cdb0: 2020 2d20 2020 2020 2020 2020 2020 2020    -             
│ │ │ │ +0005cdc0: 2020 2020 2020 2020 2020 207c 0a7c 3334             |.|34
│ │ │ │ +0005cdd0: 3239 3231 3630 2020 3120 3320 3420 2020  292160  1 3 4   
│ │ │ │ +0005cde0: 2031 3333 3335 3834 3030 3020 2032 2033   1333584000  2 3
│ │ │ │ +0005cdf0: 2034 2020 2020 3633 3530 3430 3020 3320   4    6350400 3 
│ │ │ │ +0005ce00: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0005ce10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ce60: 2020 2020 207c 0a7c 3134 3330 3139 3034       |.|14301904
│ │ │ │ -0005ce70: 3831 3720 2020 3220 2020 2020 3232 3034  817   2     2204
│ │ │ │ -0005ce80: 3131 3539 3839 3320 2020 3220 2020 2020  1159893   2     
│ │ │ │ -0005ce90: 3236 3136 3931 3930 3720 3320 2020 2020  261691907 3     
│ │ │ │ -0005cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ceb0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ -0005cee0: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020 2020  ---------x x    
│ │ │ │ -0005cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf00: 2020 2020 207c 0a7c 2020 3336 3238 3830       |.|  362880
│ │ │ │ -0005cf10: 3030 2020 3120 3320 3420 2020 2039 3532  00  1 3 4    952
│ │ │ │ -0005cf20: 3536 3030 3030 2020 3220 3320 3420 2020  560000  2 3 4   
│ │ │ │ -0005cf30: 2031 3336 3038 3030 3020 3320 3420 2020   13608000 3 4   
│ │ │ │ -0005cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005ce60: 2020 2020 2020 2020 2020 207c 0a7c 3134             |.|14
│ │ │ │ +0005ce70: 3330 3139 3034 3831 3720 2020 3220 2020  301904817   2   
│ │ │ │ +0005ce80: 2020 3232 3034 3131 3539 3839 3320 2020    22041159893   
│ │ │ │ +0005ce90: 3220 2020 2020 3236 3136 3931 3930 3720  2     261691907 
│ │ │ │ +0005cea0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0005ceb0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +0005cec0: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2020  ---------x x x  
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│ │ │ │ +0005cee0: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78   x  + ---------x
│ │ │ │ +0005cef0: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ +0005cf00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005cf10: 3336 3238 3830 3030 2020 3120 3320 3420  36288000  1 3 4 
│ │ │ │ +0005cf20: 2020 2039 3532 3536 3030 3030 2020 3220     952560000  2 
│ │ │ │ +0005cf30: 3320 3420 2020 2031 3336 3038 3030 3020  3 4    13608000 
│ │ │ │ +0005cf40: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
│ │ │ │ +0005cf50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cfa0: 2020 2020 207c 0a7c 2020 3220 2020 2020       |.|  2     
│ │ │ │ -0005cfb0: 3432 3533 3536 3136 3632 3320 2020 3220  42535616623   2 
│ │ │ │ -0005cfc0: 2020 2020 3135 3734 3332 3136 3920 3320      157432169 3 
│ │ │ │ -0005cfd0: 2020 2020 3134 3537 3332 3535 3633 2020      1457325563  
│ │ │ │ -0005cfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cff0: 2020 2020 207c 0a7c 2078 2078 2020 2d20       |.| x x  - 
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│ │ │ │ -0005d010: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78 2078    + ---------x x
│ │ │ │ -0005d020: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2020    + ----------  
│ │ │ │ -0005d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d040: 2020 2020 207c 0a7c 3120 3320 3420 2020       |.|1 3 4   
│ │ │ │ -0005d050: 2035 3731 3533 3630 3030 2020 3220 3320   571536000  2 3 
│ │ │ │ -0005d060: 3420 2020 2031 3633 3239 3630 2020 3320  4    1632960  3 
│ │ │ │ -0005d070: 3420 2020 2033 3137 3532 3030 3020 2020  4    31752000   
│ │ │ │ -0005d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005cfa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005cfb0: 3220 2020 2020 3432 3533 3536 3136 3632  2     4253561662
│ │ │ │ +0005cfc0: 3320 2020 3220 2020 2020 3135 3734 3332  3   2     157432
│ │ │ │ +0005cfd0: 3136 3920 3320 2020 2020 3134 3537 3332  169 3     145732
│ │ │ │ +0005cfe0: 3535 3633 2020 2020 2020 2020 2020 2020  5563            
│ │ │ │ +0005cff0: 2020 2020 2020 2020 2020 207c 0a7c 2078             |.| x
│ │ │ │ +0005d000: 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d   x  - ----------
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│ │ │ │ +0005d020: 2d2d 2d78 2078 2020 2b20 2d2d 2d2d 2d2d  ---x x  + ------
│ │ │ │ +0005d030: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │ +0005d040: 2020 2020 2020 2020 2020 207c 0a7c 3120             |.|1 
│ │ │ │ +0005d050: 3320 3420 2020 2035 3731 3533 3630 3030  3 4    571536000
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│ │ │ │ +0005d070: 3630 2020 3320 3420 2020 2033 3137 3532  60  3 4    31752
│ │ │ │ +0005d080: 3030 3020 2020 2020 2020 2020 2020 2020  000             
│ │ │ │ +0005d090: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d0e0: 2020 2020 207c 0a7c 3220 2020 2020 3136       |.|2     16
│ │ │ │ -0005d0f0: 3338 3334 3731 3433 2020 2032 2020 2020  38347143   2    
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│ │ │ │ -0005d110: 2020 2020 3839 3033 3437 3535 2033 2020      89034755 3  
│ │ │ │ -0005d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d130: 2020 2020 207c 0a7c 2078 2020 2b20 2d2d       |.| x  + --
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│ │ │ │ -0005d160: 2020 2b20 2d2d 2d2d 2d2d 2d2d 7820 2020    + --------x   
│ │ │ │ -0005d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d180: 2020 2020 207c 0a7c 3320 3420 2020 2037       |.|3 4    7
│ │ │ │ -0005d190: 3632 3034 3830 3020 2031 2033 2034 2020  6204800  1 3 4  
│ │ │ │ -0005d1a0: 2020 3338 3130 3234 3030 3020 3220 3320    381024000 2 3 
│ │ │ │ -0005d1b0: 3420 2020 2033 3034 3831 3932 2033 2020  4    3048192 3  
│ │ │ │ -0005d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d1d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005d0e0: 2020 2020 2020 2020 2020 207c 0a7c 3220             |.|2 
│ │ │ │ +0005d0f0: 2020 2020 3136 3338 3334 3731 3433 2020      1638347143  
│ │ │ │ +0005d100: 2032 2020 2020 2034 3936 3035 3732 3736   2     496057276
│ │ │ │ +0005d110: 3920 2020 3220 2020 2020 3839 3033 3437  9   2     890347
│ │ │ │ +0005d120: 3535 2033 2020 2020 2020 2020 2020 2020  55 3            
│ │ │ │ +0005d130: 2020 2020 2020 2020 2020 207c 0a7c 2078             |.| x
│ │ │ │ +0005d140: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 7820    + ----------x 
│ │ │ │ +0005d150: 7820 7820 202d 202d 2d2d 2d2d 2d2d 2d2d  x x  - ---------
│ │ │ │ +0005d160: 2d78 2078 2078 2020 2b20 2d2d 2d2d 2d2d  -x x x  + ------
│ │ │ │ +0005d170: 2d2d 7820 2020 2020 2020 2020 2020 2020  --x             
│ │ │ │ +0005d180: 2020 2020 2020 2020 2020 207c 0a7c 3320             |.|3 
│ │ │ │ +0005d190: 3420 2020 2037 3632 3034 3830 3020 2031  4    76204800  1
│ │ │ │ +0005d1a0: 2033 2034 2020 2020 3338 3130 3234 3030   3 4    38102400
│ │ │ │ +0005d1b0: 3020 3220 3320 3420 2020 2033 3034 3831  0 2 3 4    30481
│ │ │ │ +0005d1c0: 3932 2033 2020 2020 2020 2020 2020 2020  92 3            
│ │ │ │ +0005d1d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d220: 2020 2020 207c 0a7c 3739 3733 3734 3331       |.|79737431
│ │ │ │ -0005d230: 3920 2020 3220 2020 2020 3339 3539 3635  9   2     395965
│ │ │ │ -0005d240: 3937 3137 3637 2020 2032 2020 2020 2034  971767   2     4
│ │ │ │ -0005d250: 3931 3736 3438 3135 3738 3120 2020 2020  91764815781     
│ │ │ │ -0005d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d270: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005d280: 2d78 2078 2078 2020 2d20 2d2d 2d2d 2d2d  -x x x  - ------
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│ │ │ │ -0005d2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  -----------x x  
│ │ │ │ -0005d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d2c0: 2020 2020 207c 0a7c 3030 3735 3230 3030       |.|00752000
│ │ │ │ -0005d2d0: 2020 3020 3320 3420 2020 2031 3532 3430    0 3 4    15240
│ │ │ │ -0005d2e0: 3936 3030 3020 2031 2033 2034 2020 2020  96000  1 3 4    
│ │ │ │ -0005d2f0: 3236 3637 3136 3830 3030 2020 3220 2020  2667168000  2   
│ │ │ │ -0005d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d310: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ -0005e1e0: 3838 3020 2033 2034 2020 2020 3330 3234  880  3 4    3024
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│ │ │ │ -0005e260: 2020 2020 207c 0a7c 2020 3220 2020 3532       |.|  2   52
│ │ │ │ -0005e270: 3536 3433 3720 3220 3220 2020 3637 3031  56437 2 2   6701
│ │ │ │ -0005e280: 3036 3837 3320 2020 3320 2020 3131 3338  06873   3   1138
│ │ │ │ -0005e290: 3933 3439 2020 2033 2020 2031 3739 3333  9349   3   17933
│ │ │ │ -0005e2a0: 3137 3720 2020 3320 2020 3137 3832 3839  177   3   178289
│ │ │ │ -0005e2b0: 3939 2020 207c 0a7c 2078 2020 2d20 2d2d  99   |.| x  - --
│ │ │ │ -0005e2c0: 2d2d 2d2d 2d78 2078 2020 2b20 2d2d 2d2d  -----x x  + ----
│ │ │ │ -0005e2d0: 2d2d 2d2d 2d78 2078 2020 2b20 2d2d 2d2d  -----x x  + ----
│ │ │ │ -0005e2e0: 2d2d 2d2d 7820 7820 202b 202d 2d2d 2d2d  ----x x  + -----
│ │ │ │ -0005e2f0: 2d2d 2d78 2078 2020 2d20 2d2d 2d2d 2d2d  ---x x  - ------
│ │ │ │ -0005e300: 2d2d 7820 787c 0a7c 3320 3420 2020 2031  --x x|.|3 4    1
│ │ │ │ -0005e310: 3038 3836 3420 3320 3420 2020 2035 3239  08864 3 4    529
│ │ │ │ -0005e320: 3230 3030 2020 3020 3420 2020 2033 3032  2000  0 4    302
│ │ │ │ -0005e330: 3430 3020 2031 2034 2020 2020 2039 3830  400  1 4     980
│ │ │ │ -0005e340: 3030 2020 3220 3420 2020 2032 3236 3830  00  2 4    22680
│ │ │ │ -0005e350: 3020 2033 207c 0a7c 2020 2020 2020 2020  0  3 |.|        
│ │ │ │ +0005e260: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005e270: 3220 2020 3532 3536 3433 3720 3220 3220  2   5256437 2 2 
│ │ │ │ +0005e280: 2020 3637 3031 3036 3837 3320 2020 3320    670106873   3 
│ │ │ │ +0005e290: 2020 3131 3338 3933 3439 2020 2033 2020    11389349   3  
│ │ │ │ +0005e2a0: 2031 3739 3333 3137 3720 2020 3320 2020   17933177   3   
│ │ │ │ +0005e2b0: 3137 3832 3839 3939 2020 207c 0a7c 2078  17828999   |.| x
│ │ │ │ +0005e2c0: 2020 2d20 2d2d 2d2d 2d2d 2d78 2078 2020    - -------x x  
│ │ │ │ +0005e2d0: 2b20 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  + ---------x x  
│ │ │ │ +0005e2e0: 2b20 2d2d 2d2d 2d2d 2d2d 7820 7820 202b  + --------x x  +
│ │ │ │ +0005e2f0: 202d 2d2d 2d2d 2d2d 2d78 2078 2020 2d20   --------x x  - 
│ │ │ │ +0005e300: 2d2d 2d2d 2d2d 2d2d 7820 787c 0a7c 3320  --------x x|.|3 
│ │ │ │ +0005e310: 3420 2020 2031 3038 3836 3420 3320 3420  4    108864 3 4 
│ │ │ │ +0005e320: 2020 2035 3239 3230 3030 2020 3020 3420     5292000  0 4 
│ │ │ │ +0005e330: 2020 2033 3032 3430 3020 2031 2034 2020     302400  1 4  
│ │ │ │ +0005e340: 2020 2039 3830 3030 2020 3220 3420 2020     98000  2 4   
│ │ │ │ +0005e350: 2032 3236 3830 3020 2033 207c 0a7c 2020   226800  3 |.|  
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e3a0: 2020 2020 207c 0a7c 2020 2020 3220 2020       |.|    2   
│ │ │ │ -0005e3b0: 3132 3430 3134 3530 3631 3720 2020 2020  12401450617     
│ │ │ │ -0005e3c0: 3220 2020 3839 3834 3338 3532 3320 3220  2   898438523 2 
│ │ │ │ -0005e3d0: 3220 2020 3131 3437 3536 3038 3233 2020  2   1147560823  
│ │ │ │ -0005e3e0: 2033 2020 2039 3132 3138 3231 3739 2020   3   912182179  
│ │ │ │ -0005e3f0: 2033 2020 207c 0a7c 2078 2078 2020 2d20   3   |.| x x  - 
│ │ │ │ -0005e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078  -----------x x x
│ │ │ │ -0005e410: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d78 2078    - ---------x x
│ │ │ │ -0005e420: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 7820    - ----------x 
│ │ │ │ -0005e430: 7820 202b 202d 2d2d 2d2d 2d2d 2d2d 7820  x  + ---------x 
│ │ │ │ -0005e440: 7820 202d 207c 0a7c 3120 3320 3420 2020  x  - |.|1 3 4   
│ │ │ │ -0005e450: 2031 3930 3531 3230 3030 2020 3220 3320   190512000  2 3 
│ │ │ │ -0005e460: 3420 2020 2033 3831 3032 3430 3020 3320  4    38102400 3 
│ │ │ │ -0005e470: 3420 2020 2036 3335 3034 3030 3020 2030  4    63504000  0
│ │ │ │ -0005e480: 2034 2020 2020 3132 3730 3038 3030 2031   4    12700800 1
│ │ │ │ -0005e490: 2034 2020 207c 0a7c 2020 2020 2020 2020   4   |.|        
│ │ │ │ +0005e3a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005e3b0: 2020 3220 2020 3132 3430 3134 3530 3631    2   1240145061
│ │ │ │ +0005e3c0: 3720 2020 2020 3220 2020 3839 3834 3338  7     2   898438
│ │ │ │ +0005e3d0: 3532 3320 3220 3220 2020 3131 3437 3536  523 2 2   114756
│ │ │ │ +0005e3e0: 3038 3233 2020 2033 2020 2039 3132 3138  0823   3   91218
│ │ │ │ +0005e3f0: 3231 3739 2020 2033 2020 207c 0a7c 2078  2179   3   |.| x
│ │ │ │ +0005e400: 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d   x  - ----------
│ │ │ │ +0005e410: 2d78 2078 2078 2020 2d20 2d2d 2d2d 2d2d  -x x x  - ------
│ │ │ │ +0005e420: 2d2d 2d78 2078 2020 2d20 2d2d 2d2d 2d2d  ---x x  - ------
│ │ │ │ +0005e430: 2d2d 2d2d 7820 7820 202b 202d 2d2d 2d2d  ----x x  + -----
│ │ │ │ +0005e440: 2d2d 2d2d 7820 7820 202d 207c 0a7c 3120  ----x x  - |.|1 
│ │ │ │ +0005e450: 3320 3420 2020 2031 3930 3531 3230 3030  3 4    190512000
│ │ │ │ +0005e460: 2020 3220 3320 3420 2020 2033 3831 3032    2 3 4    38102
│ │ │ │ +0005e470: 3430 3020 3320 3420 2020 2036 3335 3034  400 3 4    63504
│ │ │ │ +0005e480: 3030 3020 2030 2034 2020 2020 3132 3730  000  0 4    1270
│ │ │ │ +0005e490: 3038 3030 2031 2034 2020 207c 0a7c 2020  0800 1 4   |.|  
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e4e0: 2020 2020 207c 0a7c 2020 2020 2032 2020       |.|     2  
│ │ │ │ -0005e4f0: 2036 3938 3834 3533 3720 2020 2020 3220   69884537     2 
│ │ │ │ -0005e500: 2020 3231 3631 3531 3632 3138 3137 2020    216151621817  
│ │ │ │ -0005e510: 2020 2032 2020 2032 3938 3831 3936 3431     2   298819641
│ │ │ │ -0005e520: 3120 3220 3220 2020 3937 3235 3130 3031  1 2 2   97251001
│ │ │ │ -0005e530: 2020 2033 207c 0a7c 7820 7820 7820 202d     3 |.|x x x  -
│ │ │ │ -0005e540: 202d 2d2d 2d2d 2d2d 2d78 2078 2078 2020   --------x x x  
│ │ │ │ -0005e550: 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820  + ------------x 
│ │ │ │ -0005e560: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d2d  x x  + ---------
│ │ │ │ -0005e570: 2d78 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d  -x x  - --------
│ │ │ │ -0005e580: 7820 7820 207c 0a7c 2030 2033 2034 2020  x x  |.| 0 3 4  
│ │ │ │ -0005e590: 2037 3134 3432 3030 3020 3120 3320 3420   71442000 1 3 4 
│ │ │ │ -0005e5a0: 2020 2031 3333 3335 3834 3030 3020 2032     1333584000  2
│ │ │ │ -0005e5b0: 2033 2034 2020 2020 3935 3235 3630 3030   3 4    95256000
│ │ │ │ -0005e5c0: 2020 3320 3420 2020 2033 3936 3930 3030    3 4    3969000
│ │ │ │ -0005e5d0: 2030 2034 207c 0a7c 2d2d 2d2d 2d2d 2d2d   0 4 |.|--------
│ │ │ │ +0005e4e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005e4f0: 2020 2032 2020 2036 3938 3834 3533 3720     2   69884537 
│ │ │ │ +0005e500: 2020 2020 3220 2020 3231 3631 3531 3632      2   21615162
│ │ │ │ +0005e510: 3138 3137 2020 2020 2032 2020 2032 3938  1817     2   298
│ │ │ │ +0005e520: 3831 3936 3431 3120 3220 3220 2020 3937  8196411 2 2   97
│ │ │ │ +0005e530: 3235 3130 3031 2020 2033 207c 0a7c 7820  251001   3 |.|x 
│ │ │ │ +0005e540: 7820 7820 202d 202d 2d2d 2d2d 2d2d 2d78  x x  - --------x
│ │ │ │ +0005e550: 2078 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d   x x  + --------
│ │ │ │ +0005e560: 2d2d 2d2d 7820 7820 7820 202b 202d 2d2d  ----x x x  + ---
│ │ │ │ +0005e570: 2d2d 2d2d 2d2d 2d78 2078 2020 2d20 2d2d  -------x x  - --
│ │ │ │ +0005e580: 2d2d 2d2d 2d2d 7820 7820 207c 0a7c 2030  ------x x  |.| 0
│ │ │ │ +0005e590: 2033 2034 2020 2037 3134 3432 3030 3020   3 4   71442000 
│ │ │ │ +0005e5a0: 3120 3320 3420 2020 2031 3333 3335 3834  1 3 4    1333584
│ │ │ │ +0005e5b0: 3030 3020 2032 2033 2034 2020 2020 3935  000  2 3 4    95
│ │ │ │ +0005e5c0: 3235 3630 3030 2020 3320 3420 2020 2033  256000  3 4    3
│ │ │ │ +0005e5d0: 3936 3930 3030 2030 2034 207c 0a7c 2d2d  969000 0 4 |.|--
│ │ │ │  0005e5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e620: 2d2d 2d2d 2d7c 0a7c 3734 3120 2020 3320  -----|.|741   3 
│ │ │ │ -0005e630: 2020 3433 3335 3231 3520 2020 3320 2020    4335215   3   
│ │ │ │ -0005e640: 3438 3931 3733 2034 2020 2020 2020 2020  489173 4        
│ │ │ │ +0005e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3734  -----------|.|74
│ │ │ │ +0005e630: 3120 2020 3320 2020 3433 3335 3231 3520  1   3   4335215 
│ │ │ │ +0005e640: 2020 3320 2020 3438 3931 3733 2034 2020    3   489173 4  
│ │ │ │  0005e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e670: 2020 2020 207c 0a7c 2d2d 2d78 2078 2020       |.|---x x  
│ │ │ │ -0005e680: 2b20 2d2d 2d2d 2d2d 2d78 2078 2020 2d20  + -------x x  - 
│ │ │ │ -0005e690: 2d2d 2d2d 2d2d 7820 2c20 2020 2020 2020  ------x ,       
│ │ │ │ +0005e670: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +0005e680: 2d78 2078 2020 2b20 2d2d 2d2d 2d2d 2d78  -x x  + -------x
│ │ │ │ +0005e690: 2078 2020 2d20 2d2d 2d2d 2d2d 7820 2c20   x  - ------x , 
│ │ │ │  0005e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e6c0: 2020 2020 207c 0a7c 3030 2020 3220 3420       |.|00  2 4 
│ │ │ │ -0005e6d0: 2020 2032 3534 3031 3620 3320 3420 2020     254016 3 4   
│ │ │ │ -0005e6e0: 3132 3630 3030 2034 2020 2020 2020 2020  126000 4        
│ │ │ │ +0005e6c0: 2020 2020 2020 2020 2020 207c 0a7c 3030             |.|00
│ │ │ │ +0005e6d0: 2020 3220 3420 2020 2032 3534 3031 3620    2 4    254016 
│ │ │ │ +0005e6e0: 3320 3420 2020 3132 3630 3030 2034 2020  3 4   126000 4  
│ │ │ │  0005e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e710: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e710: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e760: 2020 2020 207c 0a7c 2031 3738 3130 3833       |.| 1781083
│ │ │ │ -0005e770: 3720 2020 3320 2020 3132 3437 2034 2020  7   3   1247 4  
│ │ │ │ -0005e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e760: 2020 2020 2020 2020 2020 207c 0a7c 2031             |.| 1
│ │ │ │ +0005e770: 3738 3130 3833 3720 2020 3320 2020 3132  7810837   3   12
│ │ │ │ +0005e780: 3437 2034 2020 2020 2020 2020 2020 2020  47 4            
│ │ │ │  0005e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e7b0: 2020 2020 207c 0a7c 202d 2d2d 2d2d 2d2d       |.| -------
│ │ │ │ -0005e7c0: 2d78 2078 2020 2d20 2d2d 2d2d 7820 2c20  -x x  - ----x , 
│ │ │ │ -0005e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e7b0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +0005e7c0: 2d2d 2d2d 2d2d 2d78 2078 2020 2d20 2d2d  -------x x  - --
│ │ │ │ +0005e7d0: 2d2d 7820 2c20 2020 2020 2020 2020 2020  --x ,           
│ │ │ │  0005e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e800: 2020 2020 207c 0a7c 2020 3136 3830 3030       |.|  168000
│ │ │ │ -0005e810: 2020 3320 3420 2020 2033 3030 2034 2020    3 4    300 4  
│ │ │ │ -0005e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e800: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005e810: 3136 3830 3030 2020 3320 3420 2020 2033  168000  3 4    3
│ │ │ │ +0005e820: 3030 2034 2020 2020 2020 2020 2020 2020  00 4            
│ │ │ │  0005e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e850: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e8a0: 2020 2020 207c 0a7c 3320 2020 3139 3631       |.|3   1961
│ │ │ │ -0005e8b0: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
│ │ │ │ +0005e8a0: 2020 2020 2020 2020 2020 207c 0a7c 3320             |.|3 
│ │ │ │ +0005e8b0: 2020 3139 3631 3320 3420 2020 2020 2020    19613 4       
│ │ │ │  0005e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e8f0: 2020 2020 207c 0a7c 2020 2b20 2d2d 2d2d       |.|  + ----
│ │ │ │ -0005e900: 2d78 202c 2020 2020 2020 2020 2020 2020  -x ,            
│ │ │ │ +0005e8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005e900: 2b20 2d2d 2d2d 2d78 202c 2020 2020 2020  + -----x ,      
│ │ │ │  0005e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e940: 2020 2020 207c 0a7c 3420 2020 2031 3830       |.|4    180
│ │ │ │ -0005e950: 3020 3420 2020 2020 2020 2020 2020 2020  0 4             
│ │ │ │ +0005e940: 2020 2020 2020 2020 2020 207c 0a7c 3420             |.|4 
│ │ │ │ +0005e950: 2020 2031 3830 3020 3420 2020 2020 2020     1800 4       
│ │ │ │  0005e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e990: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e9e0: 2020 2020 207c 0a7c 3230 3632 3136 3333       |.|20621633
│ │ │ │ -0005e9f0: 2020 2033 2020 2032 3636 3134 3434 3720     3   26614447 
│ │ │ │ -0005ea00: 2020 3320 2020 2035 3633 2034 2020 2020    3    563 4    
│ │ │ │ -0005ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e9e0: 2020 2020 2020 2020 2020 207c 0a7c 3230             |.|20
│ │ │ │ +0005e9f0: 3632 3136 3333 2020 2033 2020 2032 3636  621633   3   266
│ │ │ │ +0005ea00: 3134 3434 3720 2020 3320 2020 2035 3633  14447   3    563
│ │ │ │ +0005ea10: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │  0005ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea30: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005ea40: 7820 7820 202d 202d 2d2d 2d2d 2d2d 2d78  x x  - --------x
│ │ │ │ -0005ea50: 2078 2020 2d20 2d2d 2d2d 7820 2c20 2020   x  - ----x ,   
│ │ │ │ -0005ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ea30: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +0005ea40: 2d2d 2d2d 2d2d 7820 7820 202d 202d 2d2d  ------x x  - ---
│ │ │ │ +0005ea50: 2d2d 2d2d 2d78 2078 2020 2d20 2d2d 2d2d  -----x x  - ----
│ │ │ │ +0005ea60: 7820 2c20 2020 2020 2020 2020 2020 2020  x ,             
│ │ │ │  0005ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea80: 2020 2020 207c 0a7c 2036 3034 3830 3020       |.| 604800 
│ │ │ │ -0005ea90: 2032 2034 2020 2020 3834 3637 3230 2020   2 4    846720  
│ │ │ │ -0005eaa0: 3320 3420 2020 3238 3030 2034 2020 2020  3 4   2800 4    
│ │ │ │ -0005eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ea80: 2020 2020 2020 2020 2020 207c 0a7c 2036             |.| 6
│ │ │ │ +0005ea90: 3034 3830 3020 2032 2034 2020 2020 3834  04800  2 4    84
│ │ │ │ +0005eaa0: 3637 3230 2020 3320 3420 2020 3238 3030  6720  3 4   2800
│ │ │ │ +0005eab0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │  0005eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ead0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005ead0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb20: 2020 2020 207c 0a7c 2020 3834 3831 3539       |.|  848159
│ │ │ │ -0005eb30: 3539 3720 2020 3320 2020 3139 3438 3330  597   3   194830
│ │ │ │ -0005eb40: 3636 3320 2020 3320 2020 3237 3634 3632  663   3   276462
│ │ │ │ -0005eb50: 3136 3720 2020 3320 2020 3532 3230 3720  167   3   52207 
│ │ │ │ -0005eb60: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0005eb70: 2020 2020 207c 0a7c 2d20 2d2d 2d2d 2d2d       |.|- ------
│ │ │ │ -0005eb80: 2d2d 2d78 2078 2020 2d20 2d2d 2d2d 2d2d  ---x x  - ------
│ │ │ │ -0005eb90: 2d2d 2d78 2078 2020 2b20 2d2d 2d2d 2d2d  ---x x  + ------
│ │ │ │ -0005eba0: 2d2d 2d78 2078 2020 2d20 2d2d 2d2d 2d78  ---x x  - -----x
│ │ │ │ -0005ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ebc0: 2020 2020 207c 0a7c 2020 3132 3730 3038       |.|  127008
│ │ │ │ -0005ebd0: 3030 3020 3120 3420 2020 2035 3239 3230  000 1 4    52920
│ │ │ │ -0005ebe0: 3030 2020 3220 3420 2020 2032 3131 3638  00  2 4    21168
│ │ │ │ -0005ebf0: 3030 3020 3320 3420 2020 3238 3030 3020  000 3 4   28000 
│ │ │ │ -0005ec00: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0005ec10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0005eb20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005eb30: 3834 3831 3539 3539 3720 2020 3320 2020  848159597   3   
│ │ │ │ +0005eb40: 3139 3438 3330 3636 3320 2020 3320 2020  194830663   3   
│ │ │ │ +0005eb50: 3237 3634 3632 3136 3720 2020 3320 2020  276462167   3   
│ │ │ │ +0005eb60: 3532 3230 3720 3420 2020 2020 2020 2020  52207 4         
│ │ │ │ +0005eb70: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ +0005eb80: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020 2d20  ---------x x  - 
│ │ │ │ +0005eb90: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020 2b20  ---------x x  + 
│ │ │ │ +0005eba0: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020 2d20  ---------x x  - 
│ │ │ │ +0005ebb0: 2d2d 2d2d 2d78 2020 2020 2020 2020 2020  -----x          
│ │ │ │ +0005ebc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005ebd0: 3132 3730 3038 3030 3020 3120 3420 2020  127008000 1 4   
│ │ │ │ +0005ebe0: 2035 3239 3230 3030 2020 3220 3420 2020   5292000  2 4   
│ │ │ │ +0005ebf0: 2032 3131 3638 3030 3020 3320 3420 2020   21168000 3 4   
│ │ │ │ +0005ec00: 3238 3030 3020 3420 2020 2020 2020 2020  28000 4         
│ │ │ │ +0005ec10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0005ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ec60: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d  -----+.|i3 : tim
│ │ │ │ -0005ec70: 6520 696e 7665 7273 6520 7068 6920 2020  e inverse phi   
│ │ │ │ -0005ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +0005ec70: 203a 2074 696d 6520 696e 7665 7273 6520   : time inverse 
│ │ │ │ +0005ec80: 7068 6920 2020 2020 2020 2020 2020 2020  phi             
│ │ │ │  0005ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ecb0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0005ecc0: 2030 2e31 3534 3533 3273 2028 6370 7529   0.154532s (cpu)
│ │ │ │ -0005ecd0: 3b20 302e 3037 3037 3436 3773 2028 7468  ; 0.0707467s (th
│ │ │ │ -0005ece0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ -0005ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005ecb0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +0005ecc0: 2d20 7573 6564 2030 2e31 3630 3273 2028  - used 0.1602s (
│ │ │ │ +0005ecd0: 6370 7529 3b20 302e 3038 3830 3531 3773  cpu); 0.0880517s
│ │ │ │ +0005ece0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +0005ecf0: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +0005ed00: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed50: 2020 2020 207c 0a7c 6f33 203d 202d 2d20       |.|o3 = -- 
│ │ │ │ -0005ed60: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ -0005ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ed50: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0005ed60: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +0005ed70: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  0005ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eda0: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -0005edb0: 7263 653a 2050 726f 6a28 5151 5b78 202c  rce: Proj(QQ[x ,
│ │ │ │ -0005edc0: 2078 202c 2078 202c 2078 202c 2078 205d   x , x , x , x ]
│ │ │ │ -0005edd0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0005eda0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005edb0: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ +0005edc0: 5151 5b78 202c 2078 202c 2078 202c 2078  QQ[x , x , x , x
│ │ │ │ +0005edd0: 202c 2078 205d 2920 2020 2020 2020 2020   , x ])         
│ │ │ │  0005ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005edf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005ee00: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0005ee10: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ -0005ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005edf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ee10: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +0005ee20: 3320 2020 3420 2020 2020 2020 2020 2020  3   4           
│ │ │ │  0005ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee40: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -0005ee50: 6765 743a 2050 726f 6a28 5151 5b78 202c  get: Proj(QQ[x ,
│ │ │ │ -0005ee60: 2078 202c 2078 202c 2078 202c 2078 205d   x , x , x , x ]
│ │ │ │ -0005ee70: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0005ee40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005ee50: 2020 2074 6172 6765 743a 2050 726f 6a28     target: Proj(
│ │ │ │ +0005ee60: 5151 5b78 202c 2078 202c 2078 202c 2078  QQ[x , x , x , x
│ │ │ │ +0005ee70: 202c 2078 205d 2920 2020 2020 2020 2020   , x ])         
│ │ │ │  0005ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005eea0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0005eeb0: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ -0005eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ee90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005eeb0: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +0005eec0: 3320 2020 3420 2020 2020 2020 2020 2020  3   4           
│ │ │ │  0005eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eee0: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -0005eef0: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ -0005ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005eee0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005eef0: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
│ │ │ │ +0005ef00: 733a 207b 2020 2020 2020 2020 2020 2020  s: {            
│ │ │ │  0005ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ef30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005ef30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ef60: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0005ef70: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -0005ef80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005ef90: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0005efa0: 3132 3238 3936 3236 3638 3232 3030 3078  122896266822000x
│ │ │ │ -0005efb0: 2020 2b20 3631 3032 3236 3031 3439 3438    + 610226014948
│ │ │ │ -0005efc0: 3230 3078 2078 2020 2020 2020 2020 2020  200x x          
│ │ │ │ -0005efd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005ef60: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +0005ef70: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +0005ef80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005efa0: 2020 2020 2d20 3132 3238 3936 3236 3638      - 1228962668
│ │ │ │ +0005efb0: 3232 3030 3078 2020 2b20 3631 3032 3236  22000x  + 610226
│ │ │ │ +0005efc0: 3031 3439 3438 3230 3078 2078 2020 2020  014948200x x    
│ │ │ │ +0005efd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f000: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -0005f010: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ -0005f020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f000: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +0005f010: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0005f020: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f070: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f090: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -0005f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f0b0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -0005f0c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005f0d0: 2020 2020 2020 2020 2020 2020 2020 3135                15
│ │ │ │ -0005f0e0: 3931 3232 3836 3138 3536 3030 7820 202d  912286185600x  -
│ │ │ │ -0005f0f0: 2033 3236 3636 3732 3937 3630 3030 3078   32666729760000x
│ │ │ │ -0005f100: 2078 2020 2d20 2020 2020 2020 2020 2020   x  -           
│ │ │ │ -0005f110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f0a0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +0005f0b0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +0005f0c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f0e0: 2020 2020 3135 3931 3232 3836 3138 3536      159122861856
│ │ │ │ +0005f0f0: 3030 7820 202d 2033 3236 3636 3732 3937  00x  - 326667297
│ │ │ │ +0005f100: 3630 3030 3078 2078 2020 2d20 2020 2020  60000x x  -     
│ │ │ │ +0005f110: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f130: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -0005f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f150: 3020 3120 2020 2020 2020 2020 2020 2020  0 1             
│ │ │ │ -0005f160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f140: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +0005f150: 2020 2020 2020 3020 3120 2020 2020 2020        0 1       
│ │ │ │ +0005f160: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f1b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1d0: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ -0005f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1f0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -0005f200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005f210: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0005f220: 3532 3239 3436 3538 3131 3932 3030 7820  52294658119200x 
│ │ │ │ -0005f230: 202d 2037 3337 3735 3738 3137 3532 3430   - 7377578175240
│ │ │ │ -0005f240: 3078 2078 2020 2020 2020 2020 2020 2020  0x x            
│ │ │ │ -0005f250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f1e0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ +0005f1f0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +0005f200: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f220: 2020 2020 2d20 3532 3239 3436 3538 3131      - 5229465811
│ │ │ │ +0005f230: 3932 3030 7820 202d 2037 3337 3735 3738  9200x  - 7377578
│ │ │ │ +0005f240: 3137 3532 3430 3078 2078 2020 2020 2020  1752400x x      
│ │ │ │ +0005f250: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f270: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -0005f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f290: 2020 3020 3120 2020 2020 2020 2020 2020    0 1           
│ │ │ │ -0005f2a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f280: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +0005f290: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ +0005f2a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f2f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f2f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f310: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ -0005f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f330: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -0005f340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005f350: 2020 2020 2020 2020 2020 2020 2020 3232                22
│ │ │ │ -0005f360: 3532 3939 3830 3535 3434 3830 3078 2020  5299805544800x  
│ │ │ │ -0005f370: 2d20 3333 3036 3432 3538 3836 3931 3830  - 33064258869180
│ │ │ │ -0005f380: 3078 2078 2020 2020 2020 2020 2020 2020  0x x            
│ │ │ │ -0005f390: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f320: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +0005f330: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +0005f340: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f360: 2020 2020 3232 3532 3939 3830 3535 3434      225299805544
│ │ │ │ +0005f370: 3830 3078 2020 2d20 3333 3036 3432 3538  800x  - 33064258
│ │ │ │ +0005f380: 3836 3931 3830 3078 2078 2020 2020 2020  8691800x x      
│ │ │ │ +0005f390: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f3b0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0005f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f3d0: 2020 3020 3120 2020 2020 2020 2020 2020    0 1           
│ │ │ │ -0005f3e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f3c0: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ +0005f3d0: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ +0005f3e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f430: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f450: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -0005f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f470: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -0005f480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005f490: 2020 2020 2020 2020 2020 2020 2020 3236                26
│ │ │ │ -0005f4a0: 3830 3032 3138 3434 3634 3030 7820 202d  800218446400x  -
│ │ │ │ -0005f4b0: 2036 3933 3135 3737 3630 3036 3430 3078   69315776006400x
│ │ │ │ -0005f4c0: 2078 2020 2d20 2020 2020 2020 2020 2020   x  -           
│ │ │ │ -0005f4d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f460: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +0005f470: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +0005f480: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f4a0: 2020 2020 3236 3830 3032 3138 3434 3634      268002184464
│ │ │ │ +0005f4b0: 3030 7820 202d 2036 3933 3135 3737 3630  00x  - 693157760
│ │ │ │ +0005f4c0: 3036 3430 3078 2078 2020 2d20 2020 2020  06400x x  -     
│ │ │ │ +0005f4d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f4f0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -0005f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f510: 3020 3120 2020 2020 2020 2020 2020 2020  0 1             
│ │ │ │ -0005f520: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005f530: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ -0005f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f500: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +0005f510: 2020 2020 2020 3020 3120 2020 2020 2020        0 1       
│ │ │ │ +0005f520: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f540: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  0005f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f570: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f5c0: 2020 2020 207c 0a7c 6f33 203a 2052 6174       |.|o3 : Rat
│ │ │ │ -0005f5d0: 696f 6e61 6c4d 6170 2028 4372 656d 6f6e  ionalMap (Cremon
│ │ │ │ -0005f5e0: 6120 7472 616e 7366 6f72 6d61 7469 6f6e  a transformation
│ │ │ │ -0005f5f0: 206f 6620 5050 5e34 2920 2020 2020 2020   of PP^4)       
│ │ │ │ +0005f5c0: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0005f5d0: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +0005f5e0: 4372 656d 6f6e 6120 7472 616e 7366 6f72  Cremona transfor
│ │ │ │ +0005f5f0: 6d61 7469 6f6e 206f 6620 5050 5e34 2920  mation of PP^4) 
│ │ │ │  0005f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f610: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0005f610: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  0005f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f660: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0005f670: 2020 2020 2020 2020 2020 2020 3220 3220              2 2 
│ │ │ │ -0005f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f690: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -0005f6a0: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ -0005f6b0: 2020 2020 207c 0a7c 2020 2d20 3537 3637       |.|  - 5767
│ │ │ │ -0005f6c0: 3533 3036 3337 3830 3630 3078 2078 2020  53063780600x x  
│ │ │ │ -0005f6d0: 2b20 3130 3237 3230 3837 3930 3135 3330  + 10272087901530
│ │ │ │ -0005f6e0: 3078 2078 2020 2d20 3934 3134 3834 3236  0x x  - 94148426
│ │ │ │ -0005f6f0: 3733 3630 3078 2020 2b20 3336 3939 3835  73600x  + 369985
│ │ │ │ -0005f700: 3230 3535 327c 0a7c 3120 2020 2020 2020  20552|.|1       
│ │ │ │ -0005f710: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -0005f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f730: 2020 3020 3120 2020 2020 2020 2020 2020    0 1           
│ │ │ │ -0005f740: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -0005f750: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0005f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f680: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +0005f690: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +0005f6a0: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +0005f6b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005f6c0: 2d20 3537 3637 3533 3036 3337 3830 3630  - 57675306378060
│ │ │ │ +0005f6d0: 3078 2078 2020 2b20 3130 3237 3230 3837  0x x  + 10272087
│ │ │ │ +0005f6e0: 3930 3135 3330 3078 2078 2020 2d20 3934  9015300x x  - 94
│ │ │ │ +0005f6f0: 3134 3834 3236 3733 3630 3078 2020 2b20  14842673600x  + 
│ │ │ │ +0005f700: 3336 3939 3835 3230 3535 327c 0a7c 3120  36998520552|.|1 
│ │ │ │ +0005f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f720: 2020 3020 3120 2020 2020 2020 2020 2020    0 1           
│ │ │ │ +0005f730: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ +0005f740: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +0005f750: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f7a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005f7b0: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ -0005f7c0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -0005f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f7e0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0005f7f0: 2020 2033 207c 0a7c 3135 3130 3830 3133     3 |.|15108013
│ │ │ │ -0005f800: 3035 3330 3030 7820 7820 202b 2031 3535  053000x x  + 155
│ │ │ │ -0005f810: 3038 3534 3935 3037 3530 3078 2078 2020  08549507500x x  
│ │ │ │ -0005f820: 2d20 3336 3335 3239 3939 3734 3630 3078  - 3635299974600x
│ │ │ │ -0005f830: 2020 2d20 3330 3035 3332 3232 3237 3731    - 300532222771
│ │ │ │ -0005f840: 3230 7820 787c 0a7c 2020 2020 2020 2020  20x x|.|        
│ │ │ │ -0005f850: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ -0005f860: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -0005f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f880: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -0005f890: 2020 2030 207c 0a7c 2020 2020 2020 2020     0 |.|        
│ │ │ │ +0005f7a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005f7b0: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
│ │ │ │ +0005f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f7d0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +0005f7e0: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +0005f7f0: 2020 2020 2020 2020 2033 207c 0a7c 3135           3 |.|15
│ │ │ │ +0005f800: 3130 3830 3133 3035 3330 3030 7820 7820  108013053000x x 
│ │ │ │ +0005f810: 202b 2031 3535 3038 3534 3935 3037 3530   + 1550854950750
│ │ │ │ +0005f820: 3078 2078 2020 2d20 3336 3335 3239 3939  0x x  - 36352999
│ │ │ │ +0005f830: 3734 3630 3078 2020 2d20 3330 3035 3332  74600x  - 300532
│ │ │ │ +0005f840: 3232 3237 3731 3230 7820 787c 0a7c 2020  22277120x x|.|  
│ │ │ │ +0005f850: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ +0005f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f870: 2020 3020 3120 2020 2020 2020 2020 2020    0 1           
│ │ │ │ +0005f880: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +0005f890: 2020 2020 2020 2020 2030 207c 0a7c 2020           0 |.|  
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f8e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005f8f0: 2020 2020 2020 2020 2020 3220 3220 2020            2 2   
│ │ │ │ -0005f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f910: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -0005f920: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -0005f930: 2020 2020 207c 0a7c 2b20 3337 3733 3735       |.|+ 377375
│ │ │ │ -0005f940: 3430 3438 3538 3230 3078 2078 2020 2d20  404858200x x  - 
│ │ │ │ -0005f950: 3734 3436 3537 3132 3737 3034 3030 7820  74465712770400x 
│ │ │ │ -0005f960: 7820 202b 2034 3430 3436 3139 3830 3030  x  + 44046198000
│ │ │ │ -0005f970: 3030 7820 202d 2035 3434 3832 3533 3130  00x  - 544825310
│ │ │ │ -0005f980: 3430 3030 307c 0a7c 2020 2020 2020 2020  40000|.|        
│ │ │ │ -0005f990: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -0005f9a0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -0005f9b0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -0005f9c0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -0005f9d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f8e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f900: 3220 3220 2020 2020 2020 2020 2020 2020  2 2             
│ │ │ │ +0005f910: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +0005f920: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ +0005f930: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ +0005f940: 3337 3733 3735 3430 3438 3538 3230 3078  377375404858200x
│ │ │ │ +0005f950: 2078 2020 2d20 3734 3436 3537 3132 3737   x  - 7446571277
│ │ │ │ +0005f960: 3034 3030 7820 7820 202b 2034 3430 3436  0400x x  + 44046
│ │ │ │ +0005f970: 3139 3830 3030 3030 7820 202d 2035 3434  19800000x  - 544
│ │ │ │ +0005f980: 3832 3533 3130 3430 3030 307c 0a7c 2020  82531040000|.|  
│ │ │ │ +0005f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f9a0: 3020 3120 2020 2020 2020 2020 2020 2020  0 1             
│ │ │ │ +0005f9b0: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │ +0005f9c0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +0005f9d0: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005fa30: 2020 2020 2020 2020 2032 2032 2020 2020           2 2    
│ │ │ │ -0005fa40: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -0005fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa60: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -0005fa70: 2020 2020 207c 0a7c 2b20 3237 3233 3538       |.|+ 272358
│ │ │ │ -0005fa80: 3933 3133 3833 3030 7820 7820 202b 2034  93138300x x  + 4
│ │ │ │ -0005fa90: 3334 3338 3630 3532 3130 3030 7820 7820  343860521000x x 
│ │ │ │ -0005faa0: 202b 2031 3632 3937 3739 3335 3638 3030   + 1629779356800
│ │ │ │ -0005fab0: 7820 202b 2031 3539 3832 3330 3037 3234  x  + 15982300724
│ │ │ │ -0005fac0: 3534 3430 787c 0a7c 2020 2020 2020 2020  5440x|.|        
│ │ │ │ -0005fad0: 2020 2020 2020 2020 2030 2031 2020 2020           0 1    
│ │ │ │ -0005fae0: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ -0005faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fb00: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -0005fb10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005fa20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005fa30: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0005fa40: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0005fa50: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +0005fa60: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +0005fa70: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ +0005fa80: 3237 3233 3538 3933 3133 3833 3030 7820  27235893138300x 
│ │ │ │ +0005fa90: 7820 202b 2034 3334 3338 3630 3532 3130  x  + 43438605210
│ │ │ │ +0005faa0: 3030 7820 7820 202b 2031 3632 3937 3739  00x x  + 1629779
│ │ │ │ +0005fab0: 3335 3638 3030 7820 202b 2031 3539 3832  356800x  + 15982
│ │ │ │ +0005fac0: 3330 3037 3234 3534 3430 787c 0a7c 2020  3007245440x|.|  
│ │ │ │ +0005fad0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +0005fae0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +0005faf0: 2020 2030 2031 2020 2020 2020 2020 2020     0 1          
│ │ │ │ +0005fb00: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +0005fb10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fb60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005fb70: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ -0005fb80: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -0005fb90: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ -0005fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fbb0: 2033 2020 207c 0a7c 3738 3738 3032 3737   3   |.|78780277
│ │ │ │ -0005fbc0: 3438 3530 3078 2078 2020 2d20 3437 3137  48500x x  - 4717
│ │ │ │ -0005fbd0: 3236 3739 3438 3035 3078 2078 2020 2b20  267948050x x  + 
│ │ │ │ -0005fbe0: 3133 3838 3737 3538 3535 3630 3030 7820  13887758556000x 
│ │ │ │ -0005fbf0: 202d 2039 3133 3438 3432 3530 3030 3030   - 9134842500000
│ │ │ │ -0005fc00: 7820 7820 207c 0a7c 2020 2020 2020 2020  x x  |.|        
│ │ │ │ -0005fc10: 2020 2020 2020 3020 3120 2020 2020 2020        0 1       
│ │ │ │ -0005fc20: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -0005fc30: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -0005fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc50: 2030 2032 207c 0a7c 2d2d 2d2d 2d2d 2d2d   0 2 |.|--------
│ │ │ │ +0005fb60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005fb70: 2020 2020 2020 2020 2020 2020 3220 3220              2 2 
│ │ │ │ +0005fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fb90: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +0005fba0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ +0005fbb0: 2020 2020 2020 2033 2020 207c 0a7c 3738         3   |.|78
│ │ │ │ +0005fbc0: 3738 3032 3737 3438 3530 3078 2078 2020  78027748500x x  
│ │ │ │ +0005fbd0: 2d20 3437 3137 3236 3739 3438 3035 3078  - 4717267948050x
│ │ │ │ +0005fbe0: 2078 2020 2b20 3133 3838 3737 3538 3535   x  + 1388775855
│ │ │ │ +0005fbf0: 3630 3030 7820 202d 2039 3133 3438 3432  6000x  - 9134842
│ │ │ │ +0005fc00: 3530 3030 3030 7820 7820 207c 0a7c 2020  500000x x  |.|  
│ │ │ │ +0005fc10: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ +0005fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fc30: 3020 3120 2020 2020 2020 2020 2020 2020  0 1             
│ │ │ │ +0005fc40: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0005fc50: 2020 2020 2020 2030 2032 207c 0a7c 2d2d         0 2 |.|--
│ │ │ │  0005fc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fca0: 2d2d 2d2d 2d7c 0a7c 2020 2033 2020 2020  -----|.|   3    
│ │ │ │ -0005fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fcc0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0005fca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0005fcb0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0005fcc0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  0005fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fcf0: 2020 2020 207c 0a7c 3030 7820 7820 202b       |.|00x x  +
│ │ │ │ -0005fd00: 2032 3934 3733 3936 3135 3137 3030 3030   294739615170000
│ │ │ │ -0005fd10: 7820 7820 7820 202d 2031 3338 3135 3333  x x x  - 1381533
│ │ │ │ -0005fd20: 3032 3936 3136 3630 7820 2020 2020 2020  02961660x       
│ │ │ │ +0005fcf0: 2020 2020 2020 2020 2020 207c 0a7c 3030             |.|00
│ │ │ │ +0005fd00: 7820 7820 202b 2032 3934 3733 3936 3135  x x  + 294739615
│ │ │ │ +0005fd10: 3137 3030 3030 7820 7820 7820 202d 2031  170000x x x  - 1
│ │ │ │ +0005fd20: 3338 3135 3333 3032 3936 3136 3630 7820  38153302961660x 
│ │ │ │  0005fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd40: 2020 2020 207c 0a7c 2020 2030 2032 2020       |.|   0 2  
│ │ │ │ -0005fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd60: 2030 2031 2032 2020 2020 2020 2020 2020   0 1 2          
│ │ │ │ +0005fd40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005fd50: 2030 2032 2020 2020 2020 2020 2020 2020   0 2            
│ │ │ │ +0005fd60: 2020 2020 2020 2030 2031 2032 2020 2020         0 1 2    
│ │ │ │  0005fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005fd90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fde0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005fdf0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0005fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe10: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0005fde0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fe00: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0005fe10: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0005fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe30: 2020 2020 207c 0a7c 2020 2b20 3637 3134       |.|  + 6714
│ │ │ │ -0005fe40: 3336 3231 3333 3739 3230 7820 7820 7820  3621337920x x x 
│ │ │ │ -0005fe50: 202d 2034 3538 3139 3431 3735 3134 3332   - 4581941751432
│ │ │ │ -0005fe60: 3078 2078 2078 2020 2b20 2020 2020 2020  0x x x  +       
│ │ │ │ +0005fe30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005fe40: 2b20 3637 3134 3336 3231 3333 3739 3230  + 67143621337920
│ │ │ │ +0005fe50: 7820 7820 7820 202d 2034 3538 3139 3431  x x x  - 4581941
│ │ │ │ +0005fe60: 3735 3134 3332 3078 2078 2078 2020 2b20  7514320x x x  + 
│ │ │ │  0005fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe80: 2020 2020 207c 0a7c 3220 2020 2020 2020       |.|2       
│ │ │ │ -0005fe90: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ -0005fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005feb0: 2020 3020 3120 3220 2020 2020 2020 2020    0 1 2         
│ │ │ │ +0005fe80: 2020 2020 2020 2020 2020 207c 0a7c 3220             |.|2 
│ │ │ │ +0005fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fea0: 2030 2031 2032 2020 2020 2020 2020 2020   0 1 2          
│ │ │ │ +0005feb0: 2020 2020 2020 2020 3020 3120 3220 2020          0 1 2   
│ │ │ │  0005fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fed0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005fed0: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff20: 2020 2020 207c 0a7c 2033 2020 2020 2020       |.| 3      
│ │ │ │ -0005ff30: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0005ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff50: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0005ff20: 2020 2020 2020 2020 2020 207c 0a7c 2033             |.| 3
│ │ │ │ +0005ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ff40: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0005ff50: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  0005ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff70: 2020 2020 207c 0a7c 7820 7820 202d 2034       |.|x x  - 4
│ │ │ │ -0005ff80: 3033 3832 3939 3332 3239 3834 3078 2078  0382993229840x x
│ │ │ │ -0005ff90: 2078 2020 2b20 3635 3339 3037 3936 3835   x  + 6539079685
│ │ │ │ -0005ffa0: 3231 3530 7820 7820 7820 2020 2020 2020  2150x x x       
│ │ │ │ +0005ff70: 2020 2020 2020 2020 2020 207c 0a7c 7820             |.|x 
│ │ │ │ +0005ff80: 7820 202d 2034 3033 3832 3939 3332 3239  x  - 40382993229
│ │ │ │ +0005ff90: 3834 3078 2078 2078 2020 2b20 3635 3339  840x x x  + 6539
│ │ │ │ +0005ffa0: 3037 3936 3835 3231 3530 7820 7820 7820  0796852150x x x 
│ │ │ │  0005ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ffc0: 2020 2020 207c 0a7c 2030 2032 2020 2020       |.| 0 2    
│ │ │ │ -0005ffd0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0005ffe0: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
│ │ │ │ -0005fff0: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │ +0005ffc0: 2020 2020 2020 2020 2020 207c 0a7c 2030             |.| 0
│ │ │ │ +0005ffd0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0005ffe0: 2020 2020 3020 3120 3220 2020 2020 2020      0 1 2       
│ │ │ │ +0005fff0: 2020 2020 2020 2020 2020 2030 2031 2020             0 1  
│ │ │ │  00060000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00060010: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00060020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060060: 2020 2020 207c 0a7c 3320 2020 2020 2020       |.|3       
│ │ │ │ -00060070: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00060080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060090: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00060060: 2020 2020 2020 2020 2020 207c 0a7c 3320             |.|3 
│ │ │ │ +00060070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060080: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00060090: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  000600a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000600b0: 2020 2020 207c 0a7c 2078 2020 2d20 3330       |.| x  - 30
│ │ │ │ -000600c0: 3736 3331 3230 3739 3330 3430 7820 7820  763120793040x x 
│ │ │ │ -000600d0: 7820 202b 2035 3330 3837 3739 3934 3230  x  + 53087799420
│ │ │ │ -000600e0: 7820 7820 7820 202b 2020 2020 2020 2020  x x x  +        
│ │ │ │ +000600b0: 2020 2020 2020 2020 2020 207c 0a7c 2078             |.| x
│ │ │ │ +000600c0: 2020 2d20 3330 3736 3331 3230 3739 3330    - 307631207930
│ │ │ │ +000600d0: 3430 7820 7820 7820 202b 2035 3330 3837  40x x x  + 53087
│ │ │ │ +000600e0: 3739 3934 3230 7820 7820 7820 202b 2020  799420x x x  +  
│ │ │ │  000600f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060100: 2020 2020 207c 0a7c 3020 3220 2020 2020       |.|0 2     
│ │ │ │ -00060110: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ -00060120: 2032 2020 2020 2020 2020 2020 2020 2020   2              
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│ │ │ │ +00060100: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00060110: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00060120: 2020 2030 2031 2032 2020 2020 2020 2020     0 1 2        
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│ │ │ │ -000601a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -000601c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000601d0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
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│ │ │ │ +000601f0: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
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│ │ │ │ -000604b0: 7820 7820 7820 202d 2031 3130 3133 3034  x x x  - 1101304
│ │ │ │ -000604c0: 3330 3231 397c 0a7c 2020 2020 2020 2020  30219|.|        
│ │ │ │ -000604d0: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ -000604e0: 2020 2020 2020 2020 2020 2020 3020 3220              0 2 
│ │ │ │ -000604f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060500: 2030 2031 2032 2020 2020 2020 2020 2020   0 1 2          
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│ │ │ │ +00060470: 2020 2020 2020 2020 2020 207c 0a7c 2033             |.| 3
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│ │ │ │ +00060490: 202d 2033 3034 3135 3532 3430 3734 3439   - 3041552407449
│ │ │ │ +000604a0: 3678 2078 2020 2b20 3237 3436 3734 3239  6x x  + 27467429
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│ │ │ │ -000605b0: 2020 2020 207c 0a7c 2020 2b20 3138 3436       |.|  + 1846
│ │ │ │ -000605c0: 3335 3337 3335 3234 3030 7820 7820 202b  3537352400x x  +
│ │ │ │ -000605d0: 2032 3235 3237 3231 3436 3638 3030 7820   2252721466800x 
│ │ │ │ -000605e0: 7820 202d 2034 3731 3235 3536 3537 3034  x  - 47125565704
│ │ │ │ -000605f0: 3536 3078 2078 2078 2020 2b20 3236 3938  560x x x  + 2698
│ │ │ │ -00060600: 3331 3235 307c 0a7c 3220 2020 2020 2020  31250|.|2       
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│ │ │ │ +000605d0: 7820 7820 202b 2032 3235 3237 3231 3436  x x  + 225272146
│ │ │ │ +000605e0: 3638 3030 7820 7820 202d 2034 3731 3235  6800x x  - 47125
│ │ │ │ +000605f0: 3536 3537 3034 3536 3078 2078 2078 2020  565704560x x x  
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│ │ │ │ +00060610: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00060700: 3532 3030 3078 2078 2020 2d20 3537 3939  52000x x  - 5799
│ │ │ │ -00060710: 3333 3133 3335 3139 3638 7820 7820 202b  3313351968x x  +
│ │ │ │ -00060720: 2038 3735 3934 3733 3032 3031 3434 3078   87594730201440x
│ │ │ │ -00060730: 2078 2078 2020 2d20 3131 3630 3139 3135   x x  - 11601915
│ │ │ │ -00060740: 3432 3234 307c 0a7c 2020 2020 2020 2020  42240|.|        
│ │ │ │ -00060750: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │ -00060760: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ -00060770: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000606d0: 2032 2032 2020 2020 2020 2020 2020 2020   2 2            
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│ │ │ │ +00060700: 3537 3639 3333 3532 3030 3078 2078 2020  57693352000x x  
│ │ │ │ +00060710: 2d20 3537 3939 3333 3133 3335 3139 3638  - 57993313351968
│ │ │ │ +00060720: 7820 7820 202b 2038 3735 3934 3733 3032  x x  + 875947302
│ │ │ │ +00060730: 3031 3434 3078 2078 2078 2020 2d20 3131  01440x x x  - 11
│ │ │ │ +00060740: 3630 3139 3135 3432 3234 307c 0a7c 2020  60191542240|.|  
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│ │ │ │ -00060840: 3238 3333 3230 3078 2078 2020 2d20 3135  2833200x x  - 15
│ │ │ │ -00060850: 3233 3935 3633 3139 3736 3936 7820 7820  239563197696x x 
│ │ │ │ -00060860: 202d 2031 3333 3833 3431 3033 3938 3936   - 1338341039896
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│ │ │ │ +00060800: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00060840: 3235 3032 3838 3238 3333 3230 3078 2078  2502882833200x x
│ │ │ │ +00060850: 2020 2d20 3135 3233 3935 3633 3139 3736    - 152395631976
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│ │ │ │ +00060870: 3033 3938 3936 3030 7820 7820 7820 202b  03989600x x x  +
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│ │ │ │ +000608a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │ -00060960: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00060980: 3630 7820 7820 202d 2032 3032 3237 3639  60x x  - 2022769
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│ │ │ │ -000609a0: 3437 3435 3733 3634 3831 3630 7820 7820  474573648160x x 
│ │ │ │ -000609b0: 202d 2031 3930 3237 3137 3733 3434 3030   - 1902717734400
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│ │ │ │ +00060980: 3636 3333 3235 3630 7820 7820 202d 2032  66332560x x  - 2
│ │ │ │ +00060990: 3032 3237 3639 3435 3138 3038 3078 2078  0227694518080x x
│ │ │ │ +000609a0: 2020 2b20 3533 3437 3435 3733 3634 3831    + 534745736481
│ │ │ │ +000609b0: 3630 7820 7820 202d 2031 3930 3237 3137  60x x  - 1902717
│ │ │ │ +000609c0: 3733 3434 3030 3078 2020 2b7c 0a7c 2020  7344000x  +|.|  
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│ │ │ │ +000609f0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │  00060a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060a60: 2020 2020 207c 0a7c 2020 2032 2032 2020       |.|   2 2  
│ │ │ │ -00060a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060a80: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00060a90: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ -00060aa0: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ -00060ab0: 2020 2020 207c 0a7c 3230 7820 7820 202b       |.|20x x  +
│ │ │ │ -00060ac0: 2037 3937 3834 3739 3831 3335 3034 7820   7978479813504x 
│ │ │ │ -00060ad0: 7820 202d 2037 3831 3834 3231 3134 3537  x  - 78184211457
│ │ │ │ -00060ae0: 3630 7820 7820 202b 2036 3738 3636 3234  60x x  + 6786624
│ │ │ │ -00060af0: 3236 3838 3030 7820 202b 2032 3835 3236  268800x  + 28526
│ │ │ │ -00060b00: 3230 3834 387c 0a7c 2020 2031 2032 2020  20848|.|   1 2  
│ │ │ │ -00060b10: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00060b20: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00060b30: 2020 2031 2032 2020 2020 2020 2020 2020     1 2          
│ │ │ │ -00060b40: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00060b50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00060a60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00060a70: 2032 2032 2020 2020 2020 2020 2020 2020   2 2            
│ │ │ │ +00060a80: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00060a90: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00060aa0: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ +00060ab0: 2020 2020 2020 2020 2020 207c 0a7c 3230             |.|20
│ │ │ │ +00060ac0: 7820 7820 202b 2037 3937 3834 3739 3831  x x  + 797847981
│ │ │ │ +00060ad0: 3335 3034 7820 7820 202d 2037 3831 3834  3504x x  - 78184
│ │ │ │ +00060ae0: 3231 3134 3537 3630 7820 7820 202b 2036  21145760x x  + 6
│ │ │ │ +00060af0: 3738 3636 3234 3236 3838 3030 7820 202b  786624268800x  +
│ │ │ │ +00060b00: 2032 3835 3236 3230 3834 387c 0a7c 2020   2852620848|.|  
│ │ │ │ +00060b10: 2031 2032 2020 2020 2020 2020 2020 2020   1 2            
│ │ │ │ +00060b20: 2020 2020 2030 2032 2020 2020 2020 2020       0 2        
│ │ │ │ +00060b30: 2020 2020 2020 2020 2031 2032 2020 2020           1 2    
│ │ │ │ +00060b40: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00060b50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00060b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ba0: 2020 2020 207c 0a7c 2020 2020 2020 3220       |.|      2 
│ │ │ │ -00060bb0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00060bc0: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ -00060bd0: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ -00060be0: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ -00060bf0: 2020 2020 207c 0a7c 3430 3230 3078 2078       |.|40200x x
│ │ │ │ -00060c00: 2020 2b20 3231 3433 3231 3539 3832 3635    + 214321598265
│ │ │ │ -00060c10: 3630 7820 7820 202d 2033 3734 3730 3530  60x x  - 3747050
│ │ │ │ -00060c20: 3430 3537 3630 3078 2078 2020 2b20 3639  4057600x x  + 69
│ │ │ │ -00060c30: 3934 3533 3631 3932 3030 3078 2020 2b20  94536192000x  + 
│ │ │ │ -00060c40: 3236 3035 397c 0a7c 2020 2020 2020 3120  26059|.|      1 
│ │ │ │ -00060c50: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00060c60: 2020 2030 2032 2020 2020 2020 2020 2020     0 2          
│ │ │ │ -00060c70: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -00060c80: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00060c90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00060ba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00060bb0: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ +00060bc0: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00060bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060be0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00060bf0: 2020 3420 2020 2020 2020 207c 0a7c 3430    4        |.|40
│ │ │ │ +00060c00: 3230 3078 2078 2020 2b20 3231 3433 3231  200x x  + 214321
│ │ │ │ +00060c10: 3539 3832 3635 3630 7820 7820 202d 2033  59826560x x  - 3
│ │ │ │ +00060c20: 3734 3730 3530 3430 3537 3630 3078 2078  7470504057600x x
│ │ │ │ +00060c30: 2020 2b20 3639 3934 3533 3631 3932 3030    + 699453619200
│ │ │ │ +00060c40: 3078 2020 2b20 3236 3035 397c 0a7c 2020  0x  + 26059|.|  
│ │ │ │ +00060c50: 2020 2020 3120 3220 2020 2020 2020 2020      1 2         
│ │ │ │ +00060c60: 2020 2020 2020 2020 2030 2032 2020 2020           0 2    
│ │ │ │ +00060c70: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00060c80: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00060c90: 2020 3220 2020 2020 2020 207c 0a7c 2020    2        |.|  
│ │ │ │  00060ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ce0: 2020 2020 207c 0a7c 2020 3220 3220 2020       |.|  2 2   
│ │ │ │ -00060cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060d00: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00060d10: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ -00060d20: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ -00060d30: 2020 2020 207c 0a7c 3078 2078 2020 2d20       |.|0x x  - 
│ │ │ │ -00060d40: 3436 3735 3730 3830 3331 3138 3430 7820  46757080311840x 
│ │ │ │ -00060d50: 7820 202b 2031 3136 3731 3834 3539 3834  x  + 11671845984
│ │ │ │ -00060d60: 3030 3078 2078 2020 2d20 3331 3935 3934  000x x  - 319594
│ │ │ │ -00060d70: 3230 3438 3030 3078 2020 2b20 3339 3534  2048000x  + 3954
│ │ │ │ -00060d80: 3130 3931 307c 0a7c 2020 3120 3220 2020  10910|.|  1 2   
│ │ │ │ -00060d90: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00060da0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00060db0: 2020 2020 3120 3220 2020 2020 2020 2020      1 2         
│ │ │ │ -00060dc0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00060dd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00060ce0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00060cf0: 3220 3220 2020 2020 2020 2020 2020 2020  2 2             
│ │ │ │ +00060d00: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00060d10: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +00060d20: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ +00060d30: 2020 2020 2020 2020 2020 207c 0a7c 3078             |.|0x
│ │ │ │ +00060d40: 2078 2020 2d20 3436 3735 3730 3830 3331   x  - 4675708031
│ │ │ │ +00060d50: 3138 3430 7820 7820 202b 2031 3136 3731  1840x x  + 11671
│ │ │ │ +00060d60: 3834 3539 3834 3030 3078 2078 2020 2d20  845984000x x  - 
│ │ │ │ +00060d70: 3331 3935 3934 3230 3438 3030 3078 2020  3195942048000x  
│ │ │ │ +00060d80: 2b20 3339 3534 3130 3931 307c 0a7c 2020  + 395410910|.|  
│ │ │ │ +00060d90: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
│ │ │ │ +00060da0: 2020 2020 2030 2032 2020 2020 2020 2020       0 2        
│ │ │ │ +00060db0: 2020 2020 2020 2020 2020 3120 3220 2020            1 2   
│ │ │ │ +00060dc0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00060dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00060de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060e20: 2020 2020 207c 0a7c 2020 2020 3220 3220       |.|    2 2 
│ │ │ │ -00060e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060e40: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00060e50: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -00060e60: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ -00060e70: 2020 2020 207c 0a7c 3030 3078 2078 2020       |.|000x x  
│ │ │ │ -00060e80: 2b20 3133 3631 3031 3533 3332 3939 3230  + 13610153329920
│ │ │ │ -00060e90: 7820 7820 202d 2031 3035 3535 3932 3633  x x  - 105559263
│ │ │ │ -00060ea0: 3630 3030 3078 2078 2020 2b20 3731 3936  60000x x  + 7196
│ │ │ │ -00060eb0: 3837 3436 3234 3030 3078 2020 2d20 3139  874624000x  - 19
│ │ │ │ -00060ec0: 3536 3132 327c 0a7c 2020 2020 3120 3220  56122|.|    1 2 
│ │ │ │ -00060ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ee0: 2030 2032 2020 2020 2020 2020 2020 2020   0 2            
│ │ │ │ -00060ef0: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │ -00060f00: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00060f10: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ +00060e30: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +00060e40: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +00060e50: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +00060e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060e70: 3420 2020 2020 2020 2020 207c 0a7c 3030  4          |.|00
│ │ │ │ +00060e80: 3078 2078 2020 2b20 3133 3631 3031 3533  0x x  + 13610153
│ │ │ │ +00060e90: 3332 3939 3230 7820 7820 202d 2031 3035  329920x x  - 105
│ │ │ │ +00060ea0: 3535 3932 3633 3630 3030 3078 2078 2020  55926360000x x  
│ │ │ │ +00060eb0: 2b20 3731 3936 3837 3436 3234 3030 3078  + 7196874624000x
│ │ │ │ +00060ec0: 2020 2d20 3139 3536 3132 327c 0a7c 2020    - 1956122|.|  
│ │ │ │ +00060ed0: 2020 3120 3220 2020 2020 2020 2020 2020    1 2           
│ │ │ │ +00060ee0: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ +00060ef0: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
│ │ │ │ +00060f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060f10: 3220 2020 2020 2020 2020 207c 0a7c 2d2d  2          |.|--
│ │ │ │  00060f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060f60: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00060f70: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ -00060f80: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00060f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060fa0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00060fb0: 2020 2020 207c 0a7c 2035 3433 3634 3738       |.| 5436478
│ │ │ │ -00060fc0: 3136 3437 3236 3030 7820 7820 202d 2037  16472600x x  - 7
│ │ │ │ -00060fd0: 3239 3633 3235 3031 3034 3738 3030 7820  29632501047800x 
│ │ │ │ -00060fe0: 7820 7820 202b 2033 3933 3838 3039 3736  x x  + 393880976
│ │ │ │ -00060ff0: 3736 3438 3030 7820 7820 7820 202b 2031  764800x x x  + 1
│ │ │ │ -00061000: 3733 3332 357c 0a7c 2020 2020 2020 2020  73325|.|        
│ │ │ │ -00061010: 2020 2020 2020 2020 2030 2033 2020 2020           0 3    
│ │ │ │ -00061020: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00061030: 2031 2033 2020 2020 2020 2020 2020 2020   1 3            
│ │ │ │ -00061040: 2020 2020 2020 2030 2031 2033 2020 2020         0 1 3    
│ │ │ │ -00061050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00060f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +00060f70: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00060f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060f90: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00060fa0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00060fb0: 2020 2020 2020 2020 2020 207c 0a7c 2035             |.| 5
│ │ │ │ +00060fc0: 3433 3634 3738 3136 3437 3236 3030 7820  43647816472600x 
│ │ │ │ +00060fd0: 7820 202d 2037 3239 3633 3235 3031 3034  x  - 72963250104
│ │ │ │ +00060fe0: 3738 3030 7820 7820 7820 202b 2033 3933  7800x x x  + 393
│ │ │ │ +00060ff0: 3838 3039 3736 3736 3438 3030 7820 7820  880976764800x x 
│ │ │ │ +00061000: 7820 202b 2031 3733 3332 357c 0a7c 2020  x  + 173325|.|  
│ │ │ │ +00061010: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00061020: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00061030: 2020 2020 2030 2031 2033 2020 2020 2020       0 1 3      
│ │ │ │ +00061040: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ +00061050: 2033 2020 2020 2020 2020 207c 0a7c 2020   3         |.|  
│ │ │ │  00061060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061080: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000610c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
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│ │ │ │ -000610e0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -000610f0: 2020 2020 207c 0a7c 3030 3078 2078 2020       |.|000x x  
│ │ │ │ -00061100: 2d20 3534 3338 3033 3739 3034 3939 3230  - 54380379049920
│ │ │ │ -00061110: 7820 7820 7820 202d 2031 3430 3539 3134  x x x  - 1405914
│ │ │ │ -00061120: 3238 3534 3236 3078 2078 2078 2020 2b20  2854260x x x  + 
│ │ │ │ -00061130: 3933 3433 3134 3638 3931 3936 3078 2078  9343146891960x x
│ │ │ │ -00061140: 2020 2b20 377c 0a7c 2020 2020 3020 3320    + 7|.|    0 3 
│ │ │ │ -00061150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061160: 2030 2031 2033 2020 2020 2020 2020 2020   0 1 3          
│ │ │ │ -00061170: 2020 2020 2020 2020 3020 3120 3320 2020          0 1 3   
│ │ │ │ -00061180: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00061190: 3320 2020 207c 0a7c 2020 2020 2020 2020  3    |.|        
│ │ │ │ +000610a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000610b0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +000610c0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000610d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000610e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000610f0: 2020 2020 3320 2020 2020 207c 0a7c 3030      3      |.|00
│ │ │ │ +00061100: 3078 2078 2020 2d20 3534 3338 3033 3739  0x x  - 54380379
│ │ │ │ +00061110: 3034 3939 3230 7820 7820 7820 202d 2031  049920x x x  - 1
│ │ │ │ +00061120: 3430 3539 3134 3238 3534 3236 3078 2078  4059142854260x x
│ │ │ │ +00061130: 2078 2020 2b20 3933 3433 3134 3638 3931   x  + 9343146891
│ │ │ │ +00061140: 3936 3078 2078 2020 2b20 377c 0a7c 2020  960x x  + 7|.|  
│ │ │ │ +00061150: 2020 3020 3320 2020 2020 2020 2020 2020    0 3           
│ │ │ │ +00061160: 2020 2020 2020 2030 2031 2033 2020 2020         0 1 3    
│ │ │ │ +00061170: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00061180: 3120 3320 2020 2020 2020 2020 2020 2020  1 3             
│ │ │ │ +00061190: 2020 2020 3120 3320 2020 207c 0a7c 2020      1 3    |.|  
│ │ │ │  000611a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000611b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000611c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000611d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000611e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000611f0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00061200: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +000611e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000611f0: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +00061200: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00061210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061220: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00061230: 2020 2020 207c 0a7c 3436 3233 3835 3034       |.|46238504
│ │ │ │ -00061240: 3030 7820 7820 202b 2032 3930 3834 3136  00x x  + 2908416
│ │ │ │ -00061250: 3039 3231 3438 3030 7820 7820 7820 202d  09214800x x x  -
│ │ │ │ -00061260: 2033 3033 3835 3037 3131 3335 3039 3030   303850711350900
│ │ │ │ -00061270: 7820 7820 7820 202d 2039 3737 3838 3438  x x x  - 9778848
│ │ │ │ -00061280: 3031 3332 307c 0a7c 2020 2020 2020 2020  01320|.|        
│ │ │ │ -00061290: 2020 2030 2033 2020 2020 2020 2020 2020     0 3          
│ │ │ │ -000612a0: 2020 2020 2020 2020 2030 2031 2033 2020           0 1 3  
│ │ │ │ -000612b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000612c0: 2030 2031 2033 2020 2020 2020 2020 2020   0 1 3          
│ │ │ │ -000612d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00061220: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00061230: 2020 2020 2020 2020 2020 207c 0a7c 3436             |.|46
│ │ │ │ +00061240: 3233 3835 3034 3030 7820 7820 202b 2032  23850400x x  + 2
│ │ │ │ +00061250: 3930 3834 3136 3039 3231 3438 3030 7820  90841609214800x 
│ │ │ │ +00061260: 7820 7820 202d 2033 3033 3835 3037 3131  x x  - 303850711
│ │ │ │ +00061270: 3335 3039 3030 7820 7820 7820 202d 2039  350900x x x  - 9
│ │ │ │ +00061280: 3737 3838 3438 3031 3332 307c 0a7c 2020  77884801320|.|  
│ │ │ │ +00061290: 2020 2020 2020 2020 2030 2033 2020 2020           0 3    
│ │ │ │ +000612a0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +000612b0: 2031 2033 2020 2020 2020 2020 2020 2020   1 3            
│ │ │ │ +000612c0: 2020 2020 2020 2030 2031 2033 2020 2020         0 1 3    
│ │ │ │ +000612d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000612e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000612f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061320: 2020 2020 207c 0a7c 2020 2020 2020 3320       |.|      3 
│ │ │ │ -00061330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061340: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00061350: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00061360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061370: 2033 2020 207c 0a7c 3538 3030 3078 2078   3   |.|58000x x
│ │ │ │ -00061380: 2020 2d20 3132 3038 3034 3537 3734 3634    - 120804577464
│ │ │ │ -00061390: 3036 3078 2078 2078 2020 2b20 3238 3833  060x x x  + 2883
│ │ │ │ -000613a0: 3530 3134 3135 3734 3630 7820 7820 7820  5014157460x x x 
│ │ │ │ -000613b0: 202d 2032 3436 3632 3738 3939 3238 3030   - 2466278992800
│ │ │ │ -000613c0: 7820 7820 207c 0a7c 2020 2020 2020 3020  x x  |.|      0 
│ │ │ │ -000613d0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -000613e0: 2020 2020 3020 3120 3320 2020 2020 2020      0 1 3       
│ │ │ │ -000613f0: 2020 2020 2020 2020 2020 2030 2031 2033             0 1 3
│ │ │ │ -00061400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061410: 2031 2033 207c 0a7c 2020 2020 2020 2020   1 3 |.|        
│ │ │ │ +00061320: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00061330: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +00061340: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00061350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061360: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00061370: 2020 2020 2020 2033 2020 207c 0a7c 3538         3   |.|58
│ │ │ │ +00061380: 3030 3078 2078 2020 2d20 3132 3038 3034  000x x  - 120804
│ │ │ │ +00061390: 3537 3734 3634 3036 3078 2078 2078 2020  577464060x x x  
│ │ │ │ +000613a0: 2b20 3238 3833 3530 3134 3135 3734 3630  + 28835014157460
│ │ │ │ +000613b0: 7820 7820 7820 202d 2032 3436 3632 3738  x x x  - 2466278
│ │ │ │ +000613c0: 3939 3238 3030 7820 7820 207c 0a7c 2020  992800x x  |.|  
│ │ │ │ +000613d0: 2020 2020 3020 3320 2020 2020 2020 2020      0 3         
│ │ │ │ +000613e0: 2020 2020 2020 2020 2020 3020 3120 3320            0 1 3 
│ │ │ │ +000613f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061400: 2030 2031 2033 2020 2020 2020 2020 2020   0 1 3          
│ │ │ │ +00061410: 2020 2020 2020 2031 2033 207c 0a7c 2020         1 3 |.|  
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│ │ │ │  00061430: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00061450: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061470: 2033 2020 2020 2020 2020 2020 2020 2020   3              
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│ │ │ │ -00061540: 2031 2033 2020 2020 2020 2020 2020 2020   1 3            
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│ │ │ │ +000614c0: 3632 3830 3030 7820 7820 202b 2032 3631  628000x x  + 261
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│ │ │ │ +000614e0: 7820 202d 2031 3537 3534 3737 3537 3736  x  - 15754775776
│ │ │ │ +000614f0: 3434 3630 7820 7820 7820 202d 2035 3539  4460x x x  - 559
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│ │ │ │ -000617b0: 3220 3320 2020 2020 2020 2020 2020 2020  2 3             
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│ │ │ │ +000617a0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ -00061c60: 3531 3034 3136 3230 7820 7820 7820 202b  51041620x x x  +
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│ │ │ │ -00061c80: 2078 2020 2b7c 0a7c 2020 2031 2032 2033   x  +|.|   1 2 3
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│ │ │ │ -00061e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00061d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061d60: 2020 3320 2020 2020 2020 2020 2020 2020    3             
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│ │ │ │ +00061df0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ -00061ec0: 3438 3133 3531 3230 7820 7820 7820 202b  48135120x x x  +
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│ │ │ │ -00061ee0: 2078 2078 2020 2d20 3132 3734 3336 3835   x x  - 12743685
│ │ │ │ -00061ef0: 3639 3630 3030 7820 7820 202b 2035 3637  696000x x  + 567
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│ │ │ │ +00061ef0: 3734 3336 3835 3639 3630 3030 7820 7820  743685696000x x 
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│ │ │ │ -00061fb0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
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│ │ │ │ -00062070: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00062170: 3339 3436 3638 3830 3078 2078 2020 2d20  394668800x x  - 
│ │ │ │ -00062180: 3539 3037 387c 0a7c 3320 2020 2020 2020  59078|.|3       
│ │ │ │ -00062190: 2020 2020 2020 2020 2020 2030 2032 2033             0 2 3
│ │ │ │ -000621a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00062150: 7820 7820 7820 202d 2031 3137 3937 3836  x x x  - 1179786
│ │ │ │ +00062160: 3838 3030 3837 3230 7820 7820 7820 202b  88008720x x x  +
│ │ │ │ +00062170: 2031 3230 3535 3339 3436 3638 3830 3078   12055394668800x
│ │ │ │ +00062180: 2078 2020 2d20 3539 3037 387c 0a7c 3320   x  - 59078|.|3 
│ │ │ │ +00062190: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000621e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00062200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00062240: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00062270: 2020 2020 207c 0a7c 2033 3838 3032 3037       |.| 3880207
│ │ │ │ -00062280: 3536 3835 3338 3030 7820 7820 202d 2034  56853800x x  - 4
│ │ │ │ -00062290: 3534 3938 3233 3331 3835 3836 3030 7820  54982331858600x 
│ │ │ │ -000622a0: 7820 7820 202b 2032 3735 3433 3933 3238  x x  + 275439328
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│ │ │ │ -000622c0: 3439 3639 367c 0a7c 2020 2020 2020 2020  49696|.|        
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│ │ │ │ -000622f0: 2031 2033 2020 2020 2020 2020 2020 2020   1 3            
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│ │ │ │ +000622a0: 3836 3030 7820 7820 7820 202b 2032 3735  8600x x x  + 275
│ │ │ │ +000622b0: 3433 3933 3238 3936 3933 3530 7820 7820  439328969350x x 
│ │ │ │ +000622c0: 202b 2033 3335 3439 3639 367c 0a7c 2020   + 33549696|.|  
│ │ │ │ +000622d0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +000622e0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
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│ │ │ │ -00062380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062390: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ -000623a0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -000623b0: 2020 2020 207c 0a7c 2d20 3132 3932 3737       |.|- 129277
│ │ │ │ -000623c0: 3238 3737 3038 3234 3078 2078 2078 2020  287708240x x x  
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│ │ │ │ -000623e0: 3078 2078 2020 2b20 3637 3031 3931 3238  0x x  + 67019128
│ │ │ │ -000623f0: 3239 3533 3932 7820 7820 7820 202d 2031  295392x x x  - 1
│ │ │ │ -00062400: 3334 3937 307c 0a7c 2020 2020 2020 2020  34970|.|        
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│ │ │ │ +000623b0: 2032 2020 2020 2020 2020 207c 0a7c 2d20   2         |.|- 
│ │ │ │ +000623c0: 3132 3932 3737 3238 3737 3038 3234 3078  129277287708240x
│ │ │ │ +000623d0: 2078 2078 2020 2b20 3130 3036 3932 3433   x x  + 10069243
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│ │ │ │ +000623f0: 3031 3931 3238 3239 3533 3932 7820 7820  019128295392x x 
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│ │ │ │ +00062450: 2033 2020 2020 2020 2020 207c 0a7c 2020   3         |.|  
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│ │ │ │ -00062540: 3830 7820 787c 0a7c 2020 2020 2020 2030  80x x|.|       0
│ │ │ │ -00062550: 2033 2020 2020 2020 2020 2020 2020 2020   3              
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│ │ │ │ -00062580: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000624c0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000624d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000624e0: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ +000624f0: 2020 2020 2020 2020 2020 207c 0a7c 3537             |.|57
│ │ │ │ +00062500: 3339 3230 7820 7820 202b 2031 3136 3532  3920x x  + 11652
│ │ │ │ +00062510: 3838 3838 3331 3639 3630 7820 7820 7820  8888316960x x x 
│ │ │ │ +00062520: 202d 2031 3332 3238 3338 3439 3036 3539   - 1322838490659
│ │ │ │ +00062530: 3030 7820 7820 202d 2031 3737 3831 3736  00x x  - 1778176
│ │ │ │ +00062540: 3030 3234 3330 3830 7820 787c 0a7c 2020  00243080x x|.|  
│ │ │ │ +00062550: 2020 2020 2030 2033 2020 2020 2020 2020       0 3        
│ │ │ │ +00062560: 2020 2020 2020 2020 2020 2030 2031 2033             0 1 3
│ │ │ │ +00062570: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000625a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000625e0: 2020 2020 207c 0a7c 2020 3220 3220 2020       |.|  2 2   
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│ │ │ │ -00062600: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00062610: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
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│ │ │ │  00062620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062630: 3220 2020 207c 0a7c 3078 2078 2020 2b20  2    |.|0x x  + 
│ │ │ │ -00062640: 3330 3539 3636 3033 3732 3233 3938 3078  305966037223980x
│ │ │ │ -00062650: 2078 2078 2020 2d20 3437 3035 3330 3130   x x  - 47053010
│ │ │ │ -00062660: 3339 3034 3030 7820 7820 202d 2037 3130  390400x x  - 710
│ │ │ │ -00062670: 3034 3238 3433 3639 3432 3478 2078 2078  04284369424x x x
│ │ │ │ -00062680: 2020 2d20 317c 0a7c 2020 3020 3320 2020    - 1|.|  0 3   
│ │ │ │ -00062690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000626a0: 3020 3120 3320 2020 2020 2020 2020 2020  0 1 3           
│ │ │ │ -000626b0: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ -000626c0: 2020 2020 2020 2020 2020 2020 3020 3220              0 2 
│ │ │ │ -000626d0: 3320 2020 207c 0a7c 2020 2020 2020 2020  3    |.|        
│ │ │ │ +00062630: 2020 2020 2020 3220 2020 207c 0a7c 3078        2    |.|0x
│ │ │ │ +00062640: 2078 2020 2b20 3330 3539 3636 3033 3732   x  + 3059660372
│ │ │ │ +00062650: 3233 3938 3078 2078 2078 2020 2d20 3437  23980x x x  - 47
│ │ │ │ +00062660: 3035 3330 3130 3339 3034 3030 7820 7820  053010390400x x 
│ │ │ │ +00062670: 202d 2037 3130 3034 3238 3433 3639 3432   - 7100428436942
│ │ │ │ +00062680: 3478 2078 2078 2020 2d20 317c 0a7c 2020  4x x x  - 1|.|  
│ │ │ │ +00062690: 3020 3320 2020 2020 2020 2020 2020 2020  0 3             
│ │ │ │ +000626a0: 2020 2020 2020 3020 3120 3320 2020 2020        0 1 3     
│ │ │ │ +000626b0: 2020 2020 2020 2020 2020 2020 2031 2033               1 3
│ │ │ │ +000626c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000626e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000626f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062710: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00062730: 2032 2032 2020 2020 2020 2020 2020 2020   2 2            
│ │ │ │ -00062740: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00062750: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00062760: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00062770: 2020 2020 207c 0a7c 3234 3734 3830 3030       |.|24748000
│ │ │ │ -00062780: 7820 7820 202b 2031 3536 3038 3233 3438  x x  + 156082348
│ │ │ │ -00062790: 3939 3438 3830 7820 7820 7820 202d 2032  994880x x x  - 2
│ │ │ │ -000627a0: 3536 3338 3139 3235 3237 3639 3235 7820  56381925276925x 
│ │ │ │ -000627b0: 7820 202d 2032 3534 3731 3935 3235 3233  x  - 25471952523
│ │ │ │ -000627c0: 3139 3230 787c 0a7c 2020 2020 2020 2020  1920x|.|        
│ │ │ │ -000627d0: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
│ │ │ │ -000627e0: 2020 2020 2020 2030 2031 2033 2020 2020         0 1 3    
│ │ │ │ -000627f0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00062800: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00062810: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00062720: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00062730: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ +00062740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062750: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00062760: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ +00062770: 2020 2020 2020 2020 2020 207c 0a7c 3234             |.|24
│ │ │ │ +00062780: 3734 3830 3030 7820 7820 202b 2031 3536  748000x x  + 156
│ │ │ │ +00062790: 3038 3233 3438 3939 3438 3830 7820 7820  082348994880x x 
│ │ │ │ +000627a0: 7820 202d 2032 3536 3338 3139 3235 3237  x  - 25638192527
│ │ │ │ +000627b0: 3639 3235 7820 7820 202d 2032 3534 3731  6925x x  - 25471
│ │ │ │ +000627c0: 3935 3235 3233 3139 3230 787c 0a7c 2020  9525231920x|.|  
│ │ │ │ +000627d0: 2020 2020 2020 2030 2033 2020 2020 2020         0 3      
│ │ │ │ +000627e0: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ +000627f0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00062800: 2020 2020 2031 2033 2020 2020 2020 2020       1 3        
│ │ │ │ +00062810: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  00062820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062860: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00062870: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00062880: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00062890: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -000628a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000628b0: 2020 2020 207c 0a7c 3636 3533 3234 3078       |.|6653240x
│ │ │ │ -000628c0: 2078 2078 2020 2d20 3333 3435 3234 3431   x x  - 33452441
│ │ │ │ -000628d0: 3830 3034 3730 3078 2078 2078 2020 2b20  8004700x x x  + 
│ │ │ │ -000628e0: 3734 3939 3539 3733 3733 3037 3230 7820  74995973730720x 
│ │ │ │ -000628f0: 7820 202d 2035 3731 3933 3132 3239 3834  x  - 57193122984
│ │ │ │ -00062900: 3438 3030 787c 0a7c 2020 2020 2020 2020  4800x|.|        
│ │ │ │ -00062910: 3020 3220 3320 2020 2020 2020 2020 2020  0 2 3           
│ │ │ │ -00062920: 2020 2020 2020 2020 3120 3220 3320 2020          1 2 3   
│ │ │ │ -00062930: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00062940: 2033 2020 2020 2020 2020 2020 2020 2020   3              
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│ │ │ │ +00062860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +00062870: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00062880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062890: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000628a0: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ +000628b0: 2020 2020 2020 2020 2020 207c 0a7c 3636             |.|66
│ │ │ │ +000628c0: 3533 3234 3078 2078 2078 2020 2d20 3333  53240x x x  - 33
│ │ │ │ +000628d0: 3435 3234 3431 3830 3034 3730 3078 2078  4524418004700x x
│ │ │ │ +000628e0: 2078 2020 2b20 3734 3939 3539 3733 3733   x  + 7499597373
│ │ │ │ +000628f0: 3037 3230 7820 7820 202d 2035 3731 3933  0720x x  - 57193
│ │ │ │ +00062900: 3132 3239 3834 3438 3030 787c 0a7c 2020  1229844800x|.|  
│ │ │ │ +00062910: 2020 2020 2020 3020 3220 3320 2020 2020        0 2 3     
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│ │ │ │ +00062930: 3220 3320 2020 2020 2020 2020 2020 2020  2 3             
│ │ │ │ +00062940: 2020 2020 2032 2033 2020 2020 2020 2020       2 3        
│ │ │ │ +00062950: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00062960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000629a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000629b0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
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│ │ │ │ -000629d0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -000629e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00062a10: 3639 3233 3132 3332 7820 7820 202d 2031  69231232x x  - 1
│ │ │ │ -00062a20: 3432 3535 3830 3838 3036 3430 7820 7820  425580880640x x 
│ │ │ │ -00062a30: 202b 2038 3934 3534 3030 3030 3331 3230   + 8945400003120
│ │ │ │ -00062a40: 3078 2078 207c 0a7c 2020 2020 2020 2020  0x x |.|        
│ │ │ │ -00062a50: 2020 3120 3220 3320 2020 2020 2020 2020    1 2 3         
│ │ │ │ -00062a60: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -00062a70: 2020 2020 2020 2020 2020 2020 2030 2033               0 3
│ │ │ │ -00062a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062a90: 2020 3120 337c 0a7c 2020 2020 2020 2020    1 3|.|        
│ │ │ │ +000629a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000629b0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000629c0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000629d0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000629e0: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +000629f0: 2020 2020 2020 2020 2020 337c 0a7c 3031            3|.|01
│ │ │ │ +00062a00: 3130 3837 3238 3078 2078 2078 2020 2b20  1087280x x x  + 
│ │ │ │ +00062a10: 3236 3439 3530 3639 3233 3132 3332 7820  26495069231232x 
│ │ │ │ +00062a20: 7820 202d 2031 3432 3535 3830 3838 3036  x  - 14255808806
│ │ │ │ +00062a30: 3430 7820 7820 202b 2038 3934 3534 3030  40x x  + 8945400
│ │ │ │ +00062a40: 3030 3331 3230 3078 2078 207c 0a7c 2020  0031200x x |.|  
│ │ │ │ +00062a50: 2020 2020 2020 2020 3120 3220 3320 2020          1 2 3   
│ │ │ │ +00062a60: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00062a70: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00062a80: 2020 2030 2033 2020 2020 2020 2020 2020     0 3          
│ │ │ │ +00062a90: 2020 2020 2020 2020 3120 337c 0a7c 2020          1 3|.|  
│ │ │ │  00062aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ae0: 2020 2020 207c 0a7c 2020 3220 2020 2020       |.|  2     
│ │ │ │ -00062af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062b00: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00062b10: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ -00062b20: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -00062b30: 2020 2020 207c 0a7c 2078 2020 2b20 3230       |.| x  + 20
│ │ │ │ -00062b40: 3434 3935 3431 3937 3632 3930 3078 2078  4495419762900x x
│ │ │ │ -00062b50: 2078 2020 2d20 3132 3239 3135 3931 3836   x  - 1229159186
│ │ │ │ -00062b60: 3831 3932 3078 2078 2020 2d20 3234 3433  81920x x  - 2443
│ │ │ │ -00062b70: 3938 3539 3338 3930 3038 3078 2078 2020  98593890080x x  
│ │ │ │ -00062b80: 2d20 3130 367c 0a7c 3220 3320 2020 2020  - 106|.|2 3     
│ │ │ │ -00062b90: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00062ba0: 3220 3320 2020 2020 2020 2020 2020 2020  2 3             
│ │ │ │ -00062bb0: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ -00062bc0: 2020 2020 2020 2020 2020 2020 3020 3320              0 3 
│ │ │ │ -00062bd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00062ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00062af0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │ +00062b10: 2020 2020 2020 2020 2020 2020 3220 3220              2 2 
│ │ │ │ +00062b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062b30: 2020 2020 3320 2020 2020 207c 0a7c 2078      3      |.| x
│ │ │ │ +00062b40: 2020 2b20 3230 3434 3935 3431 3937 3632    + 204495419762
│ │ │ │ +00062b50: 3930 3078 2078 2078 2020 2d20 3132 3239  900x x x  - 1229
│ │ │ │ +00062b60: 3135 3931 3836 3831 3932 3078 2078 2020  15918681920x x  
│ │ │ │ +00062b70: 2d20 3234 3433 3938 3539 3338 3930 3038  - 24439859389008
│ │ │ │ +00062b80: 3078 2078 2020 2d20 3130 367c 0a7c 3220  0x x  - 106|.|2 
│ │ │ │ +00062b90: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ +00062bb0: 2020 2020 2020 2020 2020 2020 3220 3320              2 3 
│ │ │ │ +00062bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00062c40: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
│ │ │ │ -00062c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062c60: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ -00062c70: 2020 2020 207c 0a7c 3631 3134 3734 3236       |.|61147426
│ │ │ │ -00062c80: 3335 3530 3078 2078 2078 2020 2b20 3233  35500x x x  + 23
│ │ │ │ -00062c90: 3236 3038 3031 3031 3638 3830 7820 7820  260801016880x x 
│ │ │ │ -00062ca0: 202b 2031 3134 3037 3235 3436 3035 3230   + 1140725460520
│ │ │ │ -00062cb0: 3830 7820 7820 202d 2031 3039 3234 3839  80x x  - 1092489
│ │ │ │ -00062cc0: 3738 3738 377c 0a7c 2020 2020 2020 2020  78787|.|        
│ │ │ │ -00062cd0: 2020 2020 2020 3120 3220 3320 2020 2020        1 2 3     
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│ │ │ │ -00062cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d00: 2020 2030 2033 2020 2020 2020 2020 2020     0 3          
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│ │ │ │ +00062c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062c40: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │ +00062c60: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00062c70: 2020 2020 2020 2020 2020 207c 0a7c 3631             |.|61
│ │ │ │ +00062c80: 3134 3734 3236 3335 3530 3078 2078 2078  14742635500x x x
│ │ │ │ +00062c90: 2020 2b20 3233 3236 3038 3031 3031 3638    + 232608010168
│ │ │ │ +00062ca0: 3830 7820 7820 202b 2031 3134 3037 3235  80x x  + 1140725
│ │ │ │ +00062cb0: 3436 3035 3230 3830 7820 7820 202d 2031  46052080x x  - 1
│ │ │ │ +00062cc0: 3039 3234 3839 3738 3738 377c 0a7c 2020  09248978787|.|  
│ │ │ │ +00062cd0: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
│ │ │ │ +00062ce0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ +00062d00: 2020 2020 2020 2020 2030 2033 2020 2020           0 3    
│ │ │ │ +00062d10: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d60: 2020 2020 207c 0a7c 2020 2020 3220 2020       |.|    2   
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│ │ │ │ -00062d80: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00062d90: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ -00062da0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -00062db0: 2020 2020 207c 0a7c 2078 2078 2020 2b20       |.| x x  + 
│ │ │ │ -00062dc0: 3336 3632 3731 3631 3032 3234 3830 3078  366271610224800x
│ │ │ │ -00062dd0: 2078 2078 2020 2d20 3731 3730 3933 3835   x x  - 71709385
│ │ │ │ -00062de0: 3638 3139 3230 7820 7820 202b 2033 3837  681920x x  + 387
│ │ │ │ -00062df0: 3436 3239 3231 3636 3534 3030 7820 7820  462921665400x x 
│ │ │ │ -00062e00: 202d 2032 317c 0a7c 3020 3220 3320 2020   - 21|.|0 2 3   
│ │ │ │ -00062e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062e20: 3120 3220 3320 2020 2020 2020 2020 2020  1 2 3           
│ │ │ │ -00062e30: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ -00062e40: 2020 2020 2020 2020 2020 2020 2030 2033               0 3
│ │ │ │ -00062e50: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ +00062d90: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
│ │ │ │ +00062da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062db0: 2020 2020 2033 2020 2020 207c 0a7c 2078       3     |.| x
│ │ │ │ +00062dc0: 2078 2020 2b20 3336 3632 3731 3631 3032   x  + 3662716102
│ │ │ │ +00062dd0: 3234 3830 3078 2078 2078 2020 2d20 3731  24800x x x  - 71
│ │ │ │ +00062de0: 3730 3933 3835 3638 3139 3230 7820 7820  709385681920x x 
│ │ │ │ +00062df0: 202b 2033 3837 3436 3239 3231 3636 3534   + 3874629216654
│ │ │ │ +00062e00: 3030 7820 7820 202d 2032 317c 0a7c 3020  00x x  - 21|.|0 
│ │ │ │ +00062e10: 3220 3320 2020 2020 2020 2020 2020 2020  2 3             
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│ │ │ │ +00062e30: 2020 2020 2020 2020 2020 2020 2032 2033               2 3
│ │ │ │ +00062e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062e50: 2020 2030 2033 2020 2020 207c 0a7c 2d2d     0 3     |.|--
│ │ │ │  00062e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062ea0: 2d2d 2d2d 2d7c 0a7c 2020 3320 2020 2020  -----|.|  3     
│ │ │ │ -00062eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ec0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00062ed0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ -00062ee0: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ -00062ef0: 2020 2020 207c 0a7c 2078 2020 2b20 3431       |.| x  + 41
│ │ │ │ -00062f00: 3133 3337 3035 3437 3839 3530 3078 2078  1337054789500x x
│ │ │ │ -00062f10: 2020 2d20 3137 3732 3935 3730 3330 3534    - 177295703054
│ │ │ │ -00062f20: 3430 3078 2078 2020 2d20 3732 3035 3333  400x x  - 720533
│ │ │ │ -00062f30: 3631 3832 3830 3030 7820 202d 2034 3730  61828000x  - 470
│ │ │ │ -00062f40: 3939 3136 377c 0a7c 3020 3320 2020 2020  99167|.|0 3     
│ │ │ │ -00062f50: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00062f60: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00062f70: 2020 2020 3220 3320 2020 2020 2020 2020      2 3         
│ │ │ │ -00062f80: 2020 2020 2020 2020 2033 2020 2020 2020           3      
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│ │ │ │ +00062ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +00062eb0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ +00062ed0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +00062ee0: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ +00062ef0: 2020 2020 2020 2020 2020 207c 0a7c 2078             |.| x
│ │ │ │ +00062f00: 2020 2b20 3431 3133 3337 3035 3437 3839    + 411337054789
│ │ │ │ +00062f10: 3530 3078 2078 2020 2d20 3137 3732 3935  500x x  - 177295
│ │ │ │ +00062f20: 3730 3330 3534 3430 3078 2078 2020 2d20  703054400x x  - 
│ │ │ │ +00062f30: 3732 3035 3333 3631 3832 3830 3030 7820  72053361828000x 
│ │ │ │ +00062f40: 202d 2034 3730 3939 3136 377c 0a7c 3020   - 47099167|.|0 
│ │ │ │ +00062f50: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ +00062f70: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ +00062f80: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00062f90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00062fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062ff0: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -00063000: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ -00063010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063020: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00063030: 2020 2020 207c 0a7c 202d 2039 3236 3435       |.| - 92645
│ │ │ │ -00063040: 3737 3133 3636 3430 7820 7820 202d 2037  77136640x x  - 7
│ │ │ │ -00063050: 3936 3731 3931 3635 3938 3430 3078 2020  9671916598400x  
│ │ │ │ -00063060: 2b20 3132 3339 3637 3830 3031 3239 3630  + 12396780012960
│ │ │ │ -00063070: 7820 7820 202b 2036 3432 3730 3339 3333  x x  + 642703933
│ │ │ │ -00063080: 3637 3034 307c 0a7c 2020 2020 2020 2020  67040|.|        
│ │ │ │ -00063090: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -000630a0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -000630b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000630c0: 2030 2034 2020 2020 2020 2020 2020 2020   0 4            
│ │ │ │ -000630d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00062fe0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00062ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063000: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00063010: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +00063020: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00063030: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00063040: 2039 3236 3435 3737 3133 3636 3430 7820   9264577136640x 
│ │ │ │ +00063050: 7820 202d 2037 3936 3731 3931 3635 3938  x  - 79671916598
│ │ │ │ +00063060: 3430 3078 2020 2b20 3132 3339 3637 3830  400x  + 12396780
│ │ │ │ +00063070: 3031 3239 3630 7820 7820 202b 2036 3432  012960x x  + 642
│ │ │ │ +00063080: 3730 3339 3333 3637 3034 307c 0a7c 2020  70393367040|.|  
│ │ │ │ +00063090: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000630a0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +000630b0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +000630c0: 2020 2020 2020 2030 2034 2020 2020 2020         0 4      
│ │ │ │ +000630d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063120: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00063130: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -00063140: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -00063150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063160: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00063170: 2020 2033 207c 0a7c 3433 3731 3030 3635     3 |.|43710065
│ │ │ │ -00063180: 3234 3030 7820 7820 202b 2036 3338 3432  2400x x  + 63842
│ │ │ │ -00063190: 3339 3235 3838 3830 3078 2078 2020 2b20  392588800x x  + 
│ │ │ │ -000631a0: 3236 3334 3435 3739 3530 3433 3230 3078  263445795043200x
│ │ │ │ -000631b0: 2020 2b20 3638 3930 3130 3139 3231 3038    + 689010192108
│ │ │ │ -000631c0: 3030 7820 787c 0a7c 2020 2020 2020 2020  00x x|.|        
│ │ │ │ -000631d0: 2020 2020 2031 2033 2020 2020 2020 2020       1 3        
│ │ │ │ -000631e0: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -000631f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063200: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00063210: 2020 2030 207c 0a7c 2020 2020 2020 2020     0 |.|        
│ │ │ │ +00063120: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00063130: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +00063140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063150: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +00063160: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +00063170: 2020 2020 2020 2020 2033 207c 0a7c 3433           3 |.|43
│ │ │ │ +00063180: 3731 3030 3635 3234 3030 7820 7820 202b  7100652400x x  +
│ │ │ │ +00063190: 2036 3338 3432 3339 3235 3838 3830 3078   63842392588800x
│ │ │ │ +000631a0: 2078 2020 2b20 3236 3334 3435 3739 3530   x  + 2634457950
│ │ │ │ +000631b0: 3433 3230 3078 2020 2b20 3638 3930 3130  43200x  + 689010
│ │ │ │ +000631c0: 3139 3231 3038 3030 7820 787c 0a7c 2020  19210800x x|.|  
│ │ │ │ +000631d0: 2020 2020 2020 2020 2020 2031 2033 2020             1 3  
│ │ │ │ +000631e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000631f0: 3220 3320 2020 2020 2020 2020 2020 2020  2 3             
│ │ │ │ +00063200: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00063210: 2020 2020 2020 2020 2030 207c 0a7c 2020           0 |.|  
│ │ │ │  00063220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063260: 2020 2020 207c 0a7c 2020 2020 2020 3320       |.|      3 
│ │ │ │ -00063270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063280: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00063290: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ -000632a0: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ -000632b0: 2020 2020 207c 0a7c 3230 3078 2078 2020       |.|200x x  
│ │ │ │ -000632c0: 2b20 3730 3939 3337 3332 3334 3031 3630  + 70993732340160
│ │ │ │ -000632d0: 7820 7820 202d 2031 3534 3931 3139 3237  x x  - 154911927
│ │ │ │ -000632e0: 3439 3230 3078 2020 2d20 3237 3533 3838  49200x  - 275388
│ │ │ │ -000632f0: 3937 3039 3638 3732 3078 2078 2020 2b20  970968720x x  + 
│ │ │ │ -00063300: 3632 3632 357c 0a7c 2020 2020 3120 3320  62625|.|    1 3 
│ │ │ │ -00063310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063320: 2032 2033 2020 2020 2020 2020 2020 2020   2 3            
│ │ │ │ -00063330: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ -00063340: 2020 2020 2020 2020 2020 3020 3420 2020            0 4   
│ │ │ │ -00063350: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063260: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00063270: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +00063280: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +00063290: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +000632a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000632b0: 3320 2020 2020 2020 2020 207c 0a7c 3230  3          |.|20
│ │ │ │ +000632c0: 3078 2078 2020 2b20 3730 3939 3337 3332  0x x  + 70993732
│ │ │ │ +000632d0: 3334 3031 3630 7820 7820 202d 2031 3534  340160x x  - 154
│ │ │ │ +000632e0: 3931 3139 3237 3439 3230 3078 2020 2d20  91192749200x  - 
│ │ │ │ +000632f0: 3237 3533 3838 3937 3039 3638 3732 3078  275388970968720x
│ │ │ │ +00063300: 2078 2020 2b20 3632 3632 357c 0a7c 2020   x  + 62625|.|  
│ │ │ │ +00063310: 2020 3120 3320 2020 2020 2020 2020 2020    1 3           
│ │ │ │ +00063320: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ +00063330: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +00063340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063350: 3020 3420 2020 2020 2020 207c 0a7c 2020  0 4        |.|  
│ │ │ │  00063360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000633a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000633b0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -000633c0: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ -000633d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000633e0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -000633f0: 2020 2020 337c 0a7c 3933 3335 3837 3439      3|.|93358749
│ │ │ │ -00063400: 3638 3730 3078 2078 2020 2b20 3135 3330  68700x x  + 1530
│ │ │ │ -00063410: 3332 3230 3433 3034 3030 7820 7820 202b  3220430400x x  +
│ │ │ │ -00063420: 2035 3932 3534 3639 3333 3434 3030 3078   59254693344000x
│ │ │ │ -00063430: 2020 2b20 3138 3537 3330 3437 3332 3134    + 185730473214
│ │ │ │ -00063440: 3732 3078 207c 0a7c 2020 2020 2020 2020  720x |.|        
│ │ │ │ -00063450: 2020 2020 2020 3120 3320 2020 2020 2020        1 3       
│ │ │ │ -00063460: 2020 2020 2020 2020 2020 2032 2033 2020             2 3  
│ │ │ │ -00063470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063480: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00063490: 2020 2020 307c 0a7c 2d2d 2d2d 2d2d 2d2d      0|.|--------
│ │ │ │ +000633a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000633b0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +000633c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000633d0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +000633e0: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +000633f0: 2020 2020 2020 2020 2020 337c 0a7c 3933            3|.|93
│ │ │ │ +00063400: 3335 3837 3439 3638 3730 3078 2078 2020  35874968700x x  
│ │ │ │ +00063410: 2b20 3135 3330 3332 3230 3433 3034 3030  + 15303220430400
│ │ │ │ +00063420: 7820 7820 202b 2035 3932 3534 3639 3333  x x  + 592546933
│ │ │ │ +00063430: 3434 3030 3078 2020 2b20 3138 3537 3330  44000x  + 185730
│ │ │ │ +00063440: 3437 3332 3134 3732 3078 207c 0a7c 2020  473214720x |.|  
│ │ │ │ +00063450: 2020 2020 2020 2020 2020 2020 3120 3320              1 3 
│ │ │ │ +00063460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063470: 2032 2033 2020 2020 2020 2020 2020 2020   2 3            
│ │ │ │ +00063480: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00063490: 2020 2020 2020 2020 2020 307c 0a7c 2d2d            0|.|--
│ │ │ │  000634a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000634b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000634c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000634d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000634e0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -000634f0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00063500: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000634e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +000634f0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00063500: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00063510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063520: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00063530: 2020 2020 207c 0a7c 3535 3538 3630 3078       |.|5558600x
│ │ │ │ -00063540: 2078 2020 2b20 3638 3435 3234 3538 3739   x  + 6845245879
│ │ │ │ -00063550: 3530 3030 3078 2078 2078 2020 2d20 3334  50000x x x  - 34
│ │ │ │ -00063560: 3333 3837 3937 3930 3036 3130 3078 2078  3387979006100x x
│ │ │ │ -00063570: 2078 2020 2b20 3131 3131 3239 3435 3533   x  + 1111294553
│ │ │ │ -00063580: 3431 3830 307c 0a7c 2020 2020 2020 2020  41800|.|        
│ │ │ │ -00063590: 3020 3420 2020 2020 2020 2020 2020 2020  0 4             
│ │ │ │ -000635a0: 2020 2020 2020 3020 3120 3420 2020 2020        0 1 4     
│ │ │ │ -000635b0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000635c0: 3120 3420 2020 2020 2020 2020 2020 2020  1 4             
│ │ │ │ -000635d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063520: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00063530: 2020 2020 2020 2020 2020 207c 0a7c 3535             |.|55
│ │ │ │ +00063540: 3538 3630 3078 2078 2020 2b20 3638 3435  58600x x  + 6845
│ │ │ │ +00063550: 3234 3538 3739 3530 3030 3078 2078 2078  24587950000x x x
│ │ │ │ +00063560: 2020 2d20 3334 3333 3837 3937 3930 3036    - 343387979006
│ │ │ │ +00063570: 3130 3078 2078 2078 2020 2b20 3131 3131  100x x x  + 1111
│ │ │ │ +00063580: 3239 3435 3533 3431 3830 307c 0a7c 2020  29455341800|.|  
│ │ │ │ +00063590: 2020 2020 2020 3020 3420 2020 2020 2020        0 4       
│ │ │ │ +000635a0: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ +000635b0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +000635c0: 2020 2020 3020 3120 3420 2020 2020 2020      0 1 4       
│ │ │ │ +000635d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000635e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000635f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063620: 2020 2020 207c 0a7c 2032 2020 2020 2020       |.| 2      
│ │ │ │ +00063620: 2020 2020 2020 2020 2020 207c 0a7c 2032             |.| 2
│ │ │ │  00063630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063640: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00063650: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -00063660: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00063670: 2020 2020 207c 0a7c 7820 7820 7820 202d       |.|x x x  -
│ │ │ │ -00063680: 2036 3030 3939 3934 3036 3933 3338 3078   60099940693380x
│ │ │ │ -00063690: 2078 2078 2020 2b20 3132 3132 3938 3131   x x  + 12129811
│ │ │ │ -000636a0: 3636 3930 3830 7820 7820 202b 2032 3330  669080x x  + 230
│ │ │ │ -000636b0: 3039 3131 3431 3133 3932 7820 7820 7820  0911411392x x x 
│ │ │ │ -000636c0: 202b 2032 337c 0a7c 2030 2031 2034 2020   + 23|.| 0 1 4  
│ │ │ │ -000636d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636e0: 3020 3120 3420 2020 2020 2020 2020 2020  0 1 4           
│ │ │ │ -000636f0: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
│ │ │ │ -00063700: 2020 2020 2020 2020 2020 2030 2032 2034             0 2 4
│ │ │ │ -00063710: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063640: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00063650: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +00063660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063670: 2032 2020 2020 2020 2020 207c 0a7c 7820   2         |.|x 
│ │ │ │ +00063680: 7820 7820 202d 2036 3030 3939 3934 3036  x x  - 600999406
│ │ │ │ +00063690: 3933 3338 3078 2078 2078 2020 2b20 3132  93380x x x  + 12
│ │ │ │ +000636a0: 3132 3938 3131 3636 3930 3830 7820 7820  129811669080x x 
│ │ │ │ +000636b0: 202b 2032 3330 3039 3131 3431 3133 3932   + 2300911411392
│ │ │ │ +000636c0: 7820 7820 7820 202b 2032 337c 0a7c 2030  x x x  + 23|.| 0
│ │ │ │ +000636d0: 2031 2034 2020 2020 2020 2020 2020 2020   1 4            
│ │ │ │ +000636e0: 2020 2020 2020 3020 3120 3420 2020 2020        0 1 4     
│ │ │ │ +000636f0: 2020 2020 2020 2020 2020 2020 2031 2034               1 4
│ │ │ │ +00063700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063710: 2030 2032 2034 2020 2020 207c 0a7c 2020   0 2 4     |.|  
│ │ │ │  00063720: 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00063770: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00063780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063790: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000637a0: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -000637b0: 2020 2020 207c 0a7c 2020 2d20 3337 3230       |.|  - 3720
│ │ │ │ -000637c0: 3437 3431 3431 3933 3030 3078 2078 2078  47414193000x x x
│ │ │ │ -000637d0: 2020 2b20 3135 3039 3033 3838 3631 3032    + 150903886102
│ │ │ │ -000637e0: 3230 3078 2078 2078 2020 2d20 3837 3031  200x x x  - 8701
│ │ │ │ -000637f0: 3934 3331 3635 3136 3030 7820 7820 202b  9431651600x x  +
│ │ │ │ -00063800: 2032 3538 397c 0a7c 3420 2020 2020 2020   2589|.|4       
│ │ │ │ -00063810: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -00063820: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00063830: 2020 2020 3020 3120 3420 2020 2020 2020      0 1 4       
│ │ │ │ -00063840: 2020 2020 2020 2020 2020 2031 2034 2020             1 4  
│ │ │ │ -00063850: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063760: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00063770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063780: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00063790: 2020 2020 2020 2020 2020 2020 3220 2020              2   
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│ │ │ │ +000637b0: 2033 2020 2020 2020 2020 207c 0a7c 2020   3         |.|  
│ │ │ │ +000637c0: 2d20 3337 3230 3437 3431 3431 3933 3030  - 37204741419300
│ │ │ │ +000637d0: 3078 2078 2078 2020 2b20 3135 3039 3033  0x x x  + 150903
│ │ │ │ +000637e0: 3838 3631 3032 3230 3078 2078 2078 2020  886102200x x x  
│ │ │ │ +000637f0: 2d20 3837 3031 3934 3331 3635 3136 3030  - 87019431651600
│ │ │ │ +00063800: 7820 7820 202b 2032 3538 397c 0a7c 3420  x x  + 2589|.|4 
│ │ │ │ +00063810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063820: 2020 3020 3120 3420 2020 2020 2020 2020    0 1 4         
│ │ │ │ +00063830: 2020 2020 2020 2020 2020 3020 3120 3420            0 1 4 
│ │ │ │ +00063840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063850: 2031 2034 2020 2020 2020 207c 0a7c 2020   1 4       |.|  
│ │ │ │  00063860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000638a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000638b0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -000638c0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -000638d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000638e0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -000638f0: 2020 2020 207c 0a7c 3635 3839 3432 3938       |.|65894298
│ │ │ │ -00063900: 3078 2078 2078 2020 2b20 3533 3235 3534  0x x x  + 532554
│ │ │ │ -00063910: 3537 3134 3833 3230 7820 7820 7820 202b  57148320x x x  +
│ │ │ │ -00063920: 2032 3931 3633 3834 3334 3635 3132 3078   29163843465120x
│ │ │ │ -00063930: 2078 2020 2d20 3537 3639 3132 3035 3337   x  - 5769120537
│ │ │ │ -00063940: 3835 3832 347c 0a7c 2020 2020 2020 2020  85824|.|        
│ │ │ │ -00063950: 2020 3020 3120 3420 2020 2020 2020 2020    0 1 4         
│ │ │ │ -00063960: 2020 2020 2020 2020 2030 2031 2034 2020           0 1 4  
│ │ │ │ -00063970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063980: 3120 3420 2020 2020 2020 2020 2020 2020  1 4             
│ │ │ │ -00063990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000638a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000638b0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000638c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000638d0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000638e0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +000638f0: 2020 2020 2020 2020 2020 207c 0a7c 3635             |.|65
│ │ │ │ +00063900: 3839 3432 3938 3078 2078 2078 2020 2b20  8942980x x x  + 
│ │ │ │ +00063910: 3533 3235 3534 3537 3134 3833 3230 7820  53255457148320x 
│ │ │ │ +00063920: 7820 7820 202b 2032 3931 3633 3834 3334  x x  + 291638434
│ │ │ │ +00063930: 3635 3132 3078 2078 2020 2d20 3537 3639  65120x x  - 5769
│ │ │ │ +00063940: 3132 3035 3337 3835 3832 347c 0a7c 2020  12053785824|.|  
│ │ │ │ +00063950: 2020 2020 2020 2020 3020 3120 3420 2020          0 1 4   
│ │ │ │ +00063960: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00063970: 2031 2034 2020 2020 2020 2020 2020 2020   1 4            
│ │ │ │ +00063980: 2020 2020 2020 3120 3420 2020 2020 2020        1 4       
│ │ │ │ +00063990: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000639a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000639b0: 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000639f0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00063a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a10: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00063a20: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -00063a30: 2020 2020 207c 0a7c 7820 202d 2034 3935       |.|x  - 495
│ │ │ │ -00063a40: 3130 3439 3132 3039 3932 3830 7820 7820  104912099280x x 
│ │ │ │ -00063a50: 7820 202b 2033 3037 3435 3831 3932 3133  x  + 30745819213
│ │ │ │ -00063a60: 3634 3130 7820 7820 7820 202d 2034 3030  6410x x x  - 400
│ │ │ │ -00063a70: 3839 3436 3038 3432 3930 3078 2078 2020  89460842900x x  
│ │ │ │ -00063a80: 2b20 3132 377c 0a7c 2034 2020 2020 2020  + 127|.| 4      
│ │ │ │ -00063a90: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ -00063aa0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -00063ab0: 2020 2020 2030 2031 2034 2020 2020 2020       0 1 4      
│ │ │ │ -00063ac0: 2020 2020 2020 2020 2020 2020 3120 3420              1 4 
│ │ │ │ -00063ad0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000639e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000639f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063a00: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00063a10: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00063a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063a30: 2020 3320 2020 2020 2020 207c 0a7c 7820    3        |.|x 
│ │ │ │ +00063a40: 202d 2034 3935 3130 3439 3132 3039 3932   - 4951049120992
│ │ │ │ +00063a50: 3830 7820 7820 7820 202b 2033 3037 3435  80x x x  + 30745
│ │ │ │ +00063a60: 3831 3932 3133 3634 3130 7820 7820 7820  8192136410x x x 
│ │ │ │ +00063a70: 202d 2034 3030 3839 3436 3038 3432 3930   - 4008946084290
│ │ │ │ +00063a80: 3078 2078 2020 2b20 3132 377c 0a7c 2034  0x x  + 127|.| 4
│ │ │ │ +00063a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063aa0: 2020 2030 2031 2034 2020 2020 2020 2020     0 1 4        
│ │ │ │ +00063ab0: 2020 2020 2020 2020 2020 2030 2031 2034             0 1 4
│ │ │ │ +00063ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063ad0: 2020 3120 3420 2020 2020 207c 0a7c 2d2d    1 4      |.|--
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│ │ │ │ -00063b30: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00063b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00063b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00063b80: 3238 3835 3837 3632 3431 3532 3078 2078  2885876241520x x
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│ │ │ │ -00063bc0: 2078 2078 207c 0a7c 2031 2034 2020 2020   x x |.| 1 4    
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│ │ │ │ -00063be0: 3220 3420 2020 2020 2020 2020 2020 2020  2 4             
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│ │ │ │ +00063bb0: 2078 2020 2d20 3236 3532 3432 3738 3834   x  - 2652427884
│ │ │ │ +00063bc0: 3338 3034 3078 2078 2078 207c 0a7c 2031  38040x x x |.| 1
│ │ │ │ +00063bd0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
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│ │ │ │  00063c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00063ca0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
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│ │ │ │ -00063cd0: 3038 3233 3433 3838 3237 3438 7820 7820  082343882748x x 
│ │ │ │ -00063ce0: 7820 202b 2037 3331 3131 3038 3836 3633  x  + 73111088663
│ │ │ │ -00063cf0: 3731 3278 2078 2078 2020 2d20 3638 3132  712x x x  - 6812
│ │ │ │ -00063d00: 3034 3836 307c 0a7c 2020 2020 2020 2020  04860|.|        
│ │ │ │ -00063d10: 2020 2020 2030 2031 2032 2034 2020 2020       0 1 2 4    
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│ │ │ │ +00063cb0: 2020 2020 2020 2020 2020 207c 0a7c 3332             |.|32
│ │ │ │ +00063cc0: 3534 3035 3936 3731 3336 7820 7820 7820  5405967136x x x 
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│ │ │ │ +00063ce0: 3438 7820 7820 7820 202b 2037 3331 3131  48x x x  + 73111
│ │ │ │ +00063cf0: 3038 3836 3633 3731 3278 2078 2078 2020  088663712x x x  
│ │ │ │ +00063d00: 2d20 3638 3132 3034 3836 307c 0a7c 2020  - 681204860|.|  
│ │ │ │ +00063d10: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
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│ │ │ │  00063d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00063d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00063e00: 3630 3078 2078 2078 2020 2d20 3131 3534  600x x x  - 1154
│ │ │ │ -00063e10: 3336 3137 3430 3436 3834 3078 2078 2078  36174046840x x x
│ │ │ │ -00063e20: 2078 2020 2b20 3139 3930 3339 3433 3633   x  + 1990394363
│ │ │ │ -00063e30: 3135 3836 3078 2078 2078 2020 2b20 3132  15860x x x  + 12
│ │ │ │ -00063e40: 3832 3235 387c 0a7c 2020 2020 2020 2020  82258|.|        
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│ │ │ │ -00063e70: 3220 3420 2020 2020 2020 2020 2020 2020  2 4             
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│ │ │ │ +00063e20: 3078 2078 2078 2078 2020 2b20 3139 3930  0x x x x  + 1990
│ │ │ │ +00063e30: 3339 3433 3633 3135 3836 3078 2078 2078  39436315860x x x
│ │ │ │ +00063e40: 2020 2b20 3132 3832 3235 387c 0a7c 2020    + 1282258|.|  
│ │ │ │ +00063e50: 2020 2020 2020 2020 2020 3020 3220 3420            0 2 4 
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│ │ │ │  00063ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00063ee0: 2020 2020 2020 2020 2020 207c 0a7c 2032             |.| 2
│ │ │ │  00063ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00063f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063f30: 2020 2020 327c 0a7c 7820 7820 7820 202b      2|.|x x x  +
│ │ │ │ -00063f40: 2034 3034 3139 3939 3639 3137 3034 3234   404199969170424
│ │ │ │ -00063f50: 7820 7820 7820 7820 202d 2036 3734 3439  x x x x  - 67449
│ │ │ │ -00063f60: 3336 3837 3935 3736 3078 2078 2078 2020  368795760x x x  
│ │ │ │ -00063f70: 2d20 3236 3636 3431 3138 3436 3030 3136  - 26664118460016
│ │ │ │ -00063f80: 3078 2078 207c 0a7c 2030 2032 2034 2020  0x x |.| 0 2 4  
│ │ │ │ -00063f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063fa0: 2030 2031 2032 2034 2020 2020 2020 2020   0 1 2 4        
│ │ │ │ -00063fb0: 2020 2020 2020 2020 2020 3120 3220 3420            1 2 4 
│ │ │ │ -00063fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063fd0: 2020 3020 327c 0a7c 2020 2020 2020 2020    0 2|.|        
│ │ │ │ +00063f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063f20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00063f30: 2020 2020 2020 2020 2020 327c 0a7c 7820            2|.|x 
│ │ │ │ +00063f40: 7820 7820 202b 2034 3034 3139 3939 3639  x x  + 404199969
│ │ │ │ +00063f50: 3137 3034 3234 7820 7820 7820 7820 202d  170424x x x x  -
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│ │ │ │ +00063f80: 3436 3030 3136 3078 2078 207c 0a7c 2030  4600160x x |.| 0
│ │ │ │ +00063f90: 2032 2034 2020 2020 2020 2020 2020 2020   2 4            
│ │ │ │ +00063fa0: 2020 2020 2020 2030 2031 2032 2034 2020         0 1 2 4  
│ │ │ │ +00063fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00063fd0: 2020 2020 2020 2020 3020 327c 0a7c 2020          0 2|.|  
│ │ │ │  00063fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00064090: 3736 3431 3034 3738 3833 3532 7820 7820  764104788352x x 
│ │ │ │ -000640a0: 7820 7820 202b 2035 3431 3534 3236 3233  x x  + 541542623
│ │ │ │ -000640b0: 3634 3732 3078 2078 2078 2020 2d20 3532  64720x x x  - 52
│ │ │ │ -000640c0: 3634 3831 347c 0a7c 2020 2020 2020 2020  64814|.|        
│ │ │ │ -000640d0: 2020 2020 2030 2032 2034 2020 2020 2020       0 2 4      
│ │ │ │ -000640e0: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ -000640f0: 2032 2034 2020 2020 2020 2020 2020 2020   2 4            
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│ │ │ │ +00064080: 3133 3733 3535 3332 3332 7820 7820 7820  1373553232x x x 
│ │ │ │ +00064090: 202d 2033 3035 3736 3431 3034 3738 3833   - 3057641047883
│ │ │ │ +000640a0: 3532 7820 7820 7820 7820 202b 2035 3431  52x x x x  + 541
│ │ │ │ +000640b0: 3534 3236 3233 3634 3732 3078 2078 2078  54262364720x x x
│ │ │ │ +000640c0: 2020 2d20 3532 3634 3831 347c 0a7c 2020    - 5264814|.|  
│ │ │ │ +000640d0: 2020 2020 2020 2020 2020 2030 2032 2034             0 2 4
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│ │ │ │ +000640f0: 2020 2030 2031 2032 2034 2020 2020 2020     0 1 2 4      
│ │ │ │ +00064100: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
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│ │ │ │ -000641f0: 3233 3839 3630 3030 7820 7820 202b 2039  23896000x x  + 9
│ │ │ │ -00064200: 3938 3533 367c 0a7c 2020 2020 2020 2020  98536|.|        
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│ │ │ │ +000641d0: 2078 2078 2020 2b20 3138 3238 3030 3438   x x  + 18280048
│ │ │ │ +000641e0: 3436 3031 3834 3078 2078 2078 2020 2d20  4601840x x x  - 
│ │ │ │ +000641f0: 3432 3038 3339 3233 3839 3630 3030 7820  42083923896000x 
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│ │ │ │ -000642e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000642f0: 2020 2020 207c 0a7c 3236 3939 3278 2078       |.|26992x x
│ │ │ │ -00064300: 2078 2020 2b20 3431 3134 3837 3235 3833   x  + 4114872583
│ │ │ │ -00064310: 3737 3630 7820 7820 202d 2039 3831 3539  7760x x  - 98159
│ │ │ │ -00064320: 3539 3332 3435 3834 3078 2078 2078 2020  593245840x x x  
│ │ │ │ -00064330: 2b20 3135 3635 3133 3237 3731 3937 3438  + 15651327719748
│ │ │ │ -00064340: 3078 2078 207c 0a7c 2020 2020 2020 3120  0x x |.|      1 
│ │ │ │ -00064350: 3220 3420 2020 2020 2020 2020 2020 2020  2 4             
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│ │ │ │ -00064370: 2020 2020 2020 2020 2020 3020 3320 3420            0 3 4 
│ │ │ │ -00064380: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000642b0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
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│ │ │ │ +000642d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000642e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │ +00064300: 3939 3278 2078 2078 2020 2b20 3431 3134  992x x x  + 4114
│ │ │ │ +00064310: 3837 3235 3833 3737 3630 7820 7820 202d  8725837760x x  -
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│ │ │ │ +00064330: 2078 2078 2020 2b20 3135 3635 3133 3237   x x  + 15651327
│ │ │ │ +00064340: 3731 3937 3438 3078 2078 207c 0a7c 2020  7197480x x |.|  
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│ │ │ │ +00064370: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000643a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000643b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000643d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000643e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000643f0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
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│ │ │ │ -00064410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064420: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ -00064450: 3837 3935 3733 3630 7820 7820 7820 202b  87957360x x x  +
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│ │ │ │ -000646b0: 2020 2020 207c 0a7c 3431 3238 3078 2078       |.|41280x x
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│ │ │ │ -000646e0: 3234 3534 3732 3030 7820 7820 202d 2034  24547200x x  - 4
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│ │ │ │ +000646a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000646e0: 2d20 3239 3832 3234 3534 3732 3030 7820  - 298224547200x 
│ │ │ │ +000646f0: 7820 202d 2034 3934 3639 3639 3835 3136  x  - 49469698516
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│ │ │ │ +00064990: 2034 2020 2020 2020 2020 2020 2020 2020   4              
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│ │ │ │ -00064aa0: 3332 3030 3630 3532 3030 7820 7820 7820  3200605200x x x 
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│ │ │ │ -00064ac0: 3238 7820 787c 0a7c 2030 2033 2034 2020  28x x|.| 0 3 4  
│ │ │ │ -00064ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064ae0: 2030 2031 2033 2034 2020 2020 2020 2020   0 1 3 4        
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│ │ │ │ -00064b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00064a80: 7820 7820 202b 2032 3537 3736 3236 3135  x x  + 257762615
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│ │ │ │ +00064ab0: 7820 7820 7820 202d 2038 3036 3231 3338  x x x  - 8062138
│ │ │ │ +00064ac0: 3539 3335 3330 3238 7820 787c 0a7c 2030  59353028x x|.| 0
│ │ │ │ +00064ad0: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
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│ │ │ │ +00064b00: 2031 2033 2034 2020 2020 2020 2020 2020   1 3 4          
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│ │ │ │  00064b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00064b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00064b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00064bc0: 3830 7820 7820 7820 7820 202d 2032 3632  80x x x x  - 262
│ │ │ │ -00064bd0: 3631 3037 3732 3239 3130 3078 2078 2078  61077229100x x x
│ │ │ │ -00064be0: 2020 2d20 3934 3335 3632 3732 3831 3830    - 943562728180
│ │ │ │ -00064bf0: 3830 7820 7820 7820 7820 202b 2031 3139  80x x x x  + 119
│ │ │ │ -00064c00: 3939 3838 377c 0a7c 2020 2020 2020 2020  99887|.|        
│ │ │ │ -00064c10: 2020 2030 2031 2033 2034 2020 2020 2020     0 1 3 4      
│ │ │ │ -00064c20: 2020 2020 2020 2020 2020 2020 3120 3320              1 3 
│ │ │ │ -00064c30: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00064c40: 2020 2030 2032 2033 2034 2020 2020 2020     0 2 3 4      
│ │ │ │ -00064c50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +00064bd0: 202d 2032 3632 3631 3037 3732 3239 3130   - 2626107722910
│ │ │ │ +00064be0: 3078 2078 2078 2020 2d20 3934 3335 3632  0x x x  - 943562
│ │ │ │ +00064bf0: 3732 3831 3830 3830 7820 7820 7820 7820  72818080x x x x 
│ │ │ │ +00064c00: 202b 2031 3139 3939 3838 377c 0a7c 2020   + 11999887|.|  
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│ │ │ │ +00064c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00064c40: 2020 2020 2020 2020 2030 2032 2033 2034           0 2 3 4
│ │ │ │ +00064c50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00064c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00064d00: 3339 3333 3631 3630 7820 7820 7820 7820  39336160x x x x 
│ │ │ │ -00064d10: 202b 2039 3230 3132 3935 3134 3936 3337   + 9201295149637
│ │ │ │ -00064d20: 3578 2078 2078 2020 2b20 3133 3436 3737  5x x x  + 134677
│ │ │ │ -00064d30: 3630 3134 3431 3238 3078 2078 2078 2078  601441280x x x x
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│ │ │ │ -00064d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00064d00: 3134 3334 3734 3339 3333 3631 3630 7820  14347439336160x 
│ │ │ │ +00064d10: 7820 7820 7820 202b 2039 3230 3132 3935  x x x  + 9201295
│ │ │ │ +00064d20: 3134 3936 3337 3578 2078 2078 2020 2b20  1496375x x x  + 
│ │ │ │ +00064d30: 3133 3436 3737 3630 3134 3431 3238 3078  134677601441280x
│ │ │ │ +00064d40: 2078 2078 2078 2020 2d20 317c 0a7c 2020   x x x  - 1|.|  
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│ │ │ │ +00065750: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00065c20: 3334 3237 3531 3838 3434 3830 7820 7820  342751884480x x 
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│ │ │ │ -00066130: 2078 2020 2b20 3735 3739 3036 3836 3738   x  + 7579068678
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│ │ │ │ +000660f0: 2020 2020 2020 2020 2020 207c 0a7c 3831             |.|81
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│ │ │ │ +00066120: 7820 7820 202b 2032 3633 3937 3935 3434  x x  + 263979544
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│ │ │ │ -00066380: 2d20 3839 3134 3730 3434 3933 3338 3030  - 89147044933800
│ │ │ │ -00066390: 7820 7820 7820 202d 2035 3333 3831 3733  x x x  - 5338173
│ │ │ │ -000663a0: 3132 3030 3030 7820 7820 202d 2031 3433  120000x x  - 143
│ │ │ │ -000663b0: 3631 3338 3136 3231 3636 3936 3078 2078  6138162166960x x
│ │ │ │ -000663c0: 2078 2020 2b7c 0a7c 2020 3020 3220 3420   x  +|.|  0 2 4 
│ │ │ │ -000663d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00066380: 2078 2078 2020 2d20 3839 3134 3730 3434   x x  - 89147044
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│ │ │ │ +000663a0: 3333 3831 3733 3132 3030 3030 7820 7820  338173120000x x 
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│ │ │ │ +000663c0: 3936 3078 2078 2078 2020 2b7c 0a7c 2020  960x x x  +|.|  
│ │ │ │ +000663d0: 3020 3220 3420 2020 2020 2020 2020 2020  0 2 4           
│ │ │ │ +000663e0: 2020 2020 2020 2031 2032 2034 2020 2020         1 2 4    
│ │ │ │ +000663f0: 2020 2020 2020 2020 2020 2020 2032 2034               2 4
│ │ │ │ +00066400: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00066480: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ -00066490: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -000664a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000664b0: 2020 2020 207c 0a7c 3937 3938 3830 7820       |.|979880x 
│ │ │ │ -000664c0: 7820 7820 202d 2036 3535 3839 3436 3731  x x  - 655894671
│ │ │ │ -000664d0: 3139 3230 3078 2078 2020 2b20 3934 3132  19200x x  + 9412
│ │ │ │ -000664e0: 3437 3131 3930 3830 3830 7820 7820 7820  4711908080x x x 
│ │ │ │ -000664f0: 202d 2036 3137 3331 3231 3331 3331 3730   - 6173121313170
│ │ │ │ -00066500: 3078 2078 207c 0a7c 2020 2020 2020 2031  0x x |.|       1
│ │ │ │ -00066510: 2032 2034 2020 2020 2020 2020 2020 2020   2 4            
│ │ │ │ -00066520: 2020 2020 2020 3220 3420 2020 2020 2020        2 4       
│ │ │ │ -00066530: 2020 2020 2020 2020 2020 2030 2033 2034             0 3 4
│ │ │ │ -00066540: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00066480: 2020 2020 2020 2020 2020 2020 3220 3220              2 2 
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│ │ │ │ +000664b0: 2020 2020 2020 2020 2020 207c 0a7c 3937             |.|97
│ │ │ │ +000664c0: 3938 3830 7820 7820 7820 202d 2036 3535  9880x x x  - 655
│ │ │ │ +000664d0: 3839 3436 3731 3139 3230 3078 2078 2020  89467119200x x  
│ │ │ │ +000664e0: 2b20 3934 3132 3437 3131 3930 3830 3830  + 94124711908080
│ │ │ │ +000664f0: 7820 7820 7820 202d 2036 3137 3331 3231  x x x  - 6173121
│ │ │ │ +00066500: 3331 3331 3730 3078 2078 207c 0a7c 2020  3131700x x |.|  
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│ │ │ │ +00066520: 2020 2020 2020 2020 2020 2020 3220 3420              2 4 
│ │ │ │ +00066530: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000665c0: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
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│ │ │ │ -00066600: 3831 3831 3630 7820 7820 7820 202b 2038  818160x x x  + 8
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│ │ │ │ -00066630: 3078 2078 2078 2020 2b20 3533 3136 3437  0x x x  + 531647
│ │ │ │ -00066640: 3736 3239 307c 0a7c 2020 2020 2020 2020  76290|.|        
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│ │ │ │ -00066670: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000665e0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000665f0: 2020 2020 2020 2020 2020 207c 0a7c 3836             |.|86
│ │ │ │ +00066600: 3434 3738 3635 3831 3831 3630 7820 7820  447865818160x x 
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│ │ │ │ +00066630: 3131 3338 3738 3078 2078 2078 2020 2b20  1138780x x x  + 
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│ │ │ │  000666a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000666b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000666c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000666d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000666e0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2032 2020  -----|.|     2  
│ │ │ │ -000666f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00066710: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00066720: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00066730: 2032 2020 207c 0a7c 7820 7820 7820 202d   2   |.|x x x  -
│ │ │ │ -00066740: 2031 3533 3439 3439 3937 3035 3631 3030   153494997056100
│ │ │ │ -00066750: 7820 7820 7820 202b 2031 3737 3339 3230  x x x  + 1773920
│ │ │ │ -00066760: 3830 3837 3434 3078 2078 2078 2020 2b20  8087440x x x  + 
│ │ │ │ -00066770: 3530 3632 3534 3838 3337 3738 3030 7820  50625488377800x 
│ │ │ │ -00066780: 7820 202d 207c 0a7c 2030 2033 2034 2020  x  - |.| 0 3 4  
│ │ │ │ -00066790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000667a0: 2031 2033 2034 2020 2020 2020 2020 2020   1 3 4          
│ │ │ │ -000667b0: 2020 2020 2020 2020 3220 3320 3420 2020          2 3 4   
│ │ │ │ -000667c0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -000667d0: 2034 2020 207c 0a7c 2020 2020 2020 2020   4   |.|        
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│ │ │ │ +00066740: 7820 7820 202d 2031 3533 3439 3439 3937  x x  - 153494997
│ │ │ │ +00066750: 3035 3631 3030 7820 7820 7820 202b 2031  056100x x x  + 1
│ │ │ │ +00066760: 3737 3339 3230 3830 3837 3434 3078 2078  7739208087440x x
│ │ │ │ +00066770: 2078 2020 2b20 3530 3632 3534 3838 3337   x  + 5062548837
│ │ │ │ +00066780: 3738 3030 7820 7820 202d 207c 0a7c 2030  7800x x  - |.| 0
│ │ │ │ +00066790: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ +000667a0: 2020 2020 2020 2031 2033 2034 2020 2020         1 3 4    
│ │ │ │ +000667b0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000667c0: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
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│ │ │ │ -00066820: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00066830: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00066840: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ -00066850: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ -00066860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066870: 2020 3320 207c 0a7c 3638 3531 3638 3078    3  |.|6851680x
│ │ │ │ -00066880: 2078 2078 2020 2b20 3930 3331 3235 3239   x x  + 90312529
│ │ │ │ -00066890: 3931 3034 3078 2078 2020 2d20 3633 3935  91040x x  - 6395
│ │ │ │ -000668a0: 3836 3337 3833 3638 3030 7820 7820 202b  8637836800x x  +
│ │ │ │ -000668b0: 2031 3534 3333 3336 3730 3635 3230 3078   15433367065200x
│ │ │ │ -000668c0: 2078 2020 2d7c 0a7c 2020 2020 2020 2020   x  -|.|        
│ │ │ │ -000668d0: 3220 3320 3420 2020 2020 2020 2020 2020  2 3 4           
│ │ │ │ -000668e0: 2020 2020 2020 3320 3420 2020 2020 2020        3 4       
│ │ │ │ -000668f0: 2020 2020 2020 2020 2020 2030 2034 2020             0 4  
│ │ │ │ -00066900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066910: 3120 3420 207c 0a7c 2020 2020 2020 2020  1 4  |.|        
│ │ │ │ +00066820: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00066830: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00066840: 2020 2020 2020 2020 2020 2020 3220 3220              2 2 
│ │ │ │ +00066850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00066860: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +00066870: 2020 2020 2020 2020 3320 207c 0a7c 3638          3  |.|68
│ │ │ │ +00066880: 3531 3638 3078 2078 2078 2020 2b20 3930  51680x x x  + 90
│ │ │ │ +00066890: 3331 3235 3239 3931 3034 3078 2078 2020  31252991040x x  
│ │ │ │ +000668a0: 2d20 3633 3935 3836 3337 3833 3638 3030  - 63958637836800
│ │ │ │ +000668b0: 7820 7820 202b 2031 3534 3333 3336 3730  x x  + 154333670
│ │ │ │ +000668c0: 3635 3230 3078 2078 2020 2d7c 0a7c 2020  65200x x  -|.|  
│ │ │ │ +000668d0: 2020 2020 2020 3220 3320 3420 2020 2020        2 3 4     
│ │ │ │ +000668e0: 2020 2020 2020 2020 2020 2020 3320 3420              3 4 
│ │ │ │ +000668f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00066900: 2030 2034 2020 2020 2020 2020 2020 2020   0 4            
│ │ │ │ +00066910: 2020 2020 2020 3120 3420 207c 0a7c 2020        1 4  |.|  
│ │ │ │  00066920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00066970: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00066980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066990: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000669a0: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ -000669b0: 2020 2020 207c 0a7c 2032 3436 3333 3130       |.| 2463310
│ │ │ │ -000669c0: 3538 3934 3138 3030 7820 7820 7820 202d  58941800x x x  -
│ │ │ │ -000669d0: 2036 3638 3730 3833 3435 3334 3834 3078   66870834534840x
│ │ │ │ -000669e0: 2078 2078 2020 2d20 3238 3535 3336 3932   x x  - 28553692
│ │ │ │ -000669f0: 3330 3536 3030 7820 7820 202b 2035 3533  305600x x  + 553
│ │ │ │ -00066a00: 3633 3830 357c 0a7c 2020 2020 2020 2020  63805|.|        
│ │ │ │ -00066a10: 2020 2020 2020 2020 2031 2033 2034 2020           1 3 4  
│ │ │ │ -00066a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066a30: 3220 3320 3420 2020 2020 2020 2020 2020  2 3 4           
│ │ │ │ -00066a40: 2020 2020 2020 2033 2034 2020 2020 2020         3 4      
│ │ │ │ -00066a50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00066960: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00066970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00066980: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00066990: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +000669a0: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
│ │ │ │ +000669b0: 2020 2020 2020 2020 2020 207c 0a7c 2032             |.| 2
│ │ │ │ +000669c0: 3436 3333 3130 3538 3934 3138 3030 7820  46331058941800x 
│ │ │ │ +000669d0: 7820 7820 202d 2036 3638 3730 3833 3435  x x  - 668708345
│ │ │ │ +000669e0: 3334 3834 3078 2078 2078 2020 2d20 3238  34840x x x  - 28
│ │ │ │ +000669f0: 3535 3336 3932 3330 3536 3030 7820 7820  553692305600x x 
│ │ │ │ +00066a00: 202b 2035 3533 3633 3830 357c 0a7c 2020   + 55363805|.|  
│ │ │ │ +00066a10: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00066a20: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ +00066a30: 2020 2020 2020 3220 3320 3420 2020 2020        2 3 4     
│ │ │ │ +00066a40: 2020 2020 2020 2020 2020 2020 2033 2034               3 4
│ │ │ │ +00066a50: 2020 2020 2020 2020 2020 207c 0a7c 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 207c 0a7c 2032 2020 2020 2020       |.| 2      
│ │ │ │ +00066aa0: 2020 2020 2020 2020 2020 207c 0a7c 2032             |.| 2
│ │ │ │  00066ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066ac0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00066ad0: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ -00066ae0: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -00066af0: 2020 2020 207c 0a7c 7820 202b 2031 3339       |.|x  + 139
│ │ │ │ -00066b00: 3132 3536 3138 3937 3536 3078 2078 2078  12561897560x x x
│ │ │ │ -00066b10: 2020 2b20 3135 3035 3531 3039 3132 3535    + 150551091255
│ │ │ │ -00066b20: 3630 7820 7820 202d 2032 3332 3938 3332  60x x  - 2329832
│ │ │ │ -00066b30: 3637 3435 3132 3030 7820 7820 202b 2031  67451200x x  + 1
│ │ │ │ -00066b40: 3231 3633 347c 0a7c 2034 2020 2020 2020  21634|.| 4      
│ │ │ │ -00066b50: 2020 2020 2020 2020 2020 2020 3220 3320              2 3 
│ │ │ │ -00066b60: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00066b70: 2020 2033 2034 2020 2020 2020 2020 2020     3 4          
│ │ │ │ -00066b80: 2020 2020 2020 2020 2030 2034 2020 2020           0 4    
│ │ │ │ -00066b90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00066ac0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00066ad0: 2020 2020 2020 2020 2032 2032 2020 2020           2 2    
│ │ │ │ +00066ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00066af0: 2033 2020 2020 2020 2020 207c 0a7c 7820   3         |.|x 
│ │ │ │ +00066b00: 202b 2031 3339 3132 3536 3138 3937 3536   + 1391256189756
│ │ │ │ +00066b10: 3078 2078 2078 2020 2b20 3135 3035 3531  0x x x  + 150551
│ │ │ │ +00066b20: 3039 3132 3535 3630 7820 7820 202d 2032  09125560x x  - 2
│ │ │ │ +00066b30: 3332 3938 3332 3637 3435 3132 3030 7820  32983267451200x 
│ │ │ │ +00066b40: 7820 202b 2031 3231 3633 347c 0a7c 2034  x  + 121634|.| 4
│ │ │ │ +00066b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00066b60: 2020 3220 3320 3420 2020 2020 2020 2020    2 3 4         
│ │ │ │ +00066b70: 2020 2020 2020 2020 2033 2034 2020 2020           3 4    
│ │ │ │ +00066b80: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00066b90: 2034 2020 2020 2020 2020 207c 0a7c 2020   4         |.|  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00066bf0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00066c00: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00066c10: 2020 2020 2020 2020 2032 2032 2020 2020           2 2    
│ │ │ │ -00066c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066c30: 3320 2020 207c 0a7c 3835 3078 2078 2078  3    |.|850x x x
│ │ │ │ -00066c40: 2020 2d20 3235 3033 3635 3130 3737 3336    - 250365107736
│ │ │ │ -00066c50: 3030 7820 7820 7820 202d 2039 3034 3532  00x x x  - 90452
│ │ │ │ -00066c60: 3336 3931 3936 3030 7820 7820 202b 2037  36919600x x  + 7
│ │ │ │ -00066c70: 3637 3934 3632 3434 3731 3630 3078 2078  6794624471600x x
│ │ │ │ -00066c80: 2020 2d20 337c 0a7c 2020 2020 3120 3320    - 3|.|    1 3 
│ │ │ │ -00066c90: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00066ca0: 2020 2032 2033 2034 2020 2020 2020 2020     2 3 4        
│ │ │ │ -00066cb0: 2020 2020 2020 2020 2033 2034 2020 2020           3 4    
│ │ │ │ -00066cc0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00066cd0: 3420 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d  4    |.|--------
│ │ │ │ +00066be0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00066bf0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00066c00: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00066c10: 2020 2020 2020 2020 2020 2020 2020 2032                 2
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│ │ │ │ +00066c30: 2020 2020 2020 3320 2020 207c 0a7c 3835        3    |.|85
│ │ │ │ +00066c40: 3078 2078 2078 2020 2d20 3235 3033 3635  0x x x  - 250365
│ │ │ │ +00066c50: 3130 3737 3336 3030 7820 7820 7820 202d  10773600x x x  -
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│ │ │ │ +00066c70: 7820 202b 2037 3637 3934 3632 3434 3731  x  + 76794624471
│ │ │ │ +00066c80: 3630 3078 2078 2020 2d20 337c 0a7c 2020  600x x  - 3|.|  
│ │ │ │ +00066c90: 2020 3120 3320 3420 2020 2020 2020 2020    1 3 4         
│ │ │ │ +00066ca0: 2020 2020 2020 2020 2032 2033 2034 2020           2 3 4  
│ │ │ │ +00066cb0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00066cc0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +00066cd0: 2020 2020 3020 3420 2020 207c 0a7c 2d2d      0 4    |.|--
│ │ │ │  00066ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066d20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00066d30: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ -00066d40: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -00066d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066d60: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00066d70: 2020 2020 207c 0a7c 3237 3131 3533 3535       |.|27115355
│ │ │ │ -00066d80: 3838 3231 3630 3078 2078 2020 2b20 3838  8821600x x  + 88
│ │ │ │ -00066d90: 3435 3831 3234 3335 3630 3078 2078 2020  45812435600x x  
│ │ │ │ -00066da0: 2b20 3739 3836 3739 3034 3235 3630 3078  + 7986790425600x
│ │ │ │ -00066db0: 2078 2020 2b20 3432 3830 3233 3632 3236   x  + 4280236226
│ │ │ │ -00066dc0: 3238 3030 787c 0a7c 2020 2020 2020 2020  2800x|.|        
│ │ │ │ -00066dd0: 2020 2020 2020 2020 3020 3420 2020 2020          0 4     
│ │ │ │ -00066de0: 2020 2020 2020 2020 2020 2020 3120 3420              1 4 
│ │ │ │ -00066df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066e00: 3220 3420 2020 2020 2020 2020 2020 2020  2 4             
│ │ │ │ -00066e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00066d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +00066d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00066d40: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00066d50: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +00066d60: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +00066d70: 2020 2020 2020 2020 2020 207c 0a7c 3237             |.|27
│ │ │ │ +00066d80: 3131 3533 3535 3838 3231 3630 3078 2078  1153558821600x x
│ │ │ │ +00066d90: 2020 2b20 3838 3435 3831 3234 3335 3630    + 884581243560
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│ │ │ │ +00066df0: 2020 3120 3420 2020 2020 2020 2020 2020    1 4           
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│ │ │ │ +00066e10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  00066e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00066e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066e60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00066e70: 2020 2020 2020 2020 2020 3320 2020 2020            3     
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│ │ │ │ -00066e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066ea0: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ -00066eb0: 2020 2020 207c 0a7c 2032 3131 3136 3432       |.| 2111642
│ │ │ │ -00066ec0: 3930 3036 3430 3078 2078 2020 2b20 3338  9006400x x  + 38
│ │ │ │ -00066ed0: 3934 3537 3634 3837 3230 3030 7820 7820  945764872000x x 
│ │ │ │ -00066ee0: 202d 2032 3235 3930 3234 3631 3638 3030   - 2259024616800
│ │ │ │ -00066ef0: 3078 202c 2020 2020 2020 2020 2020 2020  0x ,            
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│ │ │ │ -00066f10: 2020 2020 2020 2020 3220 3420 2020 2020          2 4     
│ │ │ │ -00066f20: 2020 2020 2020 2020 2020 2020 2033 2034               3 4
│ │ │ │ -00066f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066f40: 2020 3420 2020 2020 2020 2020 2020 2020    4             
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│ │ │ │ +00066e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00066e80: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ +00066ea0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +00066eb0: 2020 2020 2020 2020 2020 207c 0a7c 2032             |.| 2
│ │ │ │ +00066ec0: 3131 3136 3432 3930 3036 3430 3078 2078  1116429006400x x
│ │ │ │ +00066ed0: 2020 2b20 3338 3934 3537 3634 3837 3230    + 389457648720
│ │ │ │ +00066ee0: 3030 7820 7820 202d 2032 3235 3930 3234  00x x  - 2259024
│ │ │ │ +00066ef0: 3631 3638 3030 3078 202c 2020 2020 2020  6168000x ,      
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│ │ │ │ +00066f30: 2020 2033 2034 2020 2020 2020 2020 2020     3 4          
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│ │ │ │ +00066f50: 2020 2020 2020 2020 2020 207c 0a7c 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00066fb0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00066fc0: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -00066fd0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -00066fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00066ff0: 2020 3320 207c 0a7c 3830 3431 3630 3078    3  |.|8041600x
│ │ │ │ -00067000: 2078 2020 2d20 3535 3436 3933 3139 3339   x  - 5546931939
│ │ │ │ -00067010: 3336 3030 7820 7820 202b 2034 3838 3233  3600x x  + 48823
│ │ │ │ -00067020: 3936 3738 3538 3430 3078 2078 2020 2d20  967858400x x  - 
│ │ │ │ -00067030: 3135 3435 3832 3737 3737 3932 3830 3078  154582777792800x
│ │ │ │ -00067040: 2078 2020 2b7c 0a7c 2020 2020 2020 2020   x  +|.|        
│ │ │ │ -00067050: 3020 3420 2020 2020 2020 2020 2020 2020  0 4             
│ │ │ │ -00067060: 2020 2020 2031 2034 2020 2020 2020 2020       1 4        
│ │ │ │ -00067070: 2020 2020 2020 2020 2020 3220 3420 2020            2 4   
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│ │ │ │ -00067090: 3320 3420 207c 0a7c 2020 2020 2020 2020  3 4  |.|        
│ │ │ │ +00066fa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00066fb0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +00066fc0: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +00066fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00066fe0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +00066ff0: 2020 2020 2020 2020 3320 207c 0a7c 3830          3  |.|80
│ │ │ │ +00067000: 3431 3630 3078 2078 2020 2d20 3535 3436  41600x x  - 5546
│ │ │ │ +00067010: 3933 3139 3339 3336 3030 7820 7820 202b  9319393600x x  +
│ │ │ │ +00067020: 2034 3838 3233 3936 3738 3538 3430 3078   48823967858400x
│ │ │ │ +00067030: 2078 2020 2d20 3135 3435 3832 3737 3737   x  - 1545827777
│ │ │ │ +00067040: 3932 3830 3078 2078 2020 2b7c 0a7c 2020  92800x x  +|.|  
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│ │ │ │ +00067080: 3220 3420 2020 2020 2020 2020 2020 2020  2 4             
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│ │ │ │  000670a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000670c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000670d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00067130: 2020 2020 3420 2020 2020 207c 0a7c 3337      4      |.|37
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│ │ │ │ +00067160: 202b 2031 3130 3833 3831 3332 3937 3230   + 1108381329720
│ │ │ │ +00067170: 3078 2078 2020 2d20 3333 3732 3036 3234  0x x  - 33720624
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│ │ │ │ +000671a0: 2020 2020 2020 2020 2020 2020 2032 2034               2 4
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│ │ │ │  000671f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067220: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -00067250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067260: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00067270: 2020 2034 207c 0a7c 3839 3034 3734 3834     4 |.|89047484
│ │ │ │ -00067280: 3332 3230 3078 2078 2020 2b20 3230 3436  32200x x  + 2046
│ │ │ │ -00067290: 3435 3539 3533 3932 3030 7820 7820 202b  4559539200x x  +
│ │ │ │ -000672a0: 2037 3631 3035 3834 3630 3236 3030 7820   7610584602600x 
│ │ │ │ -000672b0: 7820 202d 2033 3233 3637 3531 3235 3230  x  - 32367512520
│ │ │ │ -000672c0: 3030 7820 207c 0a7c 2020 2020 2020 2020  00x  |.|        
│ │ │ │ -000672d0: 2020 2020 2020 3120 3420 2020 2020 2020        1 4       
│ │ │ │ -000672e0: 2020 2020 2020 2020 2020 2032 2034 2020             2 4  
│ │ │ │ -000672f0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -00067300: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -00067310: 2020 2034 207c 0a7c 2d2d 2d2d 2d2d 2d2d     4 |.|--------
│ │ │ │ +00067220: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00067230: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +00067240: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00067260: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00067270: 2020 2020 2020 2020 2034 207c 0a7c 3839           4 |.|89
│ │ │ │ +00067280: 3034 3734 3834 3332 3230 3078 2078 2020  04748432200x x  
│ │ │ │ +00067290: 2b20 3230 3436 3435 3539 3533 3932 3030  + 20464559539200
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│ │ │ │ +000672e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000672f0: 2032 2034 2020 2020 2020 2020 2020 2020   2 4            
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│ │ │ │ +00067310: 2020 2020 2020 2020 2034 207c 0a7c 2d2d           4 |.|--
│ │ │ │  00067320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00067440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -00067630: 2020 2020 207c 0a7c 2039 3037 3635 3530       |.| 9076550
│ │ │ │ -00067640: 3531 3532 3030 3078 202c 2020 2020 2020  5152000x ,      
│ │ │ │ +00067630: 2020 2020 2020 2020 2020 207c 0a7c 2039             |.| 9
│ │ │ │ +00067640: 3037 3635 3530 3531 3532 3030 3078 202c  0765505152000x ,
│ │ │ │  00067650: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000676a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000676c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000676d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000676d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  000676e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000676f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067720: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ -00067730: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00067740: 6e6f 7465 2069 6e76 6572 7365 4d61 703a  note inverseMap:
│ │ │ │ -00067750: 2069 6e76 6572 7365 4d61 702c 202d 2d20   inverseMap, -- 
│ │ │ │ -00067760: 696e 7665 7273 6520 6f66 2061 2062 6972  inverse of a bir
│ │ │ │ -00067770: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ -00067780: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ -00067790: 7020 5e20 5a5a 3a20 5261 7469 6f6e 616c  p ^ ZZ: Rational
│ │ │ │ -000677a0: 4d61 7020 5e20 5a5a 2c20 2d2d 2070 6f77  Map ^ ZZ, -- pow
│ │ │ │ -000677b0: 6572 0a20 202a 202a 6e6f 7465 2069 7349  er.  * *note isI
│ │ │ │ -000677c0: 6e76 6572 7365 4d61 7028 5261 7469 6f6e  nverseMap(Ration
│ │ │ │ -000677d0: 616c 4d61 702c 5261 7469 6f6e 616c 4d61  alMap,RationalMa
│ │ │ │ -000677e0: 7029 3a0a 2020 2020 6973 496e 7665 7273  p):.    isInvers
│ │ │ │ -000677f0: 654d 6170 5f6c 7052 6174 696f 6e61 6c4d  eMap_lpRationalM
│ │ │ │ -00067800: 6170 5f63 6d52 6174 696f 6e61 6c4d 6170  ap_cmRationalMap
│ │ │ │ -00067810: 5f72 702c 202d 2d20 6368 6563 6b73 2077  _rp, -- checks w
│ │ │ │ -00067820: 6865 7468 6572 2074 776f 2072 6174 696f  hether two ratio
│ │ │ │ -00067830: 6e61 6c0a 2020 2020 6d61 7073 2061 7265  nal.    maps are
│ │ │ │ -00067840: 206f 6e65 2074 6865 2069 6e76 6572 7365   one the inverse
│ │ │ │ -00067850: 206f 6620 7468 6520 6f74 6865 720a 0a57   of the other..W
│ │ │ │ -00067860: 6179 7320 746f 2075 7365 2074 6869 7320  ays to use this 
│ │ │ │ -00067870: 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d  method:.========
│ │ │ │ +00067720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ +00067730: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +00067740: 0a20 202a 202a 6e6f 7465 2069 6e76 6572  .  * *note inver
│ │ │ │ +00067750: 7365 4d61 703a 2069 6e76 6572 7365 4d61  seMap: inverseMa
│ │ │ │ +00067760: 702c 202d 2d20 696e 7665 7273 6520 6f66  p, -- inverse of
│ │ │ │ +00067770: 2061 2062 6972 6174 696f 6e61 6c20 6d61   a birational ma
│ │ │ │ +00067780: 700a 2020 2a20 2a6e 6f74 6520 5261 7469  p.  * *note Rati
│ │ │ │ +00067790: 6f6e 616c 4d61 7020 5e20 5a5a 3a20 5261  onalMap ^ ZZ: Ra
│ │ │ │ +000677a0: 7469 6f6e 616c 4d61 7020 5e20 5a5a 2c20  tionalMap ^ ZZ, 
│ │ │ │ +000677b0: 2d2d 2070 6f77 6572 0a20 202a 202a 6e6f  -- power.  * *no
│ │ │ │ +000677c0: 7465 2069 7349 6e76 6572 7365 4d61 7028  te isInverseMap(
│ │ │ │ +000677d0: 5261 7469 6f6e 616c 4d61 702c 5261 7469  RationalMap,Rati
│ │ │ │ +000677e0: 6f6e 616c 4d61 7029 3a0a 2020 2020 6973  onalMap):.    is
│ │ │ │ +000677f0: 496e 7665 7273 654d 6170 5f6c 7052 6174  InverseMap_lpRat
│ │ │ │ +00067800: 696f 6e61 6c4d 6170 5f63 6d52 6174 696f  ionalMap_cmRatio
│ │ │ │ +00067810: 6e61 6c4d 6170 5f72 702c 202d 2d20 6368  nalMap_rp, -- ch
│ │ │ │ +00067820: 6563 6b73 2077 6865 7468 6572 2074 776f  ecks whether two
│ │ │ │ +00067830: 2072 6174 696f 6e61 6c0a 2020 2020 6d61   rational.    ma
│ │ │ │ +00067840: 7073 2061 7265 206f 6e65 2074 6865 2069  ps are one the i
│ │ │ │ +00067850: 6e76 6572 7365 206f 6620 7468 6520 6f74  nverse of the ot
│ │ │ │ +00067860: 6865 720a 0a57 6179 7320 746f 2075 7365  her..Ways to use
│ │ │ │ +00067870: 2074 6869 7320 6d65 7468 6f64 3a0a 3d3d   this method:.==
│ │ │ │  00067880: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00067890: 0a0a 2020 2a20 2a6e 6f74 6520 696e 7665  ..  * *note inve
│ │ │ │ -000678a0: 7273 6528 5261 7469 6f6e 616c 4d61 7029  rse(RationalMap)
│ │ │ │ -000678b0: 3a20 696e 7665 7273 655f 6c70 5261 7469  : inverse_lpRati
│ │ │ │ -000678c0: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2069  onalMap_rp, -- i
│ │ │ │ -000678d0: 6e76 6572 7365 206f 6620 610a 2020 2020  nverse of a.    
│ │ │ │ -000678e0: 6269 7261 7469 6f6e 616c 206d 6170 0a20  birational map. 
│ │ │ │ -000678f0: 202a 2022 696e 7665 7273 6528 5261 7469   * "inverse(Rati
│ │ │ │ -00067900: 6f6e 616c 4d61 702c 4f70 7469 6f6e 2922  onalMap,Option)"
│ │ │ │ -00067910: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ -00067920: 2e69 6e66 6f2c 204e 6f64 653a 2069 6e76  .info, Node: inv
│ │ │ │ -00067930: 6572 7365 4d61 702c 204e 6578 743a 2069  erseMap, Next: i
│ │ │ │ -00067940: 6e76 6572 7365 4d61 705f 6c70 5f70 645f  nverseMap_lp_pd_
│ │ │ │ -00067950: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ -00067960: 3e5f 7064 5f70 645f 7064 5f72 702c 2050  >_pd_pd_pd_rp, P
│ │ │ │ -00067970: 7265 763a 2069 6e76 6572 7365 5f6c 7052  rev: inverse_lpR
│ │ │ │ -00067980: 6174 696f 6e61 6c4d 6170 5f72 702c 2055  ationalMap_rp, U
│ │ │ │ -00067990: 703a 2054 6f70 0a0a 696e 7665 7273 654d  p: Top..inverseM
│ │ │ │ -000679a0: 6170 202d 2d20 696e 7665 7273 6520 6f66  ap -- inverse of
│ │ │ │ -000679b0: 2061 2062 6972 6174 696f 6e61 6c20 6d61   a birational ma
│ │ │ │ -000679c0: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  p.**************
│ │ │ │ +00067890: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +000678a0: 6520 696e 7665 7273 6528 5261 7469 6f6e  e inverse(Ration
│ │ │ │ +000678b0: 616c 4d61 7029 3a20 696e 7665 7273 655f  alMap): inverse_
│ │ │ │ +000678c0: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ +000678d0: 2c20 2d2d 2069 6e76 6572 7365 206f 6620  , -- inverse of 
│ │ │ │ +000678e0: 610a 2020 2020 6269 7261 7469 6f6e 616c  a.    birational
│ │ │ │ +000678f0: 206d 6170 0a20 202a 2022 696e 7665 7273   map.  * "invers
│ │ │ │ +00067900: 6528 5261 7469 6f6e 616c 4d61 702c 4f70  e(RationalMap,Op
│ │ │ │ +00067910: 7469 6f6e 2922 0a1f 0a46 696c 653a 2043  tion)"...File: C
│ │ │ │ +00067920: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ +00067930: 653a 2069 6e76 6572 7365 4d61 702c 204e  e: inverseMap, N
│ │ │ │ +00067940: 6578 743a 2069 6e76 6572 7365 4d61 705f  ext: inverseMap_
│ │ │ │ +00067950: 6c70 5f70 645f 7064 5f70 645f 636d 5665  lp_pd_pd_pd_cmVe
│ │ │ │ +00067960: 7262 6f73 653d 3e5f 7064 5f70 645f 7064  rbose=>_pd_pd_pd
│ │ │ │ +00067970: 5f72 702c 2050 7265 763a 2069 6e76 6572  _rp, Prev: inver
│ │ │ │ +00067980: 7365 5f6c 7052 6174 696f 6e61 6c4d 6170  se_lpRationalMap
│ │ │ │ +00067990: 5f72 702c 2055 703a 2054 6f70 0a0a 696e  _rp, Up: Top..in
│ │ │ │ +000679a0: 7665 7273 654d 6170 202d 2d20 696e 7665  verseMap -- inve
│ │ │ │ +000679b0: 7273 6520 6f66 2061 2062 6972 6174 696f  rse of a biratio
│ │ │ │ +000679c0: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a  nal map.********
│ │ │ │  000679d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000679e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ -000679f0: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ -00067a00: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00067a10: 2020 2020 696e 7665 7273 654d 6170 2070      inverseMap p
│ │ │ │ -00067a20: 6869 0a20 202a 2049 6e70 7574 733a 0a20  hi.  * Inputs:. 
│ │ │ │ -00067a30: 2020 2020 202a 2070 6869 2c20 6120 2a6e       * phi, a *n
│ │ │ │ -00067a40: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ -00067a50: 3a20 5261 7469 6f6e 616c 4d61 702c 2c20  : RationalMap,, 
│ │ │ │ -00067a60: 6120 6269 7261 7469 6f6e 616c 206d 6170  a birational map
│ │ │ │ -00067a70: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
│ │ │ │ -00067a80: 6e61 6c20 696e 7075 7473 3a20 284d 6163  nal inputs: (Mac
│ │ │ │ -00067a90: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ -00067aa0: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
│ │ │ │ -00067ab0: 7074 696f 6e61 6c20 696e 7075 7473 2c3a  ptional inputs,:
│ │ │ │ -00067ac0: 0a20 2020 2020 202a 202a 6e6f 7465 2042  .      * *note B
│ │ │ │ -00067ad0: 6c6f 7755 7053 7472 6174 6567 793a 2042  lowUpStrategy: B
│ │ │ │ -00067ae0: 6c6f 7755 7053 7472 6174 6567 792c 203d  lowUpStrategy, =
│ │ │ │ -00067af0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -00067b00: 616c 7565 0a20 2020 2020 2020 2022 456c  alue.        "El
│ │ │ │ -00067b10: 696d 696e 6174 6522 2c0a 2020 2020 2020  iminate",.      
│ │ │ │ -00067b20: 2a20 2a6e 6f74 6520 4365 7274 6966 793a  * *note Certify:
│ │ │ │ -00067b30: 2043 6572 7469 6679 2c20 3d3e 202e 2e2e   Certify, => ...
│ │ │ │ -00067b40: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -00067b50: 6661 6c73 652c 2077 6865 7468 6572 2074  false, whether t
│ │ │ │ -00067b60: 6f20 656e 7375 7265 0a20 2020 2020 2020  o ensure.       
│ │ │ │ -00067b70: 2063 6f72 7265 6374 6e65 7373 206f 6620   correctness of 
│ │ │ │ -00067b80: 6f75 7470 7574 0a20 2020 2020 202a 202a  output.      * *
│ │ │ │ -00067b90: 6e6f 7465 2056 6572 626f 7365 3a20 696e  note Verbose: in
│ │ │ │ -00067ba0: 7665 7273 654d 6170 5f6c 705f 7064 5f70  verseMap_lp_pd_p
│ │ │ │ -00067bb0: 645f 7064 5f63 6d56 6572 626f 7365 3d3e  d_pd_cmVerbose=>
│ │ │ │ -00067bc0: 5f70 645f 7064 5f70 645f 7270 2c20 3d3e  _pd_pd_pd_rp, =>
│ │ │ │ -00067bd0: 202e 2e2e 2c0a 2020 2020 2020 2020 6465   ...,.        de
│ │ │ │ -00067be0: 6661 756c 7420 7661 6c75 6520 7472 7565  fault value true
│ │ │ │ -00067bf0: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ -00067c00: 2020 2020 202a 2061 202a 6e6f 7465 2072       * a *note r
│ │ │ │ -00067c10: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ -00067c20: 696f 6e61 6c4d 6170 2c2c 2074 6865 2069  ionalMap,, the i
│ │ │ │ -00067c30: 6e76 6572 7365 206d 6170 206f 6620 7068  nverse map of ph
│ │ │ │ -00067c40: 690a 0a44 6573 6372 6970 7469 6f6e 0a3d  i..Description.=
│ │ │ │ -00067c50: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 2074  ==========..If t
│ │ │ │ -00067c60: 6865 2073 6f75 7263 6520 7661 7269 6574  he source variet
│ │ │ │ -00067c70: 7920 6973 2061 2070 726f 6a65 6374 6976  y is a projectiv
│ │ │ │ -00067c80: 6520 7370 6163 6520 616e 6420 6966 2061  e space and if a
│ │ │ │ -00067c90: 2066 7572 7468 6572 2074 6563 686e 6963   further technic
│ │ │ │ -00067ca0: 616c 0a63 6f6e 6469 7469 6f6e 2069 7320  al.condition is 
│ │ │ │ -00067cb0: 7361 7469 7366 6965 642c 2074 6865 6e20  satisfied, then 
│ │ │ │ -00067cc0: 7468 6520 616c 676f 7269 7468 6d20 7573  the algorithm us
│ │ │ │ -00067cd0: 6564 2069 7320 7468 6174 2064 6573 6372  ed is that descr
│ │ │ │ -00067ce0: 6962 6564 2069 6e20 7468 6520 7061 7065  ibed in the pape
│ │ │ │ -00067cf0: 720a 6279 2052 7573 736f 2061 6e64 2053  r.by Russo and S
│ │ │ │ -00067d00: 696d 6973 202d 204f 6e20 6269 7261 7469  imis - On birati
│ │ │ │ -00067d10: 6f6e 616c 206d 6170 7320 616e 6420 4a61  onal maps and Ja
│ │ │ │ -00067d20: 636f 6269 616e 206d 6174 7269 6365 7320  cobian matrices 
│ │ │ │ -00067d30: 2d20 436f 6d70 6f73 2e20 4d61 7468 2e0a  - Compos. Math..
│ │ │ │ -00067d40: 3132 3620 2833 292c 2033 3335 2d33 3538  126 (3), 335-358
│ │ │ │ -00067d50: 2c20 3230 3031 2e20 466f 7220 7468 6520  , 2001. For the 
│ │ │ │ -00067d60: 6765 6e65 7261 6c20 6361 7365 2c20 7468  general case, th
│ │ │ │ -00067d70: 6520 616c 676f 7269 7468 6d20 7573 6564  e algorithm used
│ │ │ │ -00067d80: 2069 7320 7468 6520 7361 6d65 2061 730a   is the same as.
│ │ │ │ -00067d90: 666f 7220 2a6e 6f74 6520 696e 7665 7274  for *note invert
│ │ │ │ -00067da0: 4269 7261 7469 6f6e 616c 4d61 703a 2028  BirationalMap: (
│ │ │ │ -00067db0: 5061 7261 6d65 7472 697a 6174 696f 6e29  Parametrization)
│ │ │ │ -00067dc0: 696e 7665 7274 4269 7261 7469 6f6e 616c  invertBirational
│ │ │ │ -00067dd0: 4d61 702c 2069 6e20 7468 650a 7061 636b  Map, in the.pack
│ │ │ │ -00067de0: 6167 6520 2a6e 6f74 6520 5061 7261 6d65  age *note Parame
│ │ │ │ -00067df0: 7472 697a 6174 696f 6e3a 2028 5061 7261  trization: (Para
│ │ │ │ -00067e00: 6d65 7472 697a 6174 696f 6e29 546f 702c  metrization)Top,
│ │ │ │ -00067e10: 2e20 4e6f 7465 2074 6861 7420 696e 2074  . Note that in t
│ │ │ │ -00067e20: 6869 7320 6361 7365 2c0a 7468 6520 616e  his case,.the an
│ │ │ │ -00067e30: 616c 6f67 6f75 7320 6d65 7468 6f64 202a  alogous method *
│ │ │ │ -00067e40: 6e6f 7465 2069 6e76 6572 7365 4f66 4d61  note inverseOfMa
│ │ │ │ -00067e50: 703a 2028 5261 7469 6f6e 616c 4d61 7073  p: (RationalMaps
│ │ │ │ -00067e60: 2969 6e76 6572 7365 4f66 4d61 702c 2069  )inverseOfMap, i
│ │ │ │ -00067e70: 6e20 7468 650a 7061 636b 6167 6520 5261  n the.package Ra
│ │ │ │ -00067e80: 7469 6f6e 616c 4d61 7073 2028 6d69 7373  tionalMaps (miss
│ │ │ │ -00067e90: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ -00067ea0: 6e29 2067 656e 6572 616c 6c79 2074 7572  n) generally tur
│ │ │ │ -00067eb0: 6e73 206f 7574 2074 6f20 6265 2066 6173  ns out to be fas
│ │ │ │ -00067ec0: 7465 722e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ter...+---------
│ │ │ │ +000679e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000679f0: 2a0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  *..Synopsis.====
│ │ │ │ +00067a00: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ +00067a10: 200a 2020 2020 2020 2020 696e 7665 7273   .        invers
│ │ │ │ +00067a20: 654d 6170 2070 6869 0a20 202a 2049 6e70  eMap phi.  * Inp
│ │ │ │ +00067a30: 7574 733a 0a20 2020 2020 202a 2070 6869  uts:.      * phi
│ │ │ │ +00067a40: 2c20 6120 2a6e 6f74 6520 7261 7469 6f6e  , a *note ration
│ │ │ │ +00067a50: 616c 206d 6170 3a20 5261 7469 6f6e 616c  al map: Rational
│ │ │ │ +00067a60: 4d61 702c 2c20 6120 6269 7261 7469 6f6e  Map,, a biration
│ │ │ │ +00067a70: 616c 206d 6170 0a20 202a 202a 6e6f 7465  al map.  * *note
│ │ │ │ +00067a80: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ +00067a90: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00067aa0: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ +00067ab0: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
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│ │ │ │ +00067ad0: 6e6f 7465 2042 6c6f 7755 7053 7472 6174  note BlowUpStrat
│ │ │ │ +00067ae0: 6567 793a 2042 6c6f 7755 7053 7472 6174  egy: BlowUpStrat
│ │ │ │ +00067af0: 6567 792c 203d 3e20 2e2e 2e2c 2064 6566  egy, => ..., def
│ │ │ │ +00067b00: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ +00067b10: 2020 2022 456c 696d 696e 6174 6522 2c0a     "Eliminate",.
│ │ │ │ +00067b20: 2020 2020 2020 2a20 2a6e 6f74 6520 4365        * *note Ce
│ │ │ │ +00067b30: 7274 6966 793a 2043 6572 7469 6679 2c20  rtify: Certify, 
│ │ │ │ +00067b40: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +00067b50: 7661 6c75 6520 6661 6c73 652c 2077 6865  value false, whe
│ │ │ │ +00067b60: 7468 6572 2074 6f20 656e 7375 7265 0a20  ther to ensure. 
│ │ │ │ +00067b70: 2020 2020 2020 2063 6f72 7265 6374 6e65         correctne
│ │ │ │ +00067b80: 7373 206f 6620 6f75 7470 7574 0a20 2020  ss of output.   
│ │ │ │ +00067b90: 2020 202a 202a 6e6f 7465 2056 6572 626f     * *note Verbo
│ │ │ │ +00067ba0: 7365 3a20 696e 7665 7273 654d 6170 5f6c  se: inverseMap_l
│ │ │ │ +00067bb0: 705f 7064 5f70 645f 7064 5f63 6d56 6572  p_pd_pd_pd_cmVer
│ │ │ │ +00067bc0: 626f 7365 3d3e 5f70 645f 7064 5f70 645f  bose=>_pd_pd_pd_
│ │ │ │ +00067bd0: 7270 2c20 3d3e 202e 2e2e 2c0a 2020 2020  rp, => ...,.    
│ │ │ │ +00067be0: 2020 2020 6465 6661 756c 7420 7661 6c75      default valu
│ │ │ │ +00067bf0: 6520 7472 7565 2c0a 2020 2a20 4f75 7470  e true,.  * Outp
│ │ │ │ +00067c00: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +00067c10: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ +00067c20: 703a 2052 6174 696f 6e61 6c4d 6170 2c2c  p: RationalMap,,
│ │ │ │ +00067c30: 2074 6865 2069 6e76 6572 7365 206d 6170   the inverse map
│ │ │ │ +00067c40: 206f 6620 7068 690a 0a44 6573 6372 6970   of phi..Descrip
│ │ │ │ +00067c50: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +00067c60: 0a0a 4966 2074 6865 2073 6f75 7263 6520  ..If the source 
│ │ │ │ +00067c70: 7661 7269 6574 7920 6973 2061 2070 726f  variety is a pro
│ │ │ │ +00067c80: 6a65 6374 6976 6520 7370 6163 6520 616e  jective space an
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│ │ │ │ +00067ca0: 6563 686e 6963 616c 0a63 6f6e 6469 7469  echnical.conditi
│ │ │ │ +00067cb0: 6f6e 2069 7320 7361 7469 7366 6965 642c  on is satisfied,
│ │ │ │ +00067cc0: 2074 6865 6e20 7468 6520 616c 676f 7269   then the algori
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│ │ │ │ +00067cf0: 6520 7061 7065 720a 6279 2052 7573 736f  e paper.by Russo
│ │ │ │ +00067d00: 2061 6e64 2053 696d 6973 202d 204f 6e20   and Simis - On 
│ │ │ │ +00067d10: 6269 7261 7469 6f6e 616c 206d 6170 7320  birational maps 
│ │ │ │ +00067d20: 616e 6420 4a61 636f 6269 616e 206d 6174  and Jacobian mat
│ │ │ │ +00067d30: 7269 6365 7320 2d20 436f 6d70 6f73 2e20  rices - Compos. 
│ │ │ │ +00067d40: 4d61 7468 2e0a 3132 3620 2833 292c 2033  Math..126 (3), 3
│ │ │ │ +00067d50: 3335 2d33 3538 2c20 3230 3031 2e20 466f  35-358, 2001. Fo
│ │ │ │ +00067d60: 7220 7468 6520 6765 6e65 7261 6c20 6361  r the general ca
│ │ │ │ +00067d70: 7365 2c20 7468 6520 616c 676f 7269 7468  se, the algorith
│ │ │ │ +00067d80: 6d20 7573 6564 2069 7320 7468 6520 7361  m used is the sa
│ │ │ │ +00067d90: 6d65 2061 730a 666f 7220 2a6e 6f74 6520  me as.for *note 
│ │ │ │ +00067da0: 696e 7665 7274 4269 7261 7469 6f6e 616c  invertBirational
│ │ │ │ +00067db0: 4d61 703a 2028 5061 7261 6d65 7472 697a  Map: (Parametriz
│ │ │ │ +00067dc0: 6174 696f 6e29 696e 7665 7274 4269 7261  ation)invertBira
│ │ │ │ +00067dd0: 7469 6f6e 616c 4d61 702c 2069 6e20 7468  tionalMap, in th
│ │ │ │ +00067de0: 650a 7061 636b 6167 6520 2a6e 6f74 6520  e.package *note 
│ │ │ │ +00067df0: 5061 7261 6d65 7472 697a 6174 696f 6e3a  Parametrization:
│ │ │ │ +00067e00: 2028 5061 7261 6d65 7472 697a 6174 696f   (Parametrizatio
│ │ │ │ +00067e10: 6e29 546f 702c 2e20 4e6f 7465 2074 6861  n)Top,. Note tha
│ │ │ │ +00067e20: 7420 696e 2074 6869 7320 6361 7365 2c0a  t in this case,.
│ │ │ │ +00067e30: 7468 6520 616e 616c 6f67 6f75 7320 6d65  the analogous me
│ │ │ │ +00067e40: 7468 6f64 202a 6e6f 7465 2069 6e76 6572  thod *note inver
│ │ │ │ +00067e50: 7365 4f66 4d61 703a 2028 5261 7469 6f6e  seOfMap: (Ration
│ │ │ │ +00067e60: 616c 4d61 7073 2969 6e76 6572 7365 4f66  alMaps)inverseOf
│ │ │ │ +00067e70: 4d61 702c 2069 6e20 7468 650a 7061 636b  Map, in the.pack
│ │ │ │ +00067e80: 6167 6520 5261 7469 6f6e 616c 4d61 7073  age RationalMaps
│ │ │ │ +00067e90: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ +00067ea0: 6e74 6174 696f 6e29 2067 656e 6572 616c  ntation) general
│ │ │ │ +00067eb0: 6c79 2074 7572 6e73 206f 7574 2074 6f20  ly turns out to 
│ │ │ │ +00067ec0: 6265 2066 6173 7465 722e 0a0a 2b2d 2d2d  be faster...+---
│ │ │ │  00067ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067f10: 2d2d 2d2d 2b0a 7c69 3120 3a20 2d2d 2041  ----+.|i1 : -- A
│ │ │ │ -00067f20: 2043 7265 6d6f 6e61 2074 7261 6e73 666f   Cremona transfo
│ │ │ │ -00067f30: 726d 6174 696f 6e20 6f66 2050 5e32 3020  rmation of P^20 
│ │ │ │ -00067f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00067f20: 3a20 2d2d 2041 2043 7265 6d6f 6e61 2074  : -- A Cremona t
│ │ │ │ +00067f30: 7261 6e73 666f 726d 6174 696f 6e20 6f66  ransformation of
│ │ │ │ +00067f40: 2050 5e32 3020 2020 2020 2020 2020 2020   P^20           
│ │ │ │  00067f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067f60: 2020 2020 7c0a 7c20 2020 2020 7068 6920      |.|     phi 
│ │ │ │ -00067f70: 3d20 7261 7469 6f6e 616c 4d61 7020 6d61  = rationalMap ma
│ │ │ │ -00067f80: 7020 7175 6164 726f 5175 6164 7269 6343  p quadroQuadricC
│ │ │ │ -00067f90: 7265 6d6f 6e61 5472 616e 7366 6f72 6d61  remonaTransforma
│ │ │ │ -00067fa0: 7469 6f6e 2832 302c 3129 2020 2020 2020  tion(20,1)      
│ │ │ │ -00067fb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00067f60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00067f70: 2020 7068 6920 3d20 7261 7469 6f6e 616c    phi = rational
│ │ │ │ +00067f80: 4d61 7020 6d61 7020 7175 6164 726f 5175  Map map quadroQu
│ │ │ │ +00067f90: 6164 7269 6343 7265 6d6f 6e61 5472 616e  adricCremonaTran
│ │ │ │ +00067fa0: 7366 6f72 6d61 7469 6f6e 2832 302c 3129  sformation(20,1)
│ │ │ │ +00067fb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00067fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c6f 3120 3d20 2d2d 2072      |.|o1 = -- r
│ │ │ │ -00068010: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │ -00068020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068000: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00068010: 3d20 2d2d 2072 6174 696f 6e61 6c20 6d61  = -- rational ma
│ │ │ │ +00068020: 7020 2d2d 2020 2020 2020 2020 2020 2020  p --            
│ │ │ │  00068030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068050: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
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│ │ │ │ -00068070: 7720 2c20 7720 2c20 7720 2c20 7720 2c20  w , w , w , w , 
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│ │ │ │ -000680a0: 202c 2020 7c0a 7c20 2020 2020 2020 2020   ,  |.|         
│ │ │ │ -000680b0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000680c0: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -000680d0: 2035 2020 2036 2020 2037 2020 2038 2020   5   6   7   8  
│ │ │ │ -000680e0: 2039 2020 2031 3020 2020 3131 2020 2031   9   10   11   1
│ │ │ │ -000680f0: 3220 2020 7c0a 7c20 2020 2020 7461 7267  2   |.|     targ
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│ │ │ │ -00068110: 7720 2c20 7720 2c20 7720 2c20 7720 2c20  w , w , w , w , 
│ │ │ │ -00068120: 7720 2c20 7720 2c20 7720 2c20 7720 2c20  w , w , w , w , 
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│ │ │ │ -00068140: 202c 2020 7c0a 7c20 2020 2020 2020 2020   ,  |.|         
│ │ │ │ -00068150: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00068160: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -00068170: 2035 2020 2036 2020 2037 2020 2038 2020   5   6   7   8  
│ │ │ │ -00068180: 2039 2020 2031 3020 2020 3131 2020 2031   9   10   11   1
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│ │ │ │ +000680b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000680d0: 2020 2034 2020 2035 2020 2036 2020 2037     4   5   6   7
│ │ │ │ +000680e0: 2020 2038 2020 2039 2020 2031 3020 2020     8   9   10   
│ │ │ │ +000680f0: 3131 2020 2031 3220 2020 7c0a 7c20 2020  11   12   |.|   
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│ │ │ │ +00068150: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00068190: 3131 2020 2031 3220 2020 7c0a 7c20 2020  11   12   |.|   
│ │ │ │ +000681a0: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
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│ │ │ │  000681d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000681f0: 2020 2020 2020 2020 2020 2020 2077 2020               w  
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│ │ │ │ +000681f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00068240: 2020 2020 2020 2020 2020 2020 2020 3130                10
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│ │ │ │ +00068230: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00068240: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00068270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068280: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00068280: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00068290: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000682c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000682e0: 2020 2020 2020 2020 2020 2020 2077 2020               w  
│ │ │ │ -000682f0: 7720 2020 2d20 7720 7720 2020 2b20 7720  w   - w w   + w 
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│ │ │ │ +000682d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00068310: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00068320: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ +00068340: 2020 2020 3130 2031 3420 2020 2038 2031      10 14    8 1
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│ │ │ │  00068360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068370: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00068370: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00068380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000683a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000683b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000683c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000683d0: 2020 2020 2020 2020 2020 2020 2077 2077               w w
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│ │ │ │ +000683d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000683e0: 2020 2077 2077 2020 202d 2077 2077 2020     w w   - w w  
│ │ │ │ +000683f0: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
│ │ │ │  00068400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068410: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ +00068430: 2020 2020 3920 3134 2020 2020 3820 3135      9 14    8 15
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│ │ │ │  00068450: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00068470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068480: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000684a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000684c0: 2020 2020 2020 2020 2020 2020 2077 2077               w w
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│ │ │ │ +000684c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000684d0: 2020 2077 2077 2020 202d 2077 2077 2020     w w   - w w  
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│ │ │ │  000684f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00068530: 3136 2020 2020 2020 2020 2020 2020 2020  16              
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│ │ │ │ +00068c60: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
│ │ │ │  00068c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00068c90: 2020 2020 2020 2020 2020 2020 2020 3820                8 
│ │ │ │ -00068ca0: 3131 2020 2020 3320 3134 2020 2020 3020  11    3 14    0 
│ │ │ │ -00068cb0: 3230 2020 2020 2020 2020 2020 2020 2020  20              
│ │ │ │ +00068c80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00068c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00068cb0: 2020 2020 3020 3230 2020 2020 2020 2020      0 20        
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│ │ │ │ -00068d20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ -00068d40: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ -00068d50: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ -00068d60: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
│ │ │ │ -00068d70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00068d80: 2020 2020 2020 2020 2020 2020 2020 3720                7 
│ │ │ │ -00068d90: 3131 2020 2020 3320 3132 2020 2020 3220  11    3 12    2 
│ │ │ │ -00068da0: 3137 2020 2020 3120 3138 2020 2020 3020  17    1 18    0 
│ │ │ │ -00068db0: 3139 2020 2020 2020 2020 2020 2020 2020  19              
│ │ │ │ -00068dc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00068d20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ +00068d40: 2020 2077 2077 2020 202d 2077 2077 2020     w w   - w w  
│ │ │ │ +00068d50: 202b 2077 2077 2020 202d 2077 2077 2020   + w w   - w w  
│ │ │ │ +00068d60: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
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│ │ │ │ +00068d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068d90: 2020 2020 3720 3131 2020 2020 3320 3132      7 11    3 12
│ │ │ │ +00068da0: 2020 2020 3220 3137 2020 2020 3120 3138      2 17    1 18
│ │ │ │ +00068db0: 2020 2020 3020 3139 2020 2020 2020 2020      0 19        
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│ │ │ │  00068de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00068e10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00068e20: 2020 2020 2020 2020 2020 2020 2077 2077               w w
│ │ │ │ -00068e30: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ -00068e40: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
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│ │ │ │ +00068e30: 2020 2077 2077 2020 202d 2077 2077 2020     w w   - w w  
│ │ │ │ +00068e40: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
│ │ │ │  00068e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00068e70: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -00068e80: 3131 2020 2020 3220 3135 2020 2020 3120  11    2 15    1 
│ │ │ │ -00068e90: 3136 2020 2020 2020 2020 2020 2020 2020  16              
│ │ │ │ +00068e60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00068e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068e80: 2020 2020 3620 3131 2020 2020 3220 3135      6 11    2 15
│ │ │ │ +00068e90: 2020 2020 3120 3136 2020 2020 2020 2020      1 16        
│ │ │ │  00068ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068eb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00068eb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00068ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068f00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00068f10: 2020 2020 2020 2020 2020 2020 2077 2077               w w
│ │ │ │ -00068f20: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ -00068f30: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
│ │ │ │ +00068f00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00068f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068f20: 2020 2077 2077 2020 202d 2077 2077 2020     w w   - w w  
│ │ │ │ +00068f30: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
│ │ │ │  00068f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068f50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00068f60: 2020 2020 2020 2020 2020 2020 2020 3520                5 
│ │ │ │ -00068f70: 3131 2020 2020 3220 3134 2020 2020 3020  11    2 14    0 
│ │ │ │ -00068f80: 3136 2020 2020 2020 2020 2020 2020 2020  16              
│ │ │ │ +00068f50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00068f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068f70: 2020 2020 3520 3131 2020 2020 3220 3134      5 11    2 14
│ │ │ │ +00068f80: 2020 2020 3020 3136 2020 2020 2020 2020      0 16        
│ │ │ │  00068f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068fa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00068fa0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00068fb0: 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00069000: 2020 2020 2020 2020 2020 2020 2077 2077               w w
│ │ │ │ -00069010: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ -00069020: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
│ │ │ │ +00068ff0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069010: 2020 2077 2077 2020 202d 2077 2077 2020     w w   - w w  
│ │ │ │ +00069020: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
│ │ │ │  00069030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069040: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00069050: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ -00069060: 3131 2020 2020 3120 3134 2020 2020 3020  11    1 14    0 
│ │ │ │ -00069070: 3135 2020 2020 2020 2020 2020 2020 2020  15              
│ │ │ │ +00069040: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069060: 2020 2020 3420 3131 2020 2020 3120 3134      4 11    1 14
│ │ │ │ +00069070: 2020 2020 3020 3135 2020 2020 2020 2020      0 15        
│ │ │ │  00069080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069090: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069090: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000690a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000690b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000690c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000690d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000690e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000690f0: 2020 2020 2020 2020 2020 2020 2077 2077               w w
│ │ │ │ -00069100: 2020 2d20 7720 7720 202b 2077 2077 2020    - w w  + w w  
│ │ │ │ -00069110: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +000690e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000690f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069100: 2020 2077 2077 2020 2d20 7720 7720 202b     w w  - w w  +
│ │ │ │ +00069110: 2077 2077 2020 2c20 2020 2020 2020 2020   w w  ,         
│ │ │ │  00069120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069130: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00069140: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -00069150: 3820 2020 2035 2039 2020 2020 3420 3130  8    5 9    4 10
│ │ │ │ -00069160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069130: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069150: 2020 2020 3620 3820 2020 2035 2039 2020      6 8    5 9  
│ │ │ │ +00069160: 2020 3420 3130 2020 2020 2020 2020 2020    4 10          
│ │ │ │  00069170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069180: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069180: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00069190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000691a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000691b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000691c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000691d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000691e0: 2020 2020 2020 2020 2020 2020 2077 2077               w w
│ │ │ │ -000691f0: 2020 2d20 7720 7720 202b 2077 2077 2020    - w w  + w w  
│ │ │ │ -00069200: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +000691d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000691e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000691f0: 2020 2077 2077 2020 2d20 7720 7720 202b     w w  - w w  +
│ │ │ │ +00069200: 2077 2077 2020 2c20 2020 2020 2020 2020   w w  ,         
│ │ │ │  00069210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069220: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00069230: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -00069240: 3620 2020 2032 2039 2020 2020 3120 3130  6    2 9    1 10
│ │ │ │ -00069250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069220: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069240: 2020 2020 3320 3620 2020 2032 2039 2020      3 6    2 9  
│ │ │ │ +00069250: 2020 3120 3130 2020 2020 2020 2020 2020    1 10          
│ │ │ │  00069260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069270: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069270: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00069280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000692a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000692b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000692c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000692d0: 2020 2020 2020 2020 2020 2020 2077 2077               w w
│ │ │ │ -000692e0: 2020 2d20 7720 7720 202b 2077 2077 2020    - w w  + w w  
│ │ │ │ -000692f0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +000692c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000692d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000692e0: 2020 2077 2077 2020 2d20 7720 7720 202b     w w  - w w  +
│ │ │ │ +000692f0: 2077 2077 2020 2c20 2020 2020 2020 2020   w w  ,         
│ │ │ │  00069300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069310: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00069320: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -00069330: 3520 2020 2032 2038 2020 2020 3020 3130  5    2 8    0 10
│ │ │ │ -00069340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069310: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069330: 2020 2020 3320 3520 2020 2032 2038 2020      3 5    2 8  
│ │ │ │ +00069340: 2020 3020 3130 2020 2020 2020 2020 2020    0 10          
│ │ │ │  00069350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069360: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000693b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000693c0: 2020 2020 2020 2020 2020 2020 2077 2077               w w
│ │ │ │ -000693d0: 2020 2d20 7720 7720 202b 2077 2077 202c    - w w  + w w ,
│ │ │ │ -000693e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000693b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000693c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000693d0: 2020 2077 2077 2020 2d20 7720 7720 202b     w w  - w w  +
│ │ │ │ +000693e0: 2077 2077 202c 2020 2020 2020 2020 2020   w w ,          
│ │ │ │  000693f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069400: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00069410: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -00069420: 3420 2020 2031 2038 2020 2020 3020 3920  4    1 8    0 9 
│ │ │ │ -00069430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069400: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069420: 2020 2020 3320 3420 2020 2031 2038 2020      3 4    1 8  
│ │ │ │ +00069430: 2020 3020 3920 2020 2020 2020 2020 2020    0 9           
│ │ │ │  00069440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069450: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00069460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069470: 2020 2020 2020 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000694b0: 2020 2020 2020 2020 2020 2020 2077 2077               w w
│ │ │ │ -000694c0: 2020 2d20 7720 7720 202b 2077 2077 2020    - w w  + w w  
│ │ │ │ -000694d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000694a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000694b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000694c0: 2020 2077 2077 2020 2d20 7720 7720 202b     w w  - w w  +
│ │ │ │ +000694d0: 2077 2077 2020 2020 2020 2020 2020 2020   w w            
│ │ │ │  000694e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000694f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00069500: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00069510: 3420 2020 2031 2035 2020 2020 3020 3620  4    1 5    0 6 
│ │ │ │ -00069520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000694f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069510: 2020 2020 3220 3420 2020 2031 2035 2020      2 4    1 5  
│ │ │ │ +00069520: 2020 3020 3620 2020 2020 2020 2020 2020    0 6           
│ │ │ │  00069530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069540: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00069550: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ -00069560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069540: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069560: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │  00069570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069590: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069590: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000695a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000695b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000695c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000695d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000695e0: 2020 2020 7c0a 7c6f 3120 3a20 5261 7469      |.|o1 : Rati
│ │ │ │ -000695f0: 6f6e 616c 4d61 7020 2871 7561 6472 6174  onalMap (quadrat
│ │ │ │ -00069600: 6963 2043 7265 6d6f 6e61 2074 7261 6e73  ic Cremona trans
│ │ │ │ -00069610: 666f 726d 6174 696f 6e20 6f66 2050 505e  formation of PP^
│ │ │ │ -00069620: 3230 2920 2020 2020 2020 2020 2020 2020  20)             
│ │ │ │ -00069630: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +000695e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +000695f0: 3a20 5261 7469 6f6e 616c 4d61 7020 2871  : RationalMap (q
│ │ │ │ +00069600: 7561 6472 6174 6963 2043 7265 6d6f 6e61  uadratic Cremona
│ │ │ │ +00069610: 2074 7261 6e73 666f 726d 6174 696f 6e20   transformation 
│ │ │ │ +00069620: 6f66 2050 505e 3230 2920 2020 2020 2020  of PP^20)       
│ │ │ │ +00069630: 2020 2020 2020 2020 2020 7c0a 7c2d 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 2d2d 2d2d  ----------------
│ │ │ │  00069670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069680: 2d2d 2d2d 7c0a 7c77 2020 2c20 7720 202c  ----|.|w  , w  ,
│ │ │ │ -00069690: 2077 2020 2c20 7720 202c 2077 2020 2c20   w  , w  , w  , 
│ │ │ │ -000696a0: 7720 202c 2077 2020 2c20 7720 205d 2920  w  , w  , w  ]) 
│ │ │ │ -000696b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069680: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c77 2020  ----------|.|w  
│ │ │ │ +00069690: 2c20 7720 202c 2077 2020 2c20 7720 202c  , w  , w  , w  ,
│ │ │ │ +000696a0: 2077 2020 2c20 7720 202c 2077 2020 2c20   w  , w  , w  , 
│ │ │ │ +000696b0: 7720 205d 2920 2020 2020 2020 2020 2020  w  ])           
│ │ │ │  000696c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000696d0: 2020 2020 7c0a 7c20 3133 2020 2031 3420      |.| 13   14 
│ │ │ │ -000696e0: 2020 3135 2020 2031 3620 2020 3137 2020    15   16   17  
│ │ │ │ -000696f0: 2031 3820 2020 3139 2020 2032 3020 2020   18   19   20   
│ │ │ │ -00069700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000696d0: 2020 2020 2020 2020 2020 7c0a 7c20 3133            |.| 13
│ │ │ │ +000696e0: 2020 2031 3420 2020 3135 2020 2031 3620     14   15   16 
│ │ │ │ +000696f0: 2020 3137 2020 2031 3820 2020 3139 2020    17   18   19  
│ │ │ │ +00069700: 2032 3020 2020 2020 2020 2020 2020 2020   20             
│ │ │ │  00069710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069720: 2020 2020 7c0a 7c77 2020 2c20 7720 202c      |.|w  , w  ,
│ │ │ │ -00069730: 2077 2020 2c20 7720 202c 2077 2020 2c20   w  , w  , w  , 
│ │ │ │ -00069740: 7720 202c 2077 2020 2c20 7720 205d 2920  w  , w  , w  ]) 
│ │ │ │ -00069750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069720: 2020 2020 2020 2020 2020 7c0a 7c77 2020            |.|w  
│ │ │ │ +00069730: 2c20 7720 202c 2077 2020 2c20 7720 202c  , w  , w  , w  ,
│ │ │ │ +00069740: 2077 2020 2c20 7720 202c 2077 2020 2c20   w  , w  , w  , 
│ │ │ │ +00069750: 7720 205d 2920 2020 2020 2020 2020 2020  w  ])           
│ │ │ │  00069760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069770: 2020 2020 7c0a 7c20 3133 2020 2031 3420      |.| 13   14 
│ │ │ │ -00069780: 2020 3135 2020 2031 3620 2020 3137 2020    15   16   17  
│ │ │ │ -00069790: 2031 3820 2020 3139 2020 2032 3020 2020   18   19   20   
│ │ │ │ -000697a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069770: 2020 2020 2020 2020 2020 7c0a 7c20 3133            |.| 13
│ │ │ │ +00069780: 2020 2031 3420 2020 3135 2020 2031 3620     14   15   16 
│ │ │ │ +00069790: 2020 3137 2020 2031 3820 2020 3139 2020    17   18   19  
│ │ │ │ +000697a0: 2032 3020 2020 2020 2020 2020 2020 2020   20             
│ │ │ │  000697b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000697c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000697c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  000697d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000697e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000697f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069810: 2d2d 2d2d 2b0a 7c69 3220 3a20 7469 6d65  ----+.|i2 : time
│ │ │ │ -00069820: 2070 7369 203d 2069 6e76 6572 7365 4d61   psi = inverseMa
│ │ │ │ -00069830: 7020 7068 6920 2020 2020 2020 2020 2020  p phi           
│ │ │ │ +00069810: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00069820: 3a20 7469 6d65 2070 7369 203d 2069 6e76  : time psi = inv
│ │ │ │ +00069830: 6572 7365 4d61 7020 7068 6920 2020 2020  erseMap phi     
│ │ │ │  00069840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069860: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00069870: 302e 3135 3837 3436 7320 2863 7075 293b  0.158746s (cpu);
│ │ │ │ -00069880: 2030 2e30 3838 3534 3939 7320 2874 6872   0.0885499s (thr
│ │ │ │ -00069890: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -000698a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000698b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00069860: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ +00069870: 2075 7365 6420 302e 3138 3239 3936 7320   used 0.182996s 
│ │ │ │ +00069880: 2863 7075 293b 2030 2e31 3034 3437 3173  (cpu); 0.104471s
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│ │ │ │ +000698a0: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
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│ │ │ │  000698c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000698d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000698e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000698f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069900: 2020 2020 7c0a 7c6f 3220 3d20 2d2d 2072      |.|o2 = -- r
│ │ │ │ -00069910: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │ -00069920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069900: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00069910: 3d20 2d2d 2072 6174 696f 6e61 6c20 6d61  = -- rational ma
│ │ │ │ +00069920: 7020 2d2d 2020 2020 2020 2020 2020 2020  p --            
│ │ │ │  00069930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069950: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ -00069960: 6365 3a20 5072 6f6a 2851 515b 7720 2c20  ce: Proj(QQ[w , 
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│ │ │ │ -000699a0: 202c 2020 7c0a 7c20 2020 2020 2020 2020   ,  |.|         
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│ │ │ │ -00069a00: 6574 3a20 5072 6f6a 2851 515b 7720 2c20  et: Proj(QQ[w , 
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│ │ │ │ -00069a30: 7720 2c20 7720 202c 2077 2020 2c20 7720  w , w  , w  , w 
│ │ │ │ -00069a40: 202c 2020 7c0a 7c20 2020 2020 2020 2020   ,  |.|         
│ │ │ │ -00069a50: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00069a60: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
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│ │ │ │ -00069a80: 2039 2020 2031 3020 2020 3131 2020 2031   9   10   11   1
│ │ │ │ -00069a90: 3220 2020 7c0a 7c20 2020 2020 6465 6669  2   |.|     defi
│ │ │ │ -00069aa0: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │ -00069ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069950: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069960: 2020 736f 7572 6365 3a20 5072 6f6a 2851    source: Proj(Q
│ │ │ │ +00069970: 515b 7720 2c20 7720 2c20 7720 2c20 7720  Q[w , w , w , w 
│ │ │ │ +00069980: 2c20 7720 2c20 7720 2c20 7720 2c20 7720  , w , w , w , w 
│ │ │ │ +00069990: 2c20 7720 2c20 7720 2c20 7720 202c 2077  , w , w , w  , w
│ │ │ │ +000699a0: 2020 2c20 7720 202c 2020 7c0a 7c20 2020    , w  ,  |.|   
│ │ │ │ +000699b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000699c0: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
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│ │ │ │ +000699f0: 3131 2020 2031 3220 2020 7c0a 7c20 2020  11   12   |.|   
│ │ │ │ +00069a00: 2020 7461 7267 6574 3a20 5072 6f6a 2851    target: Proj(Q
│ │ │ │ +00069a10: 515b 7720 2c20 7720 2c20 7720 2c20 7720  Q[w , w , w , w 
│ │ │ │ +00069a20: 2c20 7720 2c20 7720 2c20 7720 2c20 7720  , w , w , w , w 
│ │ │ │ +00069a30: 2c20 7720 2c20 7720 2c20 7720 202c 2077  , w , w , w  , w
│ │ │ │ +00069a40: 2020 2c20 7720 202c 2020 7c0a 7c20 2020    , w  ,  |.|   
│ │ │ │ +00069a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069a60: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
│ │ │ │ +00069a70: 2020 2034 2020 2035 2020 2036 2020 2037     4   5   6   7
│ │ │ │ +00069a80: 2020 2038 2020 2039 2020 2031 3020 2020     8   9   10   
│ │ │ │ +00069a90: 3131 2020 2031 3220 2020 7c0a 7c20 2020  11   12   |.|   
│ │ │ │ +00069aa0: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ +00069ab0: 3a20 7b20 2020 2020 2020 2020 2020 2020  : {             
│ │ │ │  00069ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00069ed0: 7720 7720 202c 2020 2020 2020 2020 2020  w w  ,          
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│ │ │ │ +00069f20: 2031 3920 2020 2032 2032 3020 2020 2020   19    2 20     
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│ │ │ │ -00069fb0: 2020 7720 2020 2b20 7720 7720 2020 2d20    w   + w w   - 
│ │ │ │ -00069fc0: 7720 7720 202c 2020 2020 2020 2020 2020  w w  ,          
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│ │ │ │ +00069fb0: 2020 202d 2077 2020 7720 2020 2b20 7720     - w  w   + w 
│ │ │ │ +00069fc0: 7720 2020 2d20 7720 7720 202c 2020 2020  w   - w w  ,    
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│ │ │ │ -0006a100: 2030 2032 3020 2020 2020 2020 2020 2020   0 20           
│ │ │ │ +0006a0f0: 2020 2020 2020 3133 2031 3620 2020 2033        13 16    3
│ │ │ │ +0006a100: 2031 3720 2020 2030 2032 3020 2020 2020   17    0 20     
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│ │ │ │  0006a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a170: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a180: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006a190: 2020 7720 2020 2d20 7720 7720 2020 2b20    w   - w w   + 
│ │ │ │ -0006a1a0: 7720 7720 2020 2d20 7720 7720 2020 2d20  w w   - w w   - 
│ │ │ │ -0006a1b0: 7720 7720 202c 2020 2020 2020 2020 2020  w w  ,          
│ │ │ │ -0006a1c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a170: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a190: 2020 202d 2077 2020 7720 2020 2d20 7720     - w  w   - w 
│ │ │ │ +0006a1a0: 7720 2020 2b20 7720 7720 2020 2d20 7720  w   + w w   - w 
│ │ │ │ +0006a1b0: 7720 2020 2d20 7720 7720 202c 2020 2020  w   - w w  ,    
│ │ │ │ +0006a1c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a1e0: 3132 2031 3620 2020 2038 2031 3720 2020  12 16    8 17   
│ │ │ │ -0006a1f0: 2037 2031 3820 2020 2036 2031 3920 2020   7 18    6 19   
│ │ │ │ -0006a200: 2035 2032 3020 2020 2020 2020 2020 2020   5 20           
│ │ │ │ -0006a210: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a1e0: 2020 2020 2020 3132 2031 3620 2020 2038        12 16    8
│ │ │ │ +0006a1f0: 2031 3720 2020 2037 2031 3820 2020 2036   17    7 18    6
│ │ │ │ +0006a200: 2031 3920 2020 2035 2032 3020 2020 2020   19    5 20     
│ │ │ │ +0006a210: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a260: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a270: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006a280: 2020 7720 2020 2b20 7720 7720 2020 2d20    w   + w w   - 
│ │ │ │ -0006a290: 7720 7720 202c 2020 2020 2020 2020 2020  w w  ,          
│ │ │ │ +0006a260: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a280: 2020 202d 2077 2020 7720 2020 2b20 7720     - w  w   + w 
│ │ │ │ +0006a290: 7720 2020 2d20 7720 7720 202c 2020 2020  w   - w w  ,    
│ │ │ │  0006a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a2b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a2b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a2d0: 3131 2031 3620 2020 2032 2031 3820 2020  11 16    2 18   
│ │ │ │ -0006a2e0: 2031 2031 3920 2020 2020 2020 2020 2020   1 19           
│ │ │ │ +0006a2d0: 2020 2020 2020 3131 2031 3620 2020 2032        11 16    2
│ │ │ │ +0006a2e0: 2031 3820 2020 2031 2031 3920 2020 2020   18    1 19     
│ │ │ │  0006a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a300: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a300: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a350: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a360: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006a370: 2020 7720 2020 2b20 7720 7720 2020 2d20    w   + w w   - 
│ │ │ │ -0006a380: 7720 7720 202c 2020 2020 2020 2020 2020  w w  ,          
│ │ │ │ +0006a350: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a370: 2020 202d 2077 2020 7720 2020 2b20 7720     - w  w   + w 
│ │ │ │ +0006a380: 7720 2020 2d20 7720 7720 202c 2020 2020  w   - w w  ,    
│ │ │ │  0006a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a3a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a3c0: 3130 2031 3620 2020 2032 2031 3720 2020  10 16    2 17   
│ │ │ │ -0006a3d0: 2030 2031 3920 2020 2020 2020 2020 2020   0 19           
│ │ │ │ +0006a3c0: 2020 2020 2020 3130 2031 3620 2020 2032        10 16    2
│ │ │ │ +0006a3d0: 2031 3720 2020 2030 2031 3920 2020 2020   17    0 19     
│ │ │ │  0006a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a3f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a3f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a440: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a450: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006a460: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ -0006a470: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │ +0006a440: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a460: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +0006a470: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │  0006a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a490: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a490: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a4b0: 3920 3136 2020 2020 3120 3137 2020 2020  9 16    1 17    
│ │ │ │ -0006a4c0: 3020 3138 2020 2020 2020 2020 2020 2020  0 18            
│ │ │ │ +0006a4b0: 2020 2020 2020 3920 3136 2020 2020 3120        9 16    1 
│ │ │ │ +0006a4c0: 3137 2020 2020 3020 3138 2020 2020 2020  17    0 18      
│ │ │ │  0006a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a4e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a4e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a530: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a540: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006a550: 2020 7720 2020 2b20 7720 2077 2020 202d    w   + w  w   -
│ │ │ │ -0006a560: 2077 2077 2020 2c20 2020 2020 2020 2020   w w  ,         
│ │ │ │ +0006a530: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a550: 2020 202d 2077 2020 7720 2020 2b20 7720     - w  w   + w 
│ │ │ │ +0006a560: 2077 2020 202d 2077 2077 2020 2c20 2020   w   - w w  ,   
│ │ │ │  0006a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a580: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a580: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a5a0: 3131 2031 3320 2020 2031 3020 3134 2020  11 13    10 14  
│ │ │ │ -0006a5b0: 2020 3920 3135 2020 2020 2020 2020 2020    9 15          
│ │ │ │ +0006a5a0: 2020 2020 2020 3131 2031 3320 2020 2031        11 13    1
│ │ │ │ +0006a5b0: 3020 3134 2020 2020 3920 3135 2020 2020  0 14    9 15    
│ │ │ │  0006a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a5d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a5d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a630: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006a640: 2077 2020 202d 2077 2077 2020 202b 2077   w   - w w   + w
│ │ │ │ -0006a650: 2077 2020 202d 2077 2077 2020 202d 2077   w   - w w   - w
│ │ │ │ -0006a660: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │ -0006a670: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a620: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a640: 2020 202d 2077 2077 2020 202d 2077 2077     - w w   - w w
│ │ │ │ +0006a650: 2020 202b 2077 2077 2020 202d 2077 2077     + w w   - w w
│ │ │ │ +0006a660: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │ +0006a670: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a690: 3320 3132 2020 2020 3820 3133 2020 2020  3 12    8 13    
│ │ │ │ -0006a6a0: 3720 3134 2020 2020 3620 3135 2020 2020  7 14    6 15    
│ │ │ │ -0006a6b0: 3420 3230 2020 2020 2020 2020 2020 2020  4 20            
│ │ │ │ -0006a6c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a690: 2020 2020 2020 3320 3132 2020 2020 3820        3 12    8 
│ │ │ │ +0006a6a0: 3133 2020 2020 3720 3134 2020 2020 3620  13    7 14    6 
│ │ │ │ +0006a6b0: 3135 2020 2020 3420 3230 2020 2020 2020  15    4 20      
│ │ │ │ +0006a6c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a710: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a720: 2020 2020 2020 2020 2020 2020 2077 2077               w w
│ │ │ │ -0006a730: 2020 2b20 7720 7720 202d 2077 2077 2020    + w w  - w w  
│ │ │ │ -0006a740: 2b20 7720 7720 202d 2077 2077 2020 2c20  + w w  - w w  , 
│ │ │ │ -0006a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a760: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a770: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -0006a780: 3520 2020 2032 2036 2020 2020 3120 3720  5    2 6    1 7 
│ │ │ │ -0006a790: 2020 2030 2038 2020 2020 3420 3136 2020     0 8    4 16  
│ │ │ │ -0006a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a7b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a710: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a730: 2020 2077 2077 2020 2b20 7720 7720 202d     w w  + w w  -
│ │ │ │ +0006a740: 2077 2077 2020 2b20 7720 7720 202d 2077   w w  + w w  - w
│ │ │ │ +0006a750: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │ +0006a760: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a780: 2020 2020 3320 3520 2020 2032 2036 2020      3 5    2 6  
│ │ │ │ +0006a790: 2020 3120 3720 2020 2030 2038 2020 2020    1 7    0 8    
│ │ │ │ +0006a7a0: 3420 3136 2020 2020 2020 2020 2020 2020  4 16            
│ │ │ │ +0006a7b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a810: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006a820: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ -0006a830: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │ +0006a800: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a820: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +0006a830: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │  0006a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a850: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a870: 3320 3131 2020 2020 3220 3134 2020 2020  3 11    2 14    
│ │ │ │ -0006a880: 3120 3135 2020 2020 2020 2020 2020 2020  1 15            
│ │ │ │ +0006a870: 2020 2020 2020 3320 3131 2020 2020 3220        3 11    2 
│ │ │ │ +0006a880: 3134 2020 2020 3120 3135 2020 2020 2020  14    1 15      
│ │ │ │  0006a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a8a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a8a0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020                  
│ │ │ │ -0006a8f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a900: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006a910: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ -0006a920: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │ +0006a8f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a910: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +0006a920: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │  0006a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a940: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a940: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a960: 3320 3130 2020 2020 3220 3133 2020 2020  3 10    2 13    
│ │ │ │ -0006a970: 3020 3135 2020 2020 2020 2020 2020 2020  0 15            
│ │ │ │ +0006a960: 2020 2020 2020 3320 3130 2020 2020 3220        3 10    2 
│ │ │ │ +0006a970: 3133 2020 2020 3020 3135 2020 2020 2020  13    0 15      
│ │ │ │  0006a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a990: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006a990: 2020 2020 2020 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a9e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006a9f0: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006aa00: 2077 2020 2b20 7720 7720 2020 2d20 7720   w  + w w   - w 
│ │ │ │ -0006aa10: 7720 202c 2020 2020 2020 2020 2020 2020  w  ,            
│ │ │ │ +0006a9e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006aa00: 2020 202d 2077 2077 2020 2b20 7720 7720     - w w  + w w 
│ │ │ │ +0006aa10: 2020 2d20 7720 7720 202c 2020 2020 2020    - w w  ,      
│ │ │ │  0006aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aa30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006aa30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aa50: 3320 3920 2020 2031 2031 3320 2020 2030  3 9    1 13    0
│ │ │ │ -0006aa60: 2031 3420 2020 2020 2020 2020 2020 2020   14             
│ │ │ │ +0006aa50: 2020 2020 2020 3320 3920 2020 2031 2031        3 9    1 1
│ │ │ │ +0006aa60: 3320 2020 2030 2031 3420 2020 2020 2020  3    0 14       
│ │ │ │  0006aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aa80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006aa80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006aad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006aae0: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006aaf0: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ -0006ab00: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ -0006ab10: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │ -0006ab20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006aad0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006aaf0: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +0006ab00: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +0006ab10: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │ +0006ab20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ab40: 3820 3130 2020 2020 3720 3131 2020 2020  8 10    7 11    
│ │ │ │ -0006ab50: 3220 3132 2020 2020 3520 3135 2020 2020  2 12    5 15    
│ │ │ │ -0006ab60: 3420 3139 2020 2020 2020 2020 2020 2020  4 19            
│ │ │ │ -0006ab70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006ab40: 2020 2020 2020 3820 3130 2020 2020 3720        8 10    7 
│ │ │ │ +0006ab50: 3131 2020 2020 3220 3132 2020 2020 3520  11    2 12    5 
│ │ │ │ +0006ab60: 3135 2020 2020 3420 3139 2020 2020 2020  15    4 19      
│ │ │ │ +0006ab70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ab90: 2020 2020 2020 2020 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006abd0: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006abe0: 2077 2020 2b20 7720 7720 2020 2d20 7720   w  + w w   - w 
│ │ │ │ -0006abf0: 7720 2020 2b20 7720 7720 2020 2d20 7720  w   + w w   - w 
│ │ │ │ -0006ac00: 7720 202c 2020 2020 2020 2020 2020 2020  w  ,            
│ │ │ │ -0006ac10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006abc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006abe0: 2020 202d 2077 2077 2020 2b20 7720 7720     - w w  + w w 
│ │ │ │ +0006abf0: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +0006ac00: 2020 2d20 7720 7720 202c 2020 2020 2020    - w w  ,      
│ │ │ │ +0006ac10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ac30: 3820 3920 2020 2036 2031 3120 2020 2031  8 9    6 11    1
│ │ │ │ -0006ac40: 2031 3220 2020 2035 2031 3420 2020 2034   12    5 14    4
│ │ │ │ -0006ac50: 2031 3820 2020 2020 2020 2020 2020 2020   18             
│ │ │ │ -0006ac60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006ac30: 2020 2020 2020 3820 3920 2020 2036 2031        8 9    6 1
│ │ │ │ +0006ac40: 3120 2020 2031 2031 3220 2020 2035 2031  1    1 12    5 1
│ │ │ │ +0006ac50: 3420 2020 2034 2031 3820 2020 2020 2020  4    4 18       
│ │ │ │ +0006ac60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006acb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006acc0: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006acd0: 2077 2020 2b20 7720 7720 2020 2d20 7720   w  + w w   - w 
│ │ │ │ -0006ace0: 7720 2020 2b20 7720 7720 2020 2d20 7720  w   + w w   - w 
│ │ │ │ -0006acf0: 7720 202c 2020 2020 2020 2020 2020 2020  w  ,            
│ │ │ │ -0006ad00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006acb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006acd0: 2020 202d 2077 2077 2020 2b20 7720 7720     - w w  + w w 
│ │ │ │ +0006ace0: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +0006acf0: 2020 2d20 7720 7720 202c 2020 2020 2020    - w w  ,      
│ │ │ │ +0006ad00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ad20: 3720 3920 2020 2036 2031 3020 2020 2030  7 9    6 10    0
│ │ │ │ -0006ad30: 2031 3220 2020 2035 2031 3320 2020 2034   12    5 13    4
│ │ │ │ -0006ad40: 2031 3720 2020 2020 2020 2020 2020 2020   17             
│ │ │ │ -0006ad50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006ad20: 2020 2020 2020 3720 3920 2020 2036 2031        7 9    6 1
│ │ │ │ +0006ad30: 3020 2020 2030 2031 3220 2020 2035 2031  0    0 12    5 1
│ │ │ │ +0006ad40: 3320 2020 2034 2031 3720 2020 2020 2020  3    4 17       
│ │ │ │ +0006ad50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ada0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006adb0: 2020 2020 2020 2020 2020 2020 202d 2077               - w
│ │ │ │ -0006adc0: 2077 2020 2b20 7720 7720 2020 2d20 7720   w  + w w   - w 
│ │ │ │ -0006add0: 7720 2020 2020 2020 2020 2020 2020 2020  w               
│ │ │ │ +0006ada0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006adc0: 2020 202d 2077 2077 2020 2b20 7720 7720     - w w  + w w 
│ │ │ │ +0006add0: 2020 2d20 7720 7720 2020 2020 2020 2020    - w w         
│ │ │ │  0006ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006adf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006adf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae10: 3220 3920 2020 2031 2031 3020 2020 2030  2 9    1 10    0
│ │ │ │ -0006ae20: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │ +0006ae10: 2020 2020 2020 3220 3920 2020 2031 2031        2 9    1 1
│ │ │ │ +0006ae20: 3020 2020 2030 2031 3120 2020 2020 2020  0    0 11       
│ │ │ │  0006ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006ae50: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ -0006ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ae40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ae60: 2020 7d20 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006ae90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c6f 3220 3a20 5261 7469      |.|o2 : Rati
│ │ │ │ -0006aef0: 6f6e 616c 4d61 7020 2871 7561 6472 6174  onalMap (quadrat
│ │ │ │ -0006af00: 6963 2072 6174 696f 6e61 6c20 6d61 7020  ic rational map 
│ │ │ │ -0006af10: 6672 6f6d 2050 505e 3230 2074 6f20 5050  from PP^20 to PP
│ │ │ │ -0006af20: 5e32 3029 2020 2020 2020 2020 2020 2020  ^20)            
│ │ │ │ -0006af30: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +0006aee0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0006aef0: 3a20 5261 7469 6f6e 616c 4d61 7020 2871  : RationalMap (q
│ │ │ │ +0006af00: 7561 6472 6174 6963 2072 6174 696f 6e61  uadratic rationa
│ │ │ │ +0006af10: 6c20 6d61 7020 6672 6f6d 2050 505e 3230  l map from PP^20
│ │ │ │ +0006af20: 2074 6f20 5050 5e32 3029 2020 2020 2020   to PP^20)      
│ │ │ │ +0006af30: 2020 2020 2020 2020 2020 7c0a 7c2d 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 7c0a 7c77 2020 2c20 7720 202c  ----|.|w  , w  ,
│ │ │ │ -0006af90: 2077 2020 2c20 7720 202c 2077 2020 2c20   w  , w  , w  , 
│ │ │ │ -0006afa0: 7720 202c 2077 2020 2c20 7720 205d 2920  w  , w  , w  ]) 
│ │ │ │ -0006afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006af80: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c77 2020  ----------|.|w  
│ │ │ │ +0006af90: 2c20 7720 202c 2077 2020 2c20 7720 202c  , w  , w  , w  ,
│ │ │ │ +0006afa0: 2077 2020 2c20 7720 202c 2077 2020 2c20   w  , w  , w  , 
│ │ │ │ +0006afb0: 7720 205d 2920 2020 2020 2020 2020 2020  w  ])           
│ │ │ │  0006afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006afd0: 2020 2020 7c0a 7c20 3133 2020 2031 3420      |.| 13   14 
│ │ │ │ -0006afe0: 2020 3135 2020 2031 3620 2020 3137 2020    15   16   17  
│ │ │ │ -0006aff0: 2031 3820 2020 3139 2020 2032 3020 2020   18   19   20   
│ │ │ │ -0006b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006afd0: 2020 2020 2020 2020 2020 7c0a 7c20 3133            |.| 13
│ │ │ │ +0006afe0: 2020 2031 3420 2020 3135 2020 2031 3620     14   15   16 
│ │ │ │ +0006aff0: 2020 3137 2020 2031 3820 2020 3139 2020    17   18   19  
│ │ │ │ +0006b000: 2032 3020 2020 2020 2020 2020 2020 2020   20             
│ │ │ │  0006b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b020: 2020 2020 7c0a 7c77 2020 2c20 7720 202c      |.|w  , w  ,
│ │ │ │ -0006b030: 2077 2020 2c20 7720 202c 2077 2020 2c20   w  , w  , w  , 
│ │ │ │ -0006b040: 7720 202c 2077 2020 2c20 7720 205d 2920  w  , w  , w  ]) 
│ │ │ │ -0006b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b020: 2020 2020 2020 2020 2020 7c0a 7c77 2020            |.|w  
│ │ │ │ +0006b030: 2c20 7720 202c 2077 2020 2c20 7720 202c  , w  , w  , w  ,
│ │ │ │ +0006b040: 2077 2020 2c20 7720 202c 2077 2020 2c20   w  , w  , w  , 
│ │ │ │ +0006b050: 7720 205d 2920 2020 2020 2020 2020 2020  w  ])           
│ │ │ │  0006b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b070: 2020 2020 7c0a 7c20 3133 2020 2031 3420      |.| 13   14 
│ │ │ │ -0006b080: 2020 3135 2020 2031 3620 2020 3137 2020    15   16   17  
│ │ │ │ -0006b090: 2031 3820 2020 3139 2020 2032 3020 2020   18   19   20   
│ │ │ │ -0006b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b070: 2020 2020 2020 2020 2020 7c0a 7c20 3133            |.| 13
│ │ │ │ +0006b080: 2020 2031 3420 2020 3135 2020 2031 3620     14   15   16 
│ │ │ │ +0006b090: 2020 3137 2020 2031 3820 2020 3139 2020    17   18   19  
│ │ │ │ +0006b0a0: 2032 3020 2020 2020 2020 2020 2020 2020   20             
│ │ │ │  0006b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b0c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006b0c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  0006b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b110: 2d2d 2d2d 2b0a 7c69 3320 3a20 6173 7365  ----+.|i3 : asse
│ │ │ │ -0006b120: 7274 2870 6869 202a 2070 7369 203d 3d20  rt(phi * psi == 
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│ │ │ │ +0006b120: 3a20 6173 7365 7274 2870 6869 202a 2070  : assert(phi * p
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│ │ │ │  0006b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0006b170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b1b0: 2d2d 2d2d 2b0a 0a54 6865 206d 6574 686f  ----+..The metho
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│ │ │ │ -0006b210: 7261 7469 6f6e 616c 206d 6170 2024 5c50  rational map $\P
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│ │ │ │ -0006b240: 2e20 496e 2074 6869 7320 6361 7365 2c0a  . In this case,.
│ │ │ │ -0006b250: 7468 6520 2a6e 6f74 6520 7269 6e67 206d  the *note ring m
│ │ │ │ -0006b260: 6170 3a20 284d 6163 6175 6c61 7932 446f  ap: (Macaulay2Do
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│ │ │ │ -0006b280: 696e 6720 245c 5068 695e 7b2d 317d 2420  ing $\Phi^{-1}$ 
│ │ │ │ -0006b290: 6973 2072 6574 7572 6e65 642e 0a0a 2b2d  is returned...+-
│ │ │ │ -0006b2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6865  ----------+..The
│ │ │ │ +0006b1c0: 206d 6574 686f 6420 616c 736f 2061 6363   method also acc
│ │ │ │ +0006b1d0: 6570 7473 2061 7320 696e 7075 7420 6120  epts as input a 
│ │ │ │ +0006b1e0: 2a6e 6f74 6520 7269 6e67 206d 6170 3a20  *note ring map: 
│ │ │ │ +0006b1f0: 284d 6163 6175 6c61 7932 446f 6329 5269  (Macaulay2Doc)Ri
│ │ │ │ +0006b200: 6e67 4d61 702c 0a72 6570 7265 7365 6e74  ngMap,.represent
│ │ │ │ +0006b210: 696e 6720 6120 7261 7469 6f6e 616c 206d  ing a rational m
│ │ │ │ +0006b220: 6170 2024 5c50 6869 2420 6265 7477 6565  ap $\Phi$ betwee
│ │ │ │ +0006b230: 6e20 7072 6f6a 6563 7469 7665 2076 6172  n projective var
│ │ │ │ +0006b240: 6965 7469 6573 2e20 496e 2074 6869 7320  ieties. In this 
│ │ │ │ +0006b250: 6361 7365 2c0a 7468 6520 2a6e 6f74 6520  case,.the *note 
│ │ │ │ +0006b260: 7269 6e67 206d 6170 3a20 284d 6163 6175  ring map: (Macau
│ │ │ │ +0006b270: 6c61 7932 446f 6329 5269 6e67 4d61 702c  lay2Doc)RingMap,
│ │ │ │ +0006b280: 2064 6566 696e 696e 6720 245c 5068 695e   defining $\Phi^
│ │ │ │ +0006b290: 7b2d 317d 2420 6973 2072 6574 7572 6e65  {-1}$ is returne
│ │ │ │ +0006b2a0: 642e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  d...+-----------
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│ │ │ │  0006b2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
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│ │ │ │ -0006b300: 2074 7261 6e73 666f 726d 6174 696f 6e20   transformation 
│ │ │ │ -0006b310: 6f66 2050 5e32 3620 2020 2020 2020 2020  of P^26         
│ │ │ │ +0006b2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006b2f0: 2d2d 2b0a 7c69 3420 3a20 2d2d 2041 2043  --+.|i4 : -- A C
│ │ │ │ +0006b300: 7265 6d6f 6e61 2074 7261 6e73 666f 726d  remona transform
│ │ │ │ +0006b310: 6174 696f 6e20 6f66 2050 5e32 3620 2020  ation of P^26   
│ │ │ │  0006b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b330: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0006b340: 2020 2020 7068 6920 3d20 6d61 7020 7175      phi = map qu
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│ │ │ │ -0006b370: 2832 362c 3129 2020 2020 2020 2020 2020  (26,1)          
│ │ │ │ -0006b380: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0006b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b340: 2020 7c0a 7c20 2020 2020 7068 6920 3d20    |.|     phi = 
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│ │ │ │ +0006b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b390: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0006b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0006b3e0: 3420 3d20 6d61 7020 2851 515b 7720 2e2e  4 = map (QQ[w ..
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│ │ │ │ -0006b400: 5d2c 207b 7720 2077 2020 202d 2077 2020  ], {w  w   - w  
│ │ │ │ -0006b410: 7720 2020 2d20 7720 2077 2020 202d 2077  w   - w  w   - w
│ │ │ │ -0006b420: 2020 7720 2020 2d20 2020 2020 7c0a 7c20    w   -     |.| 
│ │ │ │ -0006b430: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -0006b440: 2032 3620 2020 2020 2020 3020 2020 3236   26       0   26
│ │ │ │ -0006b450: 2020 2020 2032 3120 3232 2020 2020 3230       21 22    20
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│ │ │ │ -0006b470: 3130 2032 3520 2020 2020 2020 7c0a 7c20  10 25       |.| 
│ │ │ │ -0006b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b3e0: 2020 7c0a 7c6f 3420 3d20 6d61 7020 2851    |.|o4 = map (Q
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│ │ │ │  0006b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0006b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0006b4d0: 3420 3a20 5269 6e67 4d61 7020 5151 5b77  4 : RingMap QQ[w
│ │ │ │ -0006b4e0: 202e 2e77 2020 5d20 3c2d 2d20 5151 5b77   ..w  ] <-- QQ[w
│ │ │ │ -0006b4f0: 202e 2e77 2020 5d20 2020 2020 2020 2020   ..w  ]         
│ │ │ │ +0006b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b4d0: 2020 7c0a 7c6f 3420 3a20 5269 6e67 4d61    |.|o4 : RingMa
│ │ │ │ +0006b4e0: 7020 5151 5b77 202e 2e77 2020 5d20 3c2d  p QQ[w ..w  ] <-
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│ │ │ │ -0006b6f0: 202d 2077 2020 7720 2020 2b20 7c0a 7c20   - w  w   + |.| 
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│ │ │ │ -0006b740: 2020 2020 3131 2032 3120 2020 7c0a 7c2d      11 21   |.|-
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│ │ │ │ +0006b6e0: 7720 2077 2020 202d 2077 2077 2020 2c20  w  w   - w w  , 
│ │ │ │ +0006b6f0: 7720 2077 2020 202d 2077 2020 7720 2020  w  w   - w  w   
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│ │ │ │ -0006b7b0: 2077 2077 2020 2c20 7720 7720 2020 2d20   w w  , w w   - 
│ │ │ │ -0006b7c0: 7720 7720 2020 2b20 7720 7720 2020 2b20  w w   + w w   + 
│ │ │ │ -0006b7d0: 7720 7720 2020 2b20 7720 7720 202c 2077  w w   + w w  , w
│ │ │ │ -0006b7e0: 2020 7720 2020 2d20 2020 2020 7c0a 7c20    w   -     |.| 
│ │ │ │ -0006b7f0: 3132 2032 3320 2020 2031 3320 3234 2020  12 23    13 24  
│ │ │ │ -0006b800: 2020 3420 3236 2020 2030 2031 3920 2020    4 26   0 19   
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│ │ │ │ -0006b820: 2033 2032 3420 2020 2034 2032 3520 2020   3 24    4 25   
│ │ │ │ -0006b830: 3135 2031 3820 2020 2020 2020 7c0a 7c2d  15 18       |.|-
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│ │ │ │ +0006b790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0006b800: 3320 3234 2020 2020 3420 3236 2020 2030  3 24    4 26   0
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│ │ │ │ -0006c410: 2020 7720 2020 2d20 2020 2020 7c0a 7c20    w   -     |.| 
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│ │ │ │ -0006c4d0: 202e 2e77 2020 5d20 3c2d 2d20 5151 5b77   ..w  ] <-- QQ[w
│ │ │ │ -0006c4e0: 202e 2e77 2020 5d20 2020 2020 2020 2020   ..w  ]         
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│ │ │ │ +0006d450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0006dec0: 7269 6e67 5f72 702c 0a20 202a 2022 6669  ring_rp,.  * "fi
│ │ │ │ -0006ded0: 6e64 5072 6f67 7261 6d28 2e2e 2e2c 5665  ndProgram(...,Ve
│ │ │ │ -0006dee0: 7262 6f73 653d 3e2e 2e2e 2922 202d 2d20  rbose=>...)" -- 
│ │ │ │ -0006def0: 7365 6520 2a6e 6f74 6520 6669 6e64 5072  see *note findPr
│ │ │ │ -0006df00: 6f67 7261 6d3a 0a20 2020 2028 4d61 6361  ogram:.    (Maca
│ │ │ │ -0006df10: 756c 6179 3244 6f63 2966 696e 6450 726f  ulay2Doc)findPro
│ │ │ │ -0006df20: 6772 616d 2c20 2d2d 206c 6f61 6420 6578  gram, -- load ex
│ │ │ │ -0006df30: 7465 726e 616c 2070 726f 6772 616d 0a20  ternal program. 
│ │ │ │ -0006df40: 202a 2022 696e 7374 616c 6c50 6163 6b61   * "installPacka
│ │ │ │ -0006df50: 6765 282e 2e2e 2c56 6572 626f 7365 3d3e  ge(...,Verbose=>
│ │ │ │ -0006df60: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f  ...)" -- see *no
│ │ │ │ -0006df70: 7465 2069 6e73 7461 6c6c 5061 636b 6167  te installPackag
│ │ │ │ -0006df80: 653a 0a20 2020 2028 4d61 6361 756c 6179  e:.    (Macaulay
│ │ │ │ -0006df90: 3244 6f63 2969 6e73 7461 6c6c 5061 636b  2Doc)installPack
│ │ │ │ -0006dfa0: 6167 652c 202d 2d20 6c6f 6164 2061 6e64  age, -- load and
│ │ │ │ -0006dfb0: 2069 6e73 7461 6c6c 2061 2070 6163 6b61   install a packa
│ │ │ │ -0006dfc0: 6765 2061 6e64 2069 7473 0a20 2020 2064  ge and its.    d
│ │ │ │ -0006dfd0: 6f63 756d 656e 7461 7469 6f6e 0a20 202a  ocumentation.  *
│ │ │ │ -0006dfe0: 2022 6170 7072 6f78 696d 6174 6549 6e76   "approximateInv
│ │ │ │ -0006dff0: 6572 7365 4d61 7028 2e2e 2e2c 5665 7262  erseMap(...,Verb
│ │ │ │ -0006e000: 6f73 653d 3e2e 2e2e 2922 0a20 202a 2022  ose=>...)".  * "
│ │ │ │ -0006e010: 4368 6572 6e53 6368 7761 7274 7a4d 6163  ChernSchwartzMac
│ │ │ │ -0006e020: 5068 6572 736f 6e28 2e2e 2e2c 5665 7262  Pherson(...,Verb
│ │ │ │ -0006e030: 6f73 653d 3e2e 2e2e 2922 0a20 202a 2022  ose=>...)".  * "
│ │ │ │ -0006e040: 6465 6772 6565 4d61 7028 2e2e 2e2c 5665  degreeMap(...,Ve
│ │ │ │ -0006e050: 7262 6f73 653d 3e2e 2e2e 2922 0a20 202a  rbose=>...)".  *
│ │ │ │ -0006e060: 2022 4575 6c65 7243 6861 7261 6374 6572   "EulerCharacter
│ │ │ │ -0006e070: 6973 7469 6328 2e2e 2e2c 5665 7262 6f73  istic(...,Verbos
│ │ │ │ -0006e080: 653d 3e2e 2e2e 2922 0a20 202a 202a 6e6f  e=>...)".  * *no
│ │ │ │ -0006e090: 7465 2069 6e76 6572 7365 4d61 7028 2e2e  te inverseMap(..
│ │ │ │ -0006e0a0: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 293a  .,Verbose=>...):
│ │ │ │ -0006e0b0: 0a20 2020 2069 6e76 6572 7365 4d61 705f  .    inverseMap_
│ │ │ │ -0006e0c0: 6c70 5f70 645f 7064 5f70 645f 636d 5665  lp_pd_pd_pd_cmVe
│ │ │ │ -0006e0d0: 7262 6f73 653d 3e5f 7064 5f70 645f 7064  rbose=>_pd_pd_pd
│ │ │ │ -0006e0e0: 5f72 702c 0a20 202a 2022 6973 4269 7261  _rp,.  * "isBira
│ │ │ │ -0006e0f0: 7469 6f6e 616c 282e 2e2e 2c56 6572 626f  tional(...,Verbo
│ │ │ │ -0006e100: 7365 3d3e 2e2e 2e29 220a 2020 2a20 2269  se=>...)".  * "i
│ │ │ │ -0006e110: 7344 6f6d 696e 616e 7428 2e2e 2e2c 5665  sDominant(...,Ve
│ │ │ │ -0006e120: 7262 6f73 653d 3e2e 2e2e 2922 0a20 202a  rbose=>...)".  *
│ │ │ │ -0006e130: 2022 7072 6f6a 6563 7469 7665 4465 6772   "projectiveDegr
│ │ │ │ -0006e140: 6565 7328 2e2e 2e2c 5665 7262 6f73 653d  ees(...,Verbose=
│ │ │ │ -0006e150: 3e2e 2e2e 2922 0a20 202a 2022 5365 6772  >...)".  * "Segr
│ │ │ │ -0006e160: 6543 6c61 7373 282e 2e2e 2c56 6572 626f  eClass(...,Verbo
│ │ │ │ -0006e170: 7365 3d3e 2e2e 2e29 220a 2020 2a20 2269  se=>...)".  * "i
│ │ │ │ -0006e180: 7349 736f 6d6f 7270 6869 6328 2e2e 2e2c  sIsomorphic(...,
│ │ │ │ -0006e190: 5665 7262 6f73 653d 3e2e 2e2e 2922 202d  Verbose=>...)" -
│ │ │ │ -0006e1a0: 2d20 7365 6520 2a6e 6f74 6520 6973 4973  - see *note isIs
│ │ │ │ -0006e1b0: 6f6d 6f72 7068 6963 3a0a 2020 2020 2849  omorphic:.    (I
│ │ │ │ -0006e1c0: 736f 6d6f 7270 6869 736d 2969 7349 736f  somorphism)isIso
│ │ │ │ -0006e1d0: 6d6f 7270 6869 632c 202d 2d20 5072 6f62  morphic, -- Prob
│ │ │ │ -0006e1e0: 6162 696c 6973 7469 6320 7465 7374 2066  abilistic test f
│ │ │ │ -0006e1f0: 6f72 2069 736f 6d6f 7270 6869 736d 206f  or isomorphism o
│ │ │ │ -0006e200: 6620 6d6f 6475 6c65 730a 2020 2a20 226d  f modules.  * "m
│ │ │ │ -0006e210: 6f76 6546 696c 6528 2e2e 2e2c 5665 7262  oveFile(...,Verb
│ │ │ │ -0006e220: 6f73 653d 3e2e 2e2e 2922 202d 2d20 7365  ose=>...)" -- se
│ │ │ │ -0006e230: 6520 2a6e 6f74 6520 6d6f 7665 4669 6c65  e *note moveFile
│ │ │ │ -0006e240: 2853 7472 696e 672c 5374 7269 6e67 293a  (String,String):
│ │ │ │ -0006e250: 0a20 2020 2028 4d61 6361 756c 6179 3244  .    (Macaulay2D
│ │ │ │ -0006e260: 6f63 296d 6f76 6546 696c 655f 6c70 5374  oc)moveFile_lpSt
│ │ │ │ -0006e270: 7269 6e67 5f63 6d53 7472 696e 675f 7270  ring_cmString_rp
│ │ │ │ -0006e280: 2c0a 2020 2a20 2272 756e 5072 6f67 7261  ,.  * "runProgra
│ │ │ │ -0006e290: 6d28 2e2e 2e2c 5665 7262 6f73 653d 3e2e  m(...,Verbose=>.
│ │ │ │ -0006e2a0: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74  ..)" -- see *not
│ │ │ │ -0006e2b0: 6520 7275 6e50 726f 6772 616d 3a0a 2020  e runProgram:.  
│ │ │ │ -0006e2c0: 2020 284d 6163 6175 6c61 7932 446f 6329    (Macaulay2Doc)
│ │ │ │ -0006e2d0: 7275 6e50 726f 6772 616d 2c20 2d2d 2072  runProgram, -- r
│ │ │ │ -0006e2e0: 756e 2061 6e20 6578 7465 726e 616c 2070  un an external p
│ │ │ │ -0006e2f0: 726f 6772 616d 0a20 202a 2022 7379 6d6c  rogram.  * "syml
│ │ │ │ -0006e300: 696e 6b44 6972 6563 746f 7279 282e 2e2e  inkDirectory(...
│ │ │ │ -0006e310: 2c56 6572 626f 7365 3d3e 2e2e 2e29 2220  ,Verbose=>...)" 
│ │ │ │ -0006e320: 2d2d 2073 6565 202a 6e6f 7465 0a20 2020  -- see *note.   
│ │ │ │ -0006e330: 2073 796d 6c69 6e6b 4469 7265 6374 6f72   symlinkDirector
│ │ │ │ -0006e340: 7928 5374 7269 6e67 2c53 7472 696e 6729  y(String,String)
│ │ │ │ -0006e350: 3a0a 2020 2020 284d 6163 6175 6c61 7932  :.    (Macaulay2
│ │ │ │ -0006e360: 446f 6329 7379 6d6c 696e 6b44 6972 6563  Doc)symlinkDirec
│ │ │ │ -0006e370: 746f 7279 5f6c 7053 7472 696e 675f 636d  tory_lpString_cm
│ │ │ │ -0006e380: 5374 7269 6e67 5f72 702c 202d 2d20 6d61  String_rp, -- ma
│ │ │ │ -0006e390: 6b65 2073 796d 626f 6c69 6320 6c69 6e6b  ke symbolic link
│ │ │ │ -0006e3a0: 730a 2020 2020 666f 7220 616c 6c20 6669  s.    for all fi
│ │ │ │ -0006e3b0: 6c65 7320 696e 2061 2064 6972 6563 746f  les in a directo
│ │ │ │ -0006e3c0: 7279 2074 7265 650a 1f0a 4669 6c65 3a20  ry tree...File: 
│ │ │ │ -0006e3d0: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ -0006e3e0: 6465 3a20 6973 4269 7261 7469 6f6e 616c  de: isBirational
│ │ │ │ -0006e3f0: 2c20 4e65 7874 3a20 6973 446f 6d69 6e61  , Next: isDomina
│ │ │ │ -0006e400: 6e74 2c20 5072 6576 3a20 696e 7665 7273  nt, Prev: invers
│ │ │ │ -0006e410: 654d 6170 5f6c 705f 7064 5f70 645f 7064  eMap_lp_pd_pd_pd
│ │ │ │ -0006e420: 5f63 6d56 6572 626f 7365 3d3e 5f70 645f  _cmVerbose=>_pd_
│ │ │ │ -0006e430: 7064 5f70 645f 7270 2c20 5570 3a20 546f  pd_pd_rp, Up: To
│ │ │ │ -0006e440: 700a 0a69 7342 6972 6174 696f 6e61 6c20  p..isBirational 
│ │ │ │ -0006e450: 2d2d 2077 6865 7468 6572 2061 2072 6174  -- whether a rat
│ │ │ │ -0006e460: 696f 6e61 6c20 6d61 7020 6973 2062 6972  ional map is bir
│ │ │ │ -0006e470: 6174 696f 6e61 6c0a 2a2a 2a2a 2a2a 2a2a  ational.********
│ │ │ │ +0006dcc0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +0006dcd0: 6368 6563 6b28 2e2e 2e2c 5665 7262 6f73  check(...,Verbos
│ │ │ │ +0006dce0: 653d 3e2e 2e2e 2922 202d 2d20 7365 6520  e=>...)" -- see 
│ │ │ │ +0006dcf0: 2a6e 6f74 6520 6368 6563 6b3a 2028 4d61  *note check: (Ma
│ │ │ │ +0006dd00: 6361 756c 6179 3244 6f63 2963 6865 636b  caulay2Doc)check
│ │ │ │ +0006dd10: 2c20 2d2d 0a20 2020 2070 6572 666f 726d  , --.    perform
│ │ │ │ +0006dd20: 2074 6573 7473 206f 6620 6120 7061 636b   tests of a pack
│ │ │ │ +0006dd30: 6167 650a 2020 2a20 2263 6865 636b 4465  age.  * "checkDe
│ │ │ │ +0006dd40: 6772 6565 7328 2e2e 2e2c 5665 7262 6f73  grees(...,Verbos
│ │ │ │ +0006dd50: 653d 3e2e 2e2e 2922 202d 2d20 7365 6520  e=>...)" -- see 
│ │ │ │ +0006dd60: 2a6e 6f74 6520 6368 6563 6b44 6567 7265  *note checkDegre
│ │ │ │ +0006dd70: 6573 3a0a 2020 2020 2849 736f 6d6f 7270  es:.    (Isomorp
│ │ │ │ +0006dd80: 6869 736d 2963 6865 636b 4465 6772 6565  hism)checkDegree
│ │ │ │ +0006dd90: 732c 202d 2d20 636f 6d70 6172 6573 2074  s, -- compares t
│ │ │ │ +0006dda0: 6865 2064 6567 7265 6573 206f 6620 6765  he degrees of ge
│ │ │ │ +0006ddb0: 6e65 7261 746f 7273 206f 6620 7477 6f0a  nerators of two.
│ │ │ │ +0006ddc0: 2020 2020 6d6f 6475 6c65 730a 2020 2a20      modules.  * 
│ │ │ │ +0006ddd0: 2263 6f70 7944 6972 6563 746f 7279 282e  "copyDirectory(.
│ │ │ │ +0006dde0: 2e2e 2c56 6572 626f 7365 3d3e 2e2e 2e29  ..,Verbose=>...)
│ │ │ │ +0006ddf0: 2220 2d2d 2073 6565 202a 6e6f 7465 0a20  " -- see *note. 
│ │ │ │ +0006de00: 2020 2063 6f70 7944 6972 6563 746f 7279     copyDirectory
│ │ │ │ +0006de10: 2853 7472 696e 672c 5374 7269 6e67 293a  (String,String):
│ │ │ │ +0006de20: 0a20 2020 2028 4d61 6361 756c 6179 3244  .    (Macaulay2D
│ │ │ │ +0006de30: 6f63 2963 6f70 7944 6972 6563 746f 7279  oc)copyDirectory
│ │ │ │ +0006de40: 5f6c 7053 7472 696e 675f 636d 5374 7269  _lpString_cmStri
│ │ │ │ +0006de50: 6e67 5f72 702c 0a20 202a 2022 636f 7079  ng_rp,.  * "copy
│ │ │ │ +0006de60: 4669 6c65 282e 2e2e 2c56 6572 626f 7365  File(...,Verbose
│ │ │ │ +0006de70: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ +0006de80: 6e6f 7465 2063 6f70 7946 696c 6528 5374  note copyFile(St
│ │ │ │ +0006de90: 7269 6e67 2c53 7472 696e 6729 3a0a 2020  ring,String):.  
│ │ │ │ +0006dea0: 2020 284d 6163 6175 6c61 7932 446f 6329    (Macaulay2Doc)
│ │ │ │ +0006deb0: 636f 7079 4669 6c65 5f6c 7053 7472 696e  copyFile_lpStrin
│ │ │ │ +0006dec0: 675f 636d 5374 7269 6e67 5f72 702c 0a20  g_cmString_rp,. 
│ │ │ │ +0006ded0: 202a 2022 6669 6e64 5072 6f67 7261 6d28   * "findProgram(
│ │ │ │ +0006dee0: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ +0006def0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ +0006df00: 6669 6e64 5072 6f67 7261 6d3a 0a20 2020  findProgram:.   
│ │ │ │ +0006df10: 2028 4d61 6361 756c 6179 3244 6f63 2966   (Macaulay2Doc)f
│ │ │ │ +0006df20: 696e 6450 726f 6772 616d 2c20 2d2d 206c  indProgram, -- l
│ │ │ │ +0006df30: 6f61 6420 6578 7465 726e 616c 2070 726f  oad external pro
│ │ │ │ +0006df40: 6772 616d 0a20 202a 2022 696e 7374 616c  gram.  * "instal
│ │ │ │ +0006df50: 6c50 6163 6b61 6765 282e 2e2e 2c56 6572  lPackage(...,Ver
│ │ │ │ +0006df60: 626f 7365 3d3e 2e2e 2e29 2220 2d2d 2073  bose=>...)" -- s
│ │ │ │ +0006df70: 6565 202a 6e6f 7465 2069 6e73 7461 6c6c  ee *note install
│ │ │ │ +0006df80: 5061 636b 6167 653a 0a20 2020 2028 4d61  Package:.    (Ma
│ │ │ │ +0006df90: 6361 756c 6179 3244 6f63 2969 6e73 7461  caulay2Doc)insta
│ │ │ │ +0006dfa0: 6c6c 5061 636b 6167 652c 202d 2d20 6c6f  llPackage, -- lo
│ │ │ │ +0006dfb0: 6164 2061 6e64 2069 6e73 7461 6c6c 2061  ad and install a
│ │ │ │ +0006dfc0: 2070 6163 6b61 6765 2061 6e64 2069 7473   package and its
│ │ │ │ +0006dfd0: 0a20 2020 2064 6f63 756d 656e 7461 7469  .    documentati
│ │ │ │ +0006dfe0: 6f6e 0a20 202a 2022 6170 7072 6f78 696d  on.  * "approxim
│ │ │ │ +0006dff0: 6174 6549 6e76 6572 7365 4d61 7028 2e2e  ateInverseMap(..
│ │ │ │ +0006e000: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 2922  .,Verbose=>...)"
│ │ │ │ +0006e010: 0a20 202a 2022 4368 6572 6e53 6368 7761  .  * "ChernSchwa
│ │ │ │ +0006e020: 7274 7a4d 6163 5068 6572 736f 6e28 2e2e  rtzMacPherson(..
│ │ │ │ +0006e030: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 2922  .,Verbose=>...)"
│ │ │ │ +0006e040: 0a20 202a 2022 6465 6772 6565 4d61 7028  .  * "degreeMap(
│ │ │ │ +0006e050: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ +0006e060: 2922 0a20 202a 2022 4575 6c65 7243 6861  )".  * "EulerCha
│ │ │ │ +0006e070: 7261 6374 6572 6973 7469 6328 2e2e 2e2c  racteristic(...,
│ │ │ │ +0006e080: 5665 7262 6f73 653d 3e2e 2e2e 2922 0a20  Verbose=>...)". 
│ │ │ │ +0006e090: 202a 202a 6e6f 7465 2069 6e76 6572 7365   * *note inverse
│ │ │ │ +0006e0a0: 4d61 7028 2e2e 2e2c 5665 7262 6f73 653d  Map(...,Verbose=
│ │ │ │ +0006e0b0: 3e2e 2e2e 293a 0a20 2020 2069 6e76 6572  >...):.    inver
│ │ │ │ +0006e0c0: 7365 4d61 705f 6c70 5f70 645f 7064 5f70  seMap_lp_pd_pd_p
│ │ │ │ +0006e0d0: 645f 636d 5665 7262 6f73 653d 3e5f 7064  d_cmVerbose=>_pd
│ │ │ │ +0006e0e0: 5f70 645f 7064 5f72 702c 0a20 202a 2022  _pd_pd_rp,.  * "
│ │ │ │ +0006e0f0: 6973 4269 7261 7469 6f6e 616c 282e 2e2e  isBirational(...
│ │ │ │ +0006e100: 2c56 6572 626f 7365 3d3e 2e2e 2e29 220a  ,Verbose=>...)".
│ │ │ │ +0006e110: 2020 2a20 2269 7344 6f6d 696e 616e 7428    * "isDominant(
│ │ │ │ +0006e120: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ +0006e130: 2922 0a20 202a 2022 7072 6f6a 6563 7469  )".  * "projecti
│ │ │ │ +0006e140: 7665 4465 6772 6565 7328 2e2e 2e2c 5665  veDegrees(...,Ve
│ │ │ │ +0006e150: 7262 6f73 653d 3e2e 2e2e 2922 0a20 202a  rbose=>...)".  *
│ │ │ │ +0006e160: 2022 5365 6772 6543 6c61 7373 282e 2e2e   "SegreClass(...
│ │ │ │ +0006e170: 2c56 6572 626f 7365 3d3e 2e2e 2e29 220a  ,Verbose=>...)".
│ │ │ │ +0006e180: 2020 2a20 2269 7349 736f 6d6f 7270 6869    * "isIsomorphi
│ │ │ │ +0006e190: 6328 2e2e 2e2c 5665 7262 6f73 653d 3e2e  c(...,Verbose=>.
│ │ │ │ +0006e1a0: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74  ..)" -- see *not
│ │ │ │ +0006e1b0: 6520 6973 4973 6f6d 6f72 7068 6963 3a0a  e isIsomorphic:.
│ │ │ │ +0006e1c0: 2020 2020 2849 736f 6d6f 7270 6869 736d      (Isomorphism
│ │ │ │ +0006e1d0: 2969 7349 736f 6d6f 7270 6869 632c 202d  )isIsomorphic, -
│ │ │ │ +0006e1e0: 2d20 5072 6f62 6162 696c 6973 7469 6320  - Probabilistic 
│ │ │ │ +0006e1f0: 7465 7374 2066 6f72 2069 736f 6d6f 7270  test for isomorp
│ │ │ │ +0006e200: 6869 736d 206f 6620 6d6f 6475 6c65 730a  hism of modules.
│ │ │ │ +0006e210: 2020 2a20 226d 6f76 6546 696c 6528 2e2e    * "moveFile(..
│ │ │ │ +0006e220: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 2922  .,Verbose=>...)"
│ │ │ │ +0006e230: 202d 2d20 7365 6520 2a6e 6f74 6520 6d6f   -- see *note mo
│ │ │ │ +0006e240: 7665 4669 6c65 2853 7472 696e 672c 5374  veFile(String,St
│ │ │ │ +0006e250: 7269 6e67 293a 0a20 2020 2028 4d61 6361  ring):.    (Maca
│ │ │ │ +0006e260: 756c 6179 3244 6f63 296d 6f76 6546 696c  ulay2Doc)moveFil
│ │ │ │ +0006e270: 655f 6c70 5374 7269 6e67 5f63 6d53 7472  e_lpString_cmStr
│ │ │ │ +0006e280: 696e 675f 7270 2c0a 2020 2a20 2272 756e  ing_rp,.  * "run
│ │ │ │ +0006e290: 5072 6f67 7261 6d28 2e2e 2e2c 5665 7262  Program(...,Verb
│ │ │ │ +0006e2a0: 6f73 653d 3e2e 2e2e 2922 202d 2d20 7365  ose=>...)" -- se
│ │ │ │ +0006e2b0: 6520 2a6e 6f74 6520 7275 6e50 726f 6772  e *note runProgr
│ │ │ │ +0006e2c0: 616d 3a0a 2020 2020 284d 6163 6175 6c61  am:.    (Macaula
│ │ │ │ +0006e2d0: 7932 446f 6329 7275 6e50 726f 6772 616d  y2Doc)runProgram
│ │ │ │ +0006e2e0: 2c20 2d2d 2072 756e 2061 6e20 6578 7465  , -- run an exte
│ │ │ │ +0006e2f0: 726e 616c 2070 726f 6772 616d 0a20 202a  rnal program.  *
│ │ │ │ +0006e300: 2022 7379 6d6c 696e 6b44 6972 6563 746f   "symlinkDirecto
│ │ │ │ +0006e310: 7279 282e 2e2e 2c56 6572 626f 7365 3d3e  ry(...,Verbose=>
│ │ │ │ +0006e320: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f  ...)" -- see *no
│ │ │ │ +0006e330: 7465 0a20 2020 2073 796d 6c69 6e6b 4469  te.    symlinkDi
│ │ │ │ +0006e340: 7265 6374 6f72 7928 5374 7269 6e67 2c53  rectory(String,S
│ │ │ │ +0006e350: 7472 696e 6729 3a0a 2020 2020 284d 6163  tring):.    (Mac
│ │ │ │ +0006e360: 6175 6c61 7932 446f 6329 7379 6d6c 696e  aulay2Doc)symlin
│ │ │ │ +0006e370: 6b44 6972 6563 746f 7279 5f6c 7053 7472  kDirectory_lpStr
│ │ │ │ +0006e380: 696e 675f 636d 5374 7269 6e67 5f72 702c  ing_cmString_rp,
│ │ │ │ +0006e390: 202d 2d20 6d61 6b65 2073 796d 626f 6c69   -- make symboli
│ │ │ │ +0006e3a0: 6320 6c69 6e6b 730a 2020 2020 666f 7220  c links.    for 
│ │ │ │ +0006e3b0: 616c 6c20 6669 6c65 7320 696e 2061 2064  all files in a d
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│ │ │ │ +0006e3f0: 7469 6f6e 616c 2c20 4e65 7874 3a20 6973  tional, Next: is
│ │ │ │ +0006e400: 446f 6d69 6e61 6e74 2c20 5072 6576 3a20  Dominant, Prev: 
│ │ │ │ +0006e410: 696e 7665 7273 654d 6170 5f6c 705f 7064  inverseMap_lp_pd
│ │ │ │ +0006e420: 5f70 645f 7064 5f63 6d56 6572 626f 7365  _pd_pd_cmVerbose
│ │ │ │ +0006e430: 3d3e 5f70 645f 7064 5f70 645f 7270 2c20  =>_pd_pd_pd_rp, 
│ │ │ │ +0006e440: 5570 3a20 546f 700a 0a69 7342 6972 6174  Up: Top..isBirat
│ │ │ │ +0006e450: 696f 6e61 6c20 2d2d 2077 6865 7468 6572  ional -- whether
│ │ │ │ +0006e460: 2061 2072 6174 696f 6e61 6c20 6d61 7020   a rational map 
│ │ │ │ +0006e470: 6973 2062 6972 6174 696f 6e61 6c0a 2a2a  is birational.**
│ │ │ │  0006e480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006e490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006e4a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -0006e4b0: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -0006e4c0: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -0006e4d0: 2020 2020 2069 7342 6972 6174 696f 6e61       isBirationa
│ │ │ │ -0006e4e0: 6c20 7068 690a 2020 2a20 496e 7075 7473  l phi.  * Inputs
│ │ │ │ -0006e4f0: 3a0a 2020 2020 2020 2a20 7068 692c 2061  :.      * phi, a
│ │ │ │ -0006e500: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ -0006e510: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ -0006e520: 2c0a 2020 2a20 2a6e 6f74 6520 4f70 7469  ,.  * *note Opti
│ │ │ │ -0006e530: 6f6e 616c 2069 6e70 7574 733a 2028 4d61  onal inputs: (Ma
│ │ │ │ -0006e540: 6361 756c 6179 3244 6f63 2975 7369 6e67  caulay2Doc)using
│ │ │ │ -0006e550: 2066 756e 6374 696f 6e73 2077 6974 6820   functions with 
│ │ │ │ -0006e560: 6f70 7469 6f6e 616c 2069 6e70 7574 732c  optional inputs,
│ │ │ │ -0006e570: 3a0a 2020 2020 2020 2a20 2a6e 6f74 6520  :.      * *note 
│ │ │ │ -0006e580: 426c 6f77 5570 5374 7261 7465 6779 3a20  BlowUpStrategy: 
│ │ │ │ -0006e590: 426c 6f77 5570 5374 7261 7465 6779 2c20  BlowUpStrategy, 
│ │ │ │ -0006e5a0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -0006e5b0: 7661 6c75 650a 2020 2020 2020 2020 2245  value.        "E
│ │ │ │ -0006e5c0: 6c69 6d69 6e61 7465 222c 0a20 2020 2020  liminate",.     
│ │ │ │ -0006e5d0: 202a 202a 6e6f 7465 2043 6572 7469 6679   * *note Certify
│ │ │ │ -0006e5e0: 3a20 4365 7274 6966 792c 203d 3e20 2e2e  : Certify, => ..
│ │ │ │ -0006e5f0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -0006e600: 2066 616c 7365 2c20 7768 6574 6865 7220   false, whether 
│ │ │ │ -0006e610: 746f 2065 6e73 7572 650a 2020 2020 2020  to ensure.      
│ │ │ │ -0006e620: 2020 636f 7272 6563 746e 6573 7320 6f66    correctness of
│ │ │ │ -0006e630: 206f 7574 7075 740a 2020 2020 2020 2a20   output.      * 
│ │ │ │ -0006e640: 2a6e 6f74 6520 5665 7262 6f73 653a 2069  *note Verbose: i
│ │ │ │ -0006e650: 6e76 6572 7365 4d61 705f 6c70 5f70 645f  nverseMap_lp_pd_
│ │ │ │ -0006e660: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ -0006e670: 3e5f 7064 5f70 645f 7064 5f72 702c 203d  >_pd_pd_pd_rp, =
│ │ │ │ -0006e680: 3e20 2e2e 2e2c 0a20 2020 2020 2020 2064  > ...,.        d
│ │ │ │ -0006e690: 6566 6175 6c74 2076 616c 7565 2074 7275  efault value tru
│ │ │ │ -0006e6a0: 652c 0a20 202a 204f 7574 7075 7473 3a0a  e,.  * Outputs:.
│ │ │ │ -0006e6b0: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ -0006e6c0: 426f 6f6c 6561 6e20 7661 6c75 653a 2028  Boolean value: (
│ │ │ │ -0006e6d0: 4d61 6361 756c 6179 3244 6f63 2942 6f6f  Macaulay2Doc)Boo
│ │ │ │ -0006e6e0: 6c65 616e 2c2c 2077 6865 7468 6572 2070  lean,, whether p
│ │ │ │ -0006e6f0: 6869 2069 730a 2020 2020 2020 2020 6269  hi is.        bi
│ │ │ │ -0006e700: 7261 7469 6f6e 616c 0a0a 4465 7363 7269  rational..Descri
│ │ │ │ -0006e710: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -0006e720: 3d0a 0a54 6865 2074 6573 7469 6e67 2070  =..The testing p
│ │ │ │ -0006e730: 6173 7365 7320 7468 726f 7567 6820 7468  asses through th
│ │ │ │ -0006e740: 6520 6d65 7468 6f64 7320 2a6e 6f74 6520  e methods *note 
│ │ │ │ -0006e750: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -0006e760: 733a 0a70 726f 6a65 6374 6976 6544 6567  s:.projectiveDeg
│ │ │ │ -0006e770: 7265 6573 2c2c 202a 6e6f 7465 2064 6567  rees,, *note deg
│ │ │ │ -0006e780: 7265 654d 6170 3a20 6465 6772 6565 4d61  reeMap: degreeMa
│ │ │ │ -0006e790: 702c 2061 6e64 202a 6e6f 7465 2069 7344  p, and *note isD
│ │ │ │ -0006e7a0: 6f6d 696e 616e 743a 0a69 7344 6f6d 696e  ominant:.isDomin
│ │ │ │ -0006e7b0: 616e 742c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ant,...+--------
│ │ │ │ +0006e4a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0006e4b0: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +0006e4c0: 3d3d 3d3d 3d0a 0a20 202a 2055 7361 6765  =====..  * Usage
│ │ │ │ +0006e4d0: 3a20 0a20 2020 2020 2020 2069 7342 6972  : .        isBir
│ │ │ │ +0006e4e0: 6174 696f 6e61 6c20 7068 690a 2020 2a20  ational phi.  * 
│ │ │ │ +0006e4f0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +0006e500: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ +0006e510: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +0006e520: 6e61 6c4d 6170 2c0a 2020 2a20 2a6e 6f74  nalMap,.  * *not
│ │ │ │ +0006e530: 6520 4f70 7469 6f6e 616c 2069 6e70 7574  e Optional input
│ │ │ │ +0006e540: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ +0006e550: 2975 7369 6e67 2066 756e 6374 696f 6e73  )using functions
│ │ │ │ +0006e560: 2077 6974 6820 6f70 7469 6f6e 616c 2069   with optional i
│ │ │ │ +0006e570: 6e70 7574 732c 3a0a 2020 2020 2020 2a20  nputs,:.      * 
│ │ │ │ +0006e580: 2a6e 6f74 6520 426c 6f77 5570 5374 7261  *note BlowUpStra
│ │ │ │ +0006e590: 7465 6779 3a20 426c 6f77 5570 5374 7261  tegy: BlowUpStra
│ │ │ │ +0006e5a0: 7465 6779 2c20 3d3e 202e 2e2e 2c20 6465  tegy, => ..., de
│ │ │ │ +0006e5b0: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ +0006e5c0: 2020 2020 2245 6c69 6d69 6e61 7465 222c      "Eliminate",
│ │ │ │ +0006e5d0: 0a20 2020 2020 202a 202a 6e6f 7465 2043  .      * *note C
│ │ │ │ +0006e5e0: 6572 7469 6679 3a20 4365 7274 6966 792c  ertify: Certify,
│ │ │ │ +0006e5f0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +0006e600: 2076 616c 7565 2066 616c 7365 2c20 7768   value false, wh
│ │ │ │ +0006e610: 6574 6865 7220 746f 2065 6e73 7572 650a  ether to ensure.
│ │ │ │ +0006e620: 2020 2020 2020 2020 636f 7272 6563 746e          correctn
│ │ │ │ +0006e630: 6573 7320 6f66 206f 7574 7075 740a 2020  ess of output.  
│ │ │ │ +0006e640: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
│ │ │ │ +0006e650: 6f73 653a 2069 6e76 6572 7365 4d61 705f  ose: inverseMap_
│ │ │ │ +0006e660: 6c70 5f70 645f 7064 5f70 645f 636d 5665  lp_pd_pd_pd_cmVe
│ │ │ │ +0006e670: 7262 6f73 653d 3e5f 7064 5f70 645f 7064  rbose=>_pd_pd_pd
│ │ │ │ +0006e680: 5f72 702c 203d 3e20 2e2e 2e2c 0a20 2020  _rp, => ...,.   
│ │ │ │ +0006e690: 2020 2020 2064 6566 6175 6c74 2076 616c       default val
│ │ │ │ +0006e6a0: 7565 2074 7275 652c 0a20 202a 204f 7574  ue true,.  * Out
│ │ │ │ +0006e6b0: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ +0006e6c0: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661  *note Boolean va
│ │ │ │ +0006e6d0: 6c75 653a 2028 4d61 6361 756c 6179 3244  lue: (Macaulay2D
│ │ │ │ +0006e6e0: 6f63 2942 6f6f 6c65 616e 2c2c 2077 6865  oc)Boolean,, whe
│ │ │ │ +0006e6f0: 7468 6572 2070 6869 2069 730a 2020 2020  ther phi is.    
│ │ │ │ +0006e700: 2020 2020 6269 7261 7469 6f6e 616c 0a0a      birational..
│ │ │ │ +0006e710: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +0006e720: 3d3d 3d3d 3d3d 3d0a 0a54 6865 2074 6573  =======..The tes
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│ │ │ │ +0006e740: 7567 6820 7468 6520 6d65 7468 6f64 7320  ugh the methods 
│ │ │ │ +0006e750: 2a6e 6f74 6520 7072 6f6a 6563 7469 7665  *note projective
│ │ │ │ +0006e760: 4465 6772 6565 733a 0a70 726f 6a65 6374  Degrees:.project
│ │ │ │ +0006e770: 6976 6544 6567 7265 6573 2c2c 202a 6e6f  iveDegrees,, *no
│ │ │ │ +0006e780: 7465 2064 6567 7265 654d 6170 3a20 6465  te degreeMap: de
│ │ │ │ +0006e790: 6772 6565 4d61 702c 2061 6e64 202a 6e6f  greeMap, and *no
│ │ │ │ +0006e7a0: 7465 2069 7344 6f6d 696e 616e 743a 0a69  te isDominant:.i
│ │ │ │ +0006e7b0: 7344 6f6d 696e 616e 742c 2e0a 0a2b 2d2d  sDominant,...+--
│ │ │ │  0006e7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e800: 2d2d 2d2d 2d2b 0a7c 6931 203a 2047 4628  -----+.|i1 : GF(
│ │ │ │ -0006e810: 3333 315e 3229 5b74 5f30 2e2e 745f 345d  331^2)[t_0..t_4]
│ │ │ │ -0006e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0006e810: 203a 2047 4628 3333 315e 3229 5b74 5f30   : GF(331^2)[t_0
│ │ │ │ +0006e820: 2e2e 745f 345d 2020 2020 2020 2020 2020  ..t_4]          
│ │ │ │  0006e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e850: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006e850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e8a0: 2020 2020 207c 0a7c 6f31 203d 2047 4620       |.|o1 = GF 
│ │ │ │ -0006e8b0: 3130 3935 3631 5b74 202e 2e74 205d 2020  109561[t ..t ]  
│ │ │ │ -0006e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e8a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0006e8b0: 203d 2047 4620 3130 3935 3631 5b74 202e   = GF 109561[t .
│ │ │ │ +0006e8c0: 2e74 205d 2020 2020 2020 2020 2020 2020  .t ]            
│ │ │ │  0006e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e8f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006e900: 2020 2020 2020 2020 3020 2020 3420 2020          0   4   
│ │ │ │ -0006e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006e900: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0006e910: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │  0006e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e940: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006e940: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e990: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c       |.|o1 : Pol
│ │ │ │ -0006e9a0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ -0006e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e990: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0006e9a0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +0006e9b0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  0006e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e9e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006e9e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0006e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ea30: 2d2d 2d2d 2d2b 0a7c 6932 203a 2070 6869  -----+.|i2 : phi
│ │ │ │ -0006ea40: 203d 2072 6174 696f 6e61 6c4d 6170 286d   = rationalMap(m
│ │ │ │ -0006ea50: 696e 6f72 7328 322c 6d61 7472 6978 7b7b  inors(2,matrix{{
│ │ │ │ -0006ea60: 745f 302e 2e74 5f33 7d2c 7b74 5f31 2e2e  t_0..t_3},{t_1..
│ │ │ │ -0006ea70: 745f 347d 7d29 2c20 2020 2020 2020 2020  t_4}}),         
│ │ │ │ -0006ea80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0006ea40: 203a 2070 6869 203d 2072 6174 696f 6e61   : phi = rationa
│ │ │ │ +0006ea50: 6c4d 6170 286d 696e 6f72 7328 322c 6d61  lMap(minors(2,ma
│ │ │ │ +0006ea60: 7472 6978 7b7b 745f 302e 2e74 5f33 7d2c  trix{{t_0..t_3},
│ │ │ │ +0006ea70: 7b74 5f31 2e2e 745f 347d 7d29 2c20 2020  {t_1..t_4}}),   
│ │ │ │ +0006ea80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ead0: 2020 2020 207c 0a7c 6f32 203d 202d 2d20       |.|o2 = -- 
│ │ │ │ -0006eae0: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ -0006eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ead0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0006eae0: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +0006eaf0: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  0006eb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006eb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006eb20: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -0006eb30: 7263 653a 2050 726f 6a28 4746 2031 3039  rce: Proj(GF 109
│ │ │ │ -0006eb40: 3536 315b 7420 2c20 7420 2c20 7420 2c20  561[t , t , t , 
│ │ │ │ -0006eb50: 7420 2c20 7420 5d29 2020 2020 2020 2020  t , t ])        
│ │ │ │ +0006eb20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006eb30: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ +0006eb40: 4746 2031 3039 3536 315b 7420 2c20 7420  GF 109561[t , t 
│ │ │ │ +0006eb50: 2c20 7420 2c20 7420 2c20 7420 5d29 2020  , t , t , t ])  
│ │ │ │  0006eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006eb70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006eb70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006eb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006eb90: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ -0006eba0: 2033 2020 2034 2020 2020 2020 2020 2020   3   4          
│ │ │ │ +0006eb90: 2020 2020 2020 2020 2020 2030 2020 2031             0   1
│ │ │ │ +0006eba0: 2020 2032 2020 2033 2020 2034 2020 2020     2   3   4    
│ │ │ │  0006ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ebc0: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -0006ebd0: 6765 743a 2073 7562 7661 7269 6574 7920  get: subvariety 
│ │ │ │ -0006ebe0: 6f66 2050 726f 6a28 4746 2031 3039 3536  of Proj(GF 10956
│ │ │ │ -0006ebf0: 315b 7820 2c20 7820 2c20 7820 2c20 7820  1[x , x , x , x 
│ │ │ │ -0006ec00: 2c20 7820 2c20 7820 2020 2020 2020 2020  , x , x         
│ │ │ │ -0006ec10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ebc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006ebd0: 2020 2074 6172 6765 743a 2073 7562 7661     target: subva
│ │ │ │ +0006ebe0: 7269 6574 7920 6f66 2050 726f 6a28 4746  riety of Proj(GF
│ │ │ │ +0006ebf0: 2031 3039 3536 315b 7820 2c20 7820 2c20   109561[x , x , 
│ │ │ │ +0006ec00: 7820 2c20 7820 2c20 7820 2c20 7820 2020  x , x , x , x   
│ │ │ │ +0006ec10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ec40: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
│ │ │ │ -0006ec50: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -0006ec60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -0006ef80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  0006f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0006f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006f260: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
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│ │ │ │  0006f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0006f340: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f360: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0006f360: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  0006f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0006f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0006f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006f4d0: 2020 2020 207c 0a7c 6f32 203a 2052 6174       |.|o2 : Rat
│ │ │ │ -0006f4e0: 696f 6e61 6c4d 6170 2028 7175 6164 7261  ionalMap (quadra
│ │ │ │ -0006f4f0: 7469 6320 646f 6d69 6e61 6e74 2072 6174  tic dominant rat
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│ │ │ │ -0006f520: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ +0006f4e0: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +0006f4f0: 7175 6164 7261 7469 6320 646f 6d69 6e61  quadratic domina
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│ │ │ │ +0006f520: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
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│ │ │ │ -0006f570: 2d2d 2d2d 2d7c 0a7c 446f 6d69 6e61 6e74  -----|.|Dominant
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│ │ │ │ +0006f570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 446f  -----------|.|Do
│ │ │ │ +0006f580: 6d69 6e61 6e74 3d3e 696e 6669 6e69 7479  minant=>infinity
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│ │ │ │  0006f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006f5c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f5c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f610: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f660: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f660: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f6b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f6b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f700: 2020 2020 207c 0a7c 205d 2920 6465 6669       |.| ]) defi
│ │ │ │ -0006f710: 6e65 6420 6279 2020 2020 2020 2020 2020  ned by          
│ │ │ │ +0006f700: 2020 2020 2020 2020 2020 207c 0a7c 205d             |.| ]
│ │ │ │ +0006f710: 2920 6465 6669 6e65 6420 6279 2020 2020  ) defined by    
│ │ │ │  0006f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f750: 2020 2020 207c 0a7c 3520 2020 2020 2020       |.|5       
│ │ │ │ +0006f750: 2020 2020 2020 2020 2020 207c 0a7c 3520             |.|5 
│ │ │ │  0006f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f7a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f7a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f7f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f7f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f840: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f840: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f890: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f890: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f8e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f8e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f930: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f930: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f980: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f9d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f9d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fa20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fa20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fa70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fa70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fac0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fac0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fb10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fb10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fb60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fb60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fbb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fbb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fc00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fc00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fc50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fc50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fca0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fca0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fcf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fcf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fd40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fd40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fd90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fd90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fde0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fde0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fe30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fe30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fe80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fe80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006fed0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006fed0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ff20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ff20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ff70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ff70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ffa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ffc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ffc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070010: 2020 2020 207c 0a7c 6879 7065 7273 7572       |.|hypersur
│ │ │ │ -00070020: 6661 6365 2069 6e20 5050 5e35 2920 2020  face in PP^5)   
│ │ │ │ -00070030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070010: 2020 2020 2020 2020 2020 207c 0a7c 6879             |.|hy
│ │ │ │ +00070020: 7065 7273 7572 6661 6365 2069 6e20 5050  persurface in PP
│ │ │ │ +00070030: 5e35 2920 2020 2020 2020 2020 2020 2020  ^5)             
│ │ │ │  00070040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070060: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00070060: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00070070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000700a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000700b0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d  -----+.|i3 : tim
│ │ │ │ -000700c0: 6520 6973 4269 7261 7469 6f6e 616c 2070  e isBirational p
│ │ │ │ -000700d0: 6869 2020 2020 2020 2020 2020 2020 2020  hi              
│ │ │ │ +000700b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +000700c0: 203a 2074 696d 6520 6973 4269 7261 7469   : time isBirati
│ │ │ │ +000700d0: 6f6e 616c 2070 6869 2020 2020 2020 2020  onal phi        
│ │ │ │  000700e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000700f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070100: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00070110: 2030 2e30 3139 3936 3231 7320 2863 7075   0.0199621s (cpu
│ │ │ │ -00070120: 293b 2030 2e30 3137 3938 3233 7320 2874  ); 0.0179823s (t
│ │ │ │ -00070130: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -00070140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070150: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00070100: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00070110: 2d20 7573 6564 2030 2e31 3134 3831 3873  - used 0.114818s
│ │ │ │ +00070120: 2028 6370 7529 3b20 302e 3034 3639 3935   (cpu); 0.046995
│ │ │ │ +00070130: 3573 2028 7468 7265 6164 293b 2030 7320  5s (thread); 0s 
│ │ │ │ +00070140: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +00070150: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00070160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000701a0: 2020 2020 207c 0a7c 6f33 203d 2074 7275       |.|o3 = tru
│ │ │ │ -000701b0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +000701a0: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +000701b0: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │  000701c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000701d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000701e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000701f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000701f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00070200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070240: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 696d  -----+.|i4 : tim
│ │ │ │ -00070250: 6520 6973 4269 7261 7469 6f6e 616c 2870  e isBirational(p
│ │ │ │ -00070260: 6869 2c43 6572 7469 6679 3d3e 7472 7565  hi,Certify=>true
│ │ │ │ -00070270: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00070240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +00070250: 203a 2074 696d 6520 6973 4269 7261 7469   : time isBirati
│ │ │ │ +00070260: 6f6e 616c 2870 6869 2c43 6572 7469 6679  onal(phi,Certify
│ │ │ │ +00070270: 3d3e 7472 7565 2920 2020 2020 2020 2020  =>true)         
│ │ │ │  00070280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070290: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000702a0: 2030 2e30 3132 3835 3839 7320 2863 7075   0.0128589s (cpu
│ │ │ │ -000702b0: 293b 2030 2e30 3134 3234 3237 7320 2874  ); 0.0142427s (t
│ │ │ │ -000702c0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -000702d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000702e0: 2020 2020 207c 0a7c 4365 7274 6966 793a       |.|Certify:
│ │ │ │ -000702f0: 206f 7574 7075 7420 6365 7274 6966 6965   output certifie
│ │ │ │ -00070300: 6421 2020 2020 2020 2020 2020 2020 2020  d!              
│ │ │ │ +00070290: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +000702a0: 2d20 7573 6564 2030 2e30 3238 3230 3631  - used 0.0282061
│ │ │ │ +000702b0: 7320 2863 7075 293b 2030 2e30 3137 3532  s (cpu); 0.01752
│ │ │ │ +000702c0: 3737 7320 2874 6872 6561 6429 3b20 3073  77s (thread); 0s
│ │ │ │ +000702d0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +000702e0: 2020 2020 2020 2020 2020 207c 0a7c 4365             |.|Ce
│ │ │ │ +000702f0: 7274 6966 793a 206f 7574 7075 7420 6365  rtify: output ce
│ │ │ │ +00070300: 7274 6966 6965 6421 2020 2020 2020 2020  rtified!        
│ │ │ │  00070310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00070330: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00070340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070380: 2020 2020 207c 0a7c 6f34 203d 2074 7275       |.|o4 = tru
│ │ │ │ -00070390: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00070380: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +00070390: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │  000703a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000703b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000703c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000703d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000703d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  000703e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000703f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070420: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ -00070430: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00070440: 6e6f 7465 2069 7344 6f6d 696e 616e 743a  note isDominant:
│ │ │ │ -00070450: 2069 7344 6f6d 696e 616e 742c 202d 2d20   isDominant, -- 
│ │ │ │ -00070460: 7768 6574 6865 7220 6120 7261 7469 6f6e  whether a ration
│ │ │ │ -00070470: 616c 206d 6170 2069 7320 646f 6d69 6e61  al map is domina
│ │ │ │ -00070480: 6e74 0a0a 5761 7973 2074 6f20 7573 6520  nt..Ways to use 
│ │ │ │ -00070490: 6973 4269 7261 7469 6f6e 616c 3a0a 3d3d  isBirational:.==
│ │ │ │ -000704a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000704b0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6973  =======..  * "is
│ │ │ │ -000704c0: 4269 7261 7469 6f6e 616c 2852 6174 696f  Birational(Ratio
│ │ │ │ -000704d0: 6e61 6c4d 6170 2922 0a20 202a 2022 6973  nalMap)".  * "is
│ │ │ │ -000704e0: 4269 7261 7469 6f6e 616c 2852 696e 674d  Birational(RingM
│ │ │ │ -000704f0: 6170 2922 0a0a 466f 7220 7468 6520 7072  ap)"..For the pr
│ │ │ │ -00070500: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -00070510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -00070520: 206f 626a 6563 7420 2a6e 6f74 6520 6973   object *note is
│ │ │ │ -00070530: 4269 7261 7469 6f6e 616c 3a20 6973 4269  Birational: isBi
│ │ │ │ -00070540: 7261 7469 6f6e 616c 2c20 6973 2061 202a  rational, is a *
│ │ │ │ -00070550: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ -00070560: 7469 6f6e 2077 6974 680a 6f70 7469 6f6e  tion with.option
│ │ │ │ -00070570: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ -00070580: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ -00070590: 6974 684f 7074 696f 6e73 2c2e 0a1f 0a46  ithOptions,....F
│ │ │ │ -000705a0: 696c 653a 2043 7265 6d6f 6e61 2e69 6e66  ile: Cremona.inf
│ │ │ │ -000705b0: 6f2c 204e 6f64 653a 2069 7344 6f6d 696e  o, Node: isDomin
│ │ │ │ -000705c0: 616e 742c 204e 6578 743a 2069 7349 6e76  ant, Next: isInv
│ │ │ │ -000705d0: 6572 7365 4d61 702c 2050 7265 763a 2069  erseMap, Prev: i
│ │ │ │ -000705e0: 7342 6972 6174 696f 6e61 6c2c 2055 703a  sBirational, Up:
│ │ │ │ -000705f0: 2054 6f70 0a0a 6973 446f 6d69 6e61 6e74   Top..isDominant
│ │ │ │ -00070600: 202d 2d20 7768 6574 6865 7220 6120 7261   -- whether a ra
│ │ │ │ -00070610: 7469 6f6e 616c 206d 6170 2069 7320 646f  tional map is do
│ │ │ │ -00070620: 6d69 6e61 6e74 0a2a 2a2a 2a2a 2a2a 2a2a  minant.*********
│ │ │ │ +00070420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ +00070430: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +00070440: 0a20 202a 202a 6e6f 7465 2069 7344 6f6d  .  * *note isDom
│ │ │ │ +00070450: 696e 616e 743a 2069 7344 6f6d 696e 616e  inant: isDominan
│ │ │ │ +00070460: 742c 202d 2d20 7768 6574 6865 7220 6120  t, -- whether a 
│ │ │ │ +00070470: 7261 7469 6f6e 616c 206d 6170 2069 7320  rational map is 
│ │ │ │ +00070480: 646f 6d69 6e61 6e74 0a0a 5761 7973 2074  dominant..Ways t
│ │ │ │ +00070490: 6f20 7573 6520 6973 4269 7261 7469 6f6e  o use isBiration
│ │ │ │ +000704a0: 616c 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  al:.============
│ │ │ │ +000704b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +000704c0: 202a 2022 6973 4269 7261 7469 6f6e 616c   * "isBirational
│ │ │ │ +000704d0: 2852 6174 696f 6e61 6c4d 6170 2922 0a20  (RationalMap)". 
│ │ │ │ +000704e0: 202a 2022 6973 4269 7261 7469 6f6e 616c   * "isBirational
│ │ │ │ +000704f0: 2852 696e 674d 6170 2922 0a0a 466f 7220  (RingMap)"..For 
│ │ │ │ +00070500: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +00070510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00070520: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +00070530: 6f74 6520 6973 4269 7261 7469 6f6e 616c  ote isBirational
│ │ │ │ +00070540: 3a20 6973 4269 7261 7469 6f6e 616c 2c20  : isBirational, 
│ │ │ │ +00070550: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ +00070560: 6420 6675 6e63 7469 6f6e 2077 6974 680a  d function with.
│ │ │ │ +00070570: 6f70 7469 6f6e 733a 2028 4d61 6361 756c  options: (Macaul
│ │ │ │ +00070580: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ +00070590: 6374 696f 6e57 6974 684f 7074 696f 6e73  ctionWithOptions
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│ │ │ │ +000705c0: 7344 6f6d 696e 616e 742c 204e 6578 743a  sDominant, Next:
│ │ │ │ +000705d0: 2069 7349 6e76 6572 7365 4d61 702c 2050   isInverseMap, P
│ │ │ │ +000705e0: 7265 763a 2069 7342 6972 6174 696f 6e61  rev: isBirationa
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│ │ │ │ +00070620: 2069 7320 646f 6d69 6e61 6e74 0a2a 2a2a   is dominant.***
│ │ │ │  00070630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00070640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00070650: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ -00070660: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ -00070670: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00070680: 6973 446f 6d69 6e61 6e74 2070 6869 0a20  isDominant phi. 
│ │ │ │ -00070690: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -000706a0: 202a 2070 6869 2c20 6120 2a6e 6f74 6520   * phi, a *note 
│ │ │ │ -000706b0: 7261 7469 6f6e 616c 206d 6170 3a20 5261  rational map: Ra
│ │ │ │ -000706c0: 7469 6f6e 616c 4d61 702c 0a20 202a 202a  tionalMap,.  * *
│ │ │ │ -000706d0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
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│ │ │ │ -000706f0: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ -00070700: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
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│ │ │ │ -00070720: 202a 202a 6e6f 7465 2043 6572 7469 6679   * *note Certify
│ │ │ │ -00070730: 3a20 4365 7274 6966 792c 203d 3e20 2e2e  : Certify, => ..
│ │ │ │ -00070740: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
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│ │ │ │ -00070790: 2a6e 6f74 6520 5665 7262 6f73 653a 2069  *note Verbose: i
│ │ │ │ -000707a0: 6e76 6572 7365 4d61 705f 6c70 5f70 645f  nverseMap_lp_pd_
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│ │ │ │ -000707c0: 3e5f 7064 5f70 645f 7064 5f72 702c 203d  >_pd_pd_pd_rp, =
│ │ │ │ -000707d0: 3e20 2e2e 2e2c 0a20 2020 2020 2020 2064  > ...,.        d
│ │ │ │ -000707e0: 6566 6175 6c74 2076 616c 7565 2074 7275  efault value tru
│ │ │ │ -000707f0: 652c 0a20 202a 204f 7574 7075 7473 3a0a  e,.  * Outputs:.
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│ │ │ │ -00070810: 426f 6f6c 6561 6e20 7661 6c75 653a 2028  Boolean value: (
│ │ │ │ -00070820: 4d61 6361 756c 6179 3244 6f63 2942 6f6f  Macaulay2Doc)Boo
│ │ │ │ -00070830: 6c65 616e 2c2c 2077 6865 7468 6572 2070  lean,, whether p
│ │ │ │ -00070840: 6869 2069 7320 646f 6d69 6e61 6e74 0a0a  hi is dominant..
│ │ │ │ -00070850: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00070860: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6d65  =======..This me
│ │ │ │ -00070870: 7468 6f64 2069 7320 6261 7365 6420 6f6e  thod is based on
│ │ │ │ -00070880: 2074 6865 2066 6962 7265 2064 696d 656e   the fibre dimen
│ │ │ │ -00070890: 7369 6f6e 2074 6865 6f72 656d 2e20 4120  sion theorem. A 
│ │ │ │ -000708a0: 6d6f 7265 2073 7461 6e64 6172 6420 7761  more standard wa
│ │ │ │ -000708b0: 7920 776f 756c 640a 6265 2074 6f20 7065  y would.be to pe
│ │ │ │ -000708c0: 7266 6f72 6d20 7468 6520 636f 6d6d 616e  rform the comman
│ │ │ │ -000708d0: 6420 6b65 726e 656c 206d 6170 2070 6869  d kernel map phi
│ │ │ │ -000708e0: 203d 3d20 302e 0a0a 2b2d 2d2d 2d2d 2d2d   == 0...+-------
│ │ │ │ +00070650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
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│ │ │ │ +00070670: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +00070680: 2020 2020 2020 6973 446f 6d69 6e61 6e74        isDominant
│ │ │ │ +00070690: 2070 6869 0a20 202a 2049 6e70 7574 733a   phi.  * Inputs:
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│ │ │ │ +000706b0: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +000706c0: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ +000706d0: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
│ │ │ │ +000706e0: 6e61 6c20 696e 7075 7473 3a20 284d 6163  nal inputs: (Mac
│ │ │ │ +000706f0: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ +00070700: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
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│ │ │ │ +00070730: 6572 7469 6679 3a20 4365 7274 6966 792c  ertify: Certify,
│ │ │ │ +00070740: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +00070750: 2076 616c 7565 2066 616c 7365 2c20 7768   value false, wh
│ │ │ │ +00070760: 6574 6865 7220 746f 2065 6e73 7572 650a  ether to ensure.
│ │ │ │ +00070770: 2020 2020 2020 2020 636f 7272 6563 746e          correctn
│ │ │ │ +00070780: 6573 7320 6f66 206f 7574 7075 740a 2020  ess of output.  
│ │ │ │ +00070790: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
│ │ │ │ +000707a0: 6f73 653a 2069 6e76 6572 7365 4d61 705f  ose: inverseMap_
│ │ │ │ +000707b0: 6c70 5f70 645f 7064 5f70 645f 636d 5665  lp_pd_pd_pd_cmVe
│ │ │ │ +000707c0: 7262 6f73 653d 3e5f 7064 5f70 645f 7064  rbose=>_pd_pd_pd
│ │ │ │ +000707d0: 5f72 702c 203d 3e20 2e2e 2e2c 0a20 2020  _rp, => ...,.   
│ │ │ │ +000707e0: 2020 2020 2064 6566 6175 6c74 2076 616c       default val
│ │ │ │ +000707f0: 7565 2074 7275 652c 0a20 202a 204f 7574  ue true,.  * Out
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│ │ │ │ +00070810: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661  *note Boolean va
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│ │ │ │ +00070830: 6f63 2942 6f6f 6c65 616e 2c2c 2077 6865  oc)Boolean,, whe
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│ │ │ │ +00070860: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
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│ │ │ │ +000708a0: 656d 2e20 4120 6d6f 7265 2073 7461 6e64  em. A more stand
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│ │ │ │ +000708c0: 2074 6f20 7065 7266 6f72 6d20 7468 6520   to perform the 
│ │ │ │ +000708d0: 636f 6d6d 616e 6420 6b65 726e 656c 206d  command kernel m
│ │ │ │ +000708e0: 6170 2070 6869 203d 3d20 302e 0a0a 2b2d  ap phi == 0...+-
│ │ │ │  000708f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070930: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5038  ------+.|i1 : P8
│ │ │ │ -00070940: 203d 205a 5a2f 3130 315b 785f 302e 2e78   = ZZ/101[x_0..x
│ │ │ │ -00070950: 5f38 5d3b 2020 2020 2020 2020 2020 2020  _8];            
│ │ │ │ +00070930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00070940: 3120 3a20 5038 203d 205a 5a2f 3130 315b  1 : P8 = ZZ/101[
│ │ │ │ +00070950: 785f 302e 2e78 5f38 5d3b 2020 2020 2020  x_0..x_8];      
│ │ │ │  00070960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070980: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00070980: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00070990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000709a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000709b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000709c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000709d0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7068  ------+.|i2 : ph
│ │ │ │ -000709e0: 6920 3d20 7261 7469 6f6e 616c 4d61 7020  i = rationalMap 
│ │ │ │ -000709f0: 6964 6561 6c20 6a61 636f 6269 616e 2069  ideal jacobian i
│ │ │ │ -00070a00: 6465 616c 2020 2020 2020 2020 2020 2020  deal            
│ │ │ │ +000709d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000709e0: 3220 3a20 7068 6920 3d20 7261 7469 6f6e  2 : phi = ration
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│ │ │ │ +00070a00: 6269 616e 2069 6465 616c 2020 2020 2020  bian ideal      
│ │ │ │  00070a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070a20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00070a20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00070a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070a70: 2020 2020 2020 7c0a 7c6f 3220 3a20 5261        |.|o2 : Ra
│ │ │ │ -00070a80: 7469 6f6e 616c 4d61 7020 2872 6174 696f  tionalMap (ratio
│ │ │ │ -00070a90: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ -00070aa0: 3820 746f 2020 2020 2020 2020 2020 2020  8 to            
│ │ │ │ +00070a70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00070a80: 3220 3a20 5261 7469 6f6e 616c 4d61 7020  2 : RationalMap 
│ │ │ │ +00070a90: 2872 6174 696f 6e61 6c20 6d61 7020 6672  (rational map fr
│ │ │ │ +00070aa0: 6f6d 2050 505e 3820 746f 2020 2020 2020  om PP^8 to      
│ │ │ │  00070ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070ac0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00070ac0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  00070ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070b10: 2d2d 2d2d 2d2d 7c0a 7c64 6574 206d 6174  ------|.|det mat
│ │ │ │ -00070b20: 7269 787b 7b78 5f30 2e2e 785f 347d 2c7b  rix{{x_0..x_4},{
│ │ │ │ -00070b30: 785f 312e 2e78 5f35 7d2c 7b78 5f32 2e2e  x_1..x_5},{x_2..
│ │ │ │ -00070b40: 785f 367d 2c7b 785f 332e 2e78 5f37 7d2c  x_6},{x_3..x_7},
│ │ │ │ -00070b50: 7b78 5f34 2e2e 785f 387d 7d3b 2020 2020  {x_4..x_8}};    
│ │ │ │ -00070b60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │ +00070b20: 6574 206d 6174 7269 787b 7b78 5f30 2e2e  et matrix{{x_0..
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│ │ │ │ +00070b40: 7b78 5f32 2e2e 785f 367d 2c7b 785f 332e  {x_2..x_6},{x_3.
│ │ │ │ +00070b50: 2e78 5f37 7d2c 7b78 5f34 2e2e 785f 387d  .x_7},{x_4..x_8}
│ │ │ │ +00070b60: 7d3b 2020 2020 2020 2020 2020 7c0a 7c20  };          |.| 
│ │ │ │  00070b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070bb0: 2020 2020 2020 7c0a 7c50 505e 3829 2020        |.|PP^8)  
│ │ │ │ -00070bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c50              |.|P
│ │ │ │ +00070bc0: 505e 3829 2020 2020 2020 2020 2020 2020  P^8)            
│ │ │ │  00070bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070c00: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00070c00: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00070c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070c50: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7469  ------+.|i3 : ti
│ │ │ │ -00070c60: 6d65 2069 7344 6f6d 696e 616e 7428 7068  me isDominant(ph
│ │ │ │ -00070c70: 692c 4365 7274 6966 793d 3e74 7275 6529  i,Certify=>true)
│ │ │ │ -00070c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00070c60: 3320 3a20 7469 6d65 2069 7344 6f6d 696e  3 : time isDomin
│ │ │ │ +00070c70: 616e 7428 7068 692c 4365 7274 6966 793d  ant(phi,Certify=
│ │ │ │ +00070c80: 3e74 7275 6529 2020 2020 2020 2020 2020  >true)          
│ │ │ │  00070c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070ca0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00070cb0: 6420 322e 3235 3133 3973 2028 6370 7529  d 2.25139s (cpu)
│ │ │ │ -00070cc0: 3b20 312e 3837 3135 3573 2028 7468 7265  ; 1.87155s (thre
│ │ │ │ -00070cd0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ -00070ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070cf0: 2020 2020 2020 7c0a 7c43 6572 7469 6679        |.|Certify
│ │ │ │ -00070d00: 3a20 6f75 7470 7574 2063 6572 7469 6669  : output certifi
│ │ │ │ -00070d10: 6564 2120 2020 2020 2020 2020 2020 2020  ed!             
│ │ │ │ +00070ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00070cb0: 2d2d 2075 7365 6420 322e 3234 3238 3173  -- used 2.24281s
│ │ │ │ +00070cc0: 2028 6370 7529 3b20 322e 3038 3831 3373   (cpu); 2.08813s
│ │ │ │ +00070cd0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +00070ce0: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +00070cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c43              |.|C
│ │ │ │ +00070d00: 6572 7469 6679 3a20 6f75 7470 7574 2063  ertify: output c
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│ │ │ │  00070d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070d40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00070d40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00070d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070d90: 2020 2020 2020 7c0a 7c6f 3320 3d20 7472        |.|o3 = tr
│ │ │ │ -00070da0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00070d90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00070da0: 3320 3d20 7472 7565 2020 2020 2020 2020  3 = true        
│ │ │ │  00070db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070de0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00070de0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00070df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070e30: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 5037  ------+.|i4 : P7
│ │ │ │ -00070e40: 203d 205a 5a2f 3130 315b 785f 302e 2e78   = ZZ/101[x_0..x
│ │ │ │ -00070e50: 5f37 5d3b 2020 2020 2020 2020 2020 2020  _7];            
│ │ │ │ +00070e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00070e40: 3420 3a20 5037 203d 205a 5a2f 3130 315b  4 : P7 = ZZ/101[
│ │ │ │ +00070e50: 785f 302e 2e78 5f37 5d3b 2020 2020 2020  x_0..x_7];      
│ │ │ │  00070e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070e80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00070e80: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00070e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070ed0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 2d2d  ------+.|i5 : --
│ │ │ │ -00070ee0: 2068 7970 6572 656c 6c69 7074 6963 2063   hyperelliptic c
│ │ │ │ -00070ef0: 7572 7665 206f 6620 6765 6e75 7320 3320  urve of genus 3 
│ │ │ │ -00070f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00070ee0: 3520 3a20 2d2d 2068 7970 6572 656c 6c69  5 : -- hyperelli
│ │ │ │ +00070ef0: 7074 6963 2063 7572 7665 206f 6620 6765  ptic curve of ge
│ │ │ │ +00070f00: 6e75 7320 3320 2020 2020 2020 2020 2020  nus 3           
│ │ │ │  00070f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070f20: 2020 2020 2020 7c0a 7c20 2020 2020 4320        |.|     C 
│ │ │ │ -00070f30: 3d20 6964 6561 6c28 785f 342a 785f 352b  = ideal(x_4*x_5+
│ │ │ │ -00070f40: 3233 2a78 5f35 5e32 2d32 332a 785f 302a  23*x_5^2-23*x_0*
│ │ │ │ -00070f50: 785f 362d 3138 2a78 5f31 2a78 5f36 2b36  x_6-18*x_1*x_6+6
│ │ │ │ -00070f60: 2a78 5f32 2a78 5f36 2b33 372a 785f 332a  *x_2*x_6+37*x_3*
│ │ │ │ -00070f70: 785f 362b 3233 7c0a 7c20 2020 2020 2020  x_6+23|.|       
│ │ │ │ +00070f20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00070f30: 2020 2020 4320 3d20 6964 6561 6c28 785f      C = ideal(x_
│ │ │ │ +00070f40: 342a 785f 352b 3233 2a78 5f35 5e32 2d32  4*x_5+23*x_5^2-2
│ │ │ │ +00070f50: 332a 785f 302a 785f 362d 3138 2a78 5f31  3*x_0*x_6-18*x_1
│ │ │ │ +00070f60: 2a78 5f36 2b36 2a78 5f32 2a78 5f36 2b33  *x_6+6*x_2*x_6+3
│ │ │ │ +00070f70: 372a 785f 332a 785f 362b 3233 7c0a 7c20  7*x_3*x_6+23|.| 
│ │ │ │  00070f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070fc0: 2020 2020 2020 7c0a 7c6f 3520 3a20 4964        |.|o5 : Id
│ │ │ │ -00070fd0: 6561 6c20 6f66 2050 3720 2020 2020 2020  eal of P7       
│ │ │ │ +00070fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00070fd0: 3520 3a20 4964 6561 6c20 6f66 2050 3720  5 : Ideal of P7 
│ │ │ │  00070fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00071000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00071010: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00071010: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  00071020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071060: 2d2d 2d2d 2d2d 7c0a 7c2a 785f 342a 785f  ------|.|*x_4*x_
│ │ │ │ -00071070: 362d 3236 2a78 5f35 2a78 5f36 2b32 2a78  6-26*x_5*x_6+2*x
│ │ │ │ -00071080: 5f36 5e32 2d32 352a 785f 302a 785f 372b  _6^2-25*x_0*x_7+
│ │ │ │ -00071090: 3435 2a78 5f31 2a78 5f37 2b33 302a 785f  45*x_1*x_7+30*x_
│ │ │ │ -000710a0: 322a 785f 372d 3439 2a78 5f33 2a78 5f37  2*x_7-49*x_3*x_7
│ │ │ │ -000710b0: 2d34 392a 785f 7c0a 7c2d 2d2d 2d2d 2d2d  -49*x_|.|-------
│ │ │ │ +00071060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a  ------------|.|*
│ │ │ │ +00071070: 785f 342a 785f 362d 3236 2a78 5f35 2a78  x_4*x_6-26*x_5*x
│ │ │ │ +00071080: 5f36 2b32 2a78 5f36 5e32 2d32 352a 785f  _6+2*x_6^2-25*x_
│ │ │ │ +00071090: 302a 785f 372b 3435 2a78 5f31 2a78 5f37  0*x_7+45*x_1*x_7
│ │ │ │ +000710a0: 2b33 302a 785f 322a 785f 372d 3439 2a78  +30*x_2*x_7-49*x
│ │ │ │ +000710b0: 5f33 2a78 5f37 2d34 392a 785f 7c0a 7c2d  _3*x_7-49*x_|.|-
│ │ │ │  000710c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000710d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000710e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000710f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071100: 2d2d 2d2d 2d2d 7c0a 7c34 2a78 5f37 2b35  ------|.|4*x_7+5
│ │ │ │ -00071110: 302a 785f 352a 785f 372c 785f 332a 785f  0*x_5*x_7,x_3*x_
│ │ │ │ -00071120: 352d 3234 2a78 5f35 5e32 2b32 312a 785f  5-24*x_5^2+21*x_
│ │ │ │ -00071130: 302a 785f 362b 785f 312a 785f 362b 3436  0*x_6+x_1*x_6+46
│ │ │ │ -00071140: 2a78 5f33 2a78 5f36 2b32 372a 785f 342a  *x_3*x_6+27*x_4*
│ │ │ │ -00071150: 785f 362b 352a 7c0a 7c2d 2d2d 2d2d 2d2d  x_6+5*|.|-------
│ │ │ │ +00071100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34  ------------|.|4
│ │ │ │ +00071110: 2a78 5f37 2b35 302a 785f 352a 785f 372c  *x_7+50*x_5*x_7,
│ │ │ │ +00071120: 785f 332a 785f 352d 3234 2a78 5f35 5e32  x_3*x_5-24*x_5^2
│ │ │ │ +00071130: 2b32 312a 785f 302a 785f 362b 785f 312a  +21*x_0*x_6+x_1*
│ │ │ │ +00071140: 785f 362b 3436 2a78 5f33 2a78 5f36 2b32  x_6+46*x_3*x_6+2
│ │ │ │ +00071150: 372a 785f 342a 785f 362b 352a 7c0a 7c2d  7*x_4*x_6+5*|.|-
│ │ │ │  00071160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000711a0: 2d2d 2d2d 2d2d 7c0a 7c78 5f35 2a78 5f36  ------|.|x_5*x_6
│ │ │ │ -000711b0: 2b33 352a 785f 365e 322b 3230 2a78 5f30  +35*x_6^2+20*x_0
│ │ │ │ -000711c0: 2a78 5f37 2d32 332a 785f 312a 785f 372b  *x_7-23*x_1*x_7+
│ │ │ │ -000711d0: 382a 785f 322a 785f 372d 3232 2a78 5f33  8*x_2*x_7-22*x_3
│ │ │ │ -000711e0: 2a78 5f37 2b32 302a 785f 342a 785f 372d  *x_7+20*x_4*x_7-
│ │ │ │ -000711f0: 3135 2a78 5f35 7c0a 7c2d 2d2d 2d2d 2d2d  15*x_5|.|-------
│ │ │ │ +000711a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c78  ------------|.|x
│ │ │ │ +000711b0: 5f35 2a78 5f36 2b33 352a 785f 365e 322b  _5*x_6+35*x_6^2+
│ │ │ │ +000711c0: 3230 2a78 5f30 2a78 5f37 2d32 332a 785f  20*x_0*x_7-23*x_
│ │ │ │ +000711d0: 312a 785f 372b 382a 785f 322a 785f 372d  1*x_7+8*x_2*x_7-
│ │ │ │ +000711e0: 3232 2a78 5f33 2a78 5f37 2b32 302a 785f  22*x_3*x_7+20*x_
│ │ │ │ +000711f0: 342a 785f 372d 3135 2a78 5f35 7c0a 7c2d  4*x_7-15*x_5|.|-
│ │ │ │  00071200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071240: 2d2d 2d2d 2d2d 7c0a 7c2a 785f 372c 785f  ------|.|*x_7,x_
│ │ │ │ -00071250: 322a 785f 352b 3437 2a78 5f35 5e32 2d34  2*x_5+47*x_5^2-4
│ │ │ │ -00071260: 302a 785f 302a 785f 362b 3337 2a78 5f31  0*x_0*x_6+37*x_1
│ │ │ │ -00071270: 2a78 5f36 2d32 352a 785f 322a 785f 362d  *x_6-25*x_2*x_6-
│ │ │ │ -00071280: 3232 2a78 5f33 2a78 5f36 2d38 2a78 5f34  22*x_3*x_6-8*x_4
│ │ │ │ -00071290: 2a78 5f36 2b20 7c0a 7c2d 2d2d 2d2d 2d2d  *x_6+ |.|-------
│ │ │ │ +00071240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a  ------------|.|*
│ │ │ │ +00071250: 785f 372c 785f 322a 785f 352b 3437 2a78  x_7,x_2*x_5+47*x
│ │ │ │ +00071260: 5f35 5e32 2d34 302a 785f 302a 785f 362b  _5^2-40*x_0*x_6+
│ │ │ │ +00071270: 3337 2a78 5f31 2a78 5f36 2d32 352a 785f  37*x_1*x_6-25*x_
│ │ │ │ +00071280: 322a 785f 362d 3232 2a78 5f33 2a78 5f36  2*x_6-22*x_3*x_6
│ │ │ │ +00071290: 2d38 2a78 5f34 2a78 5f36 2b20 7c0a 7c2d  -8*x_4*x_6+ |.|-
│ │ │ │  000712a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000712b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000712c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000712d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000712e0: 2d2d 2d2d 2d2d 7c0a 7c32 372a 785f 352a  ------|.|27*x_5*
│ │ │ │ -000712f0: 785f 362b 3135 2a78 5f36 5e32 2d32 332a  x_6+15*x_6^2-23*
│ │ │ │ -00071300: 785f 302a 785f 372d 3432 2a78 5f31 2a78  x_0*x_7-42*x_1*x
│ │ │ │ -00071310: 5f37 2b32 372a 785f 322a 785f 372b 3335  _7+27*x_2*x_7+35
│ │ │ │ -00071320: 2a78 5f33 2a78 5f37 2b33 392a 785f 342a  *x_3*x_7+39*x_4*
│ │ │ │ -00071330: 785f 372b 3234 7c0a 7c2d 2d2d 2d2d 2d2d  x_7+24|.|-------
│ │ │ │ +000712e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c32  ------------|.|2
│ │ │ │ +000712f0: 372a 785f 352a 785f 362b 3135 2a78 5f36  7*x_5*x_6+15*x_6
│ │ │ │ +00071300: 5e32 2d32 332a 785f 302a 785f 372d 3432  ^2-23*x_0*x_7-42
│ │ │ │ +00071310: 2a78 5f31 2a78 5f37 2b32 372a 785f 322a  *x_1*x_7+27*x_2*
│ │ │ │ +00071320: 785f 372b 3335 2a78 5f33 2a78 5f37 2b33  x_7+35*x_3*x_7+3
│ │ │ │ +00071330: 392a 785f 342a 785f 372b 3234 7c0a 7c2d  9*x_4*x_7+24|.|-
│ │ │ │  00071340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071380: 2d2d 2d2d 2d2d 7c0a 7c2a 785f 352a 785f  ------|.|*x_5*x_
│ │ │ │ -00071390: 372c 785f 312a 785f 352b 3135 2a78 5f35  7,x_1*x_5+15*x_5
│ │ │ │ -000713a0: 5e32 2b34 392a 785f 302a 785f 362b 382a  ^2+49*x_0*x_6+8*
│ │ │ │ -000713b0: 785f 312a 785f 362d 3331 2a78 5f32 2a78  x_1*x_6-31*x_2*x
│ │ │ │ -000713c0: 5f36 2b39 2a78 5f33 2a78 5f36 2b33 382a  _6+9*x_3*x_6+38*
│ │ │ │ -000713d0: 785f 342a 785f 7c0a 7c2d 2d2d 2d2d 2d2d  x_4*x_|.|-------
│ │ │ │ +00071380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a  ------------|.|*
│ │ │ │ +00071390: 785f 352a 785f 372c 785f 312a 785f 352b  x_5*x_7,x_1*x_5+
│ │ │ │ +000713a0: 3135 2a78 5f35 5e32 2b34 392a 785f 302a  15*x_5^2+49*x_0*
│ │ │ │ +000713b0: 785f 362b 382a 785f 312a 785f 362d 3331  x_6+8*x_1*x_6-31
│ │ │ │ +000713c0: 2a78 5f32 2a78 5f36 2b39 2a78 5f33 2a78  *x_2*x_6+9*x_3*x
│ │ │ │ +000713d0: 5f36 2b33 382a 785f 342a 785f 7c0a 7c2d  _6+38*x_4*x_|.|-
│ │ │ │  000713e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000713f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071420: 2d2d 2d2d 2d2d 7c0a 7c36 2d33 362a 785f  ------|.|6-36*x_
│ │ │ │ -00071430: 352a 785f 362d 3330 2a78 5f36 5e32 2d33  5*x_6-30*x_6^2-3
│ │ │ │ -00071440: 332a 785f 302a 785f 372b 3236 2a78 5f31  3*x_0*x_7+26*x_1
│ │ │ │ -00071450: 2a78 5f37 2b33 322a 785f 322a 785f 372b  *x_7+32*x_2*x_7+
│ │ │ │ -00071460: 3237 2a78 5f33 2a78 5f37 2b36 2a78 5f34  27*x_3*x_7+6*x_4
│ │ │ │ -00071470: 2a78 5f37 2b20 7c0a 7c2d 2d2d 2d2d 2d2d  *x_7+ |.|-------
│ │ │ │ +00071420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c36  ------------|.|6
│ │ │ │ +00071430: 2d33 362a 785f 352a 785f 362d 3330 2a78  -36*x_5*x_6-30*x
│ │ │ │ +00071440: 5f36 5e32 2d33 332a 785f 302a 785f 372b  _6^2-33*x_0*x_7+
│ │ │ │ +00071450: 3236 2a78 5f31 2a78 5f37 2b33 322a 785f  26*x_1*x_7+32*x_
│ │ │ │ +00071460: 322a 785f 372b 3237 2a78 5f33 2a78 5f37  2*x_7+27*x_3*x_7
│ │ │ │ +00071470: 2b36 2a78 5f34 2a78 5f37 2b20 7c0a 7c2d  +6*x_4*x_7+ |.|-
│ │ │ │  00071480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000714a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000714b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000714c0: 2d2d 2d2d 2d2d 7c0a 7c33 362a 785f 352a  ------|.|36*x_5*
│ │ │ │ -000714d0: 785f 372c 785f 302a 785f 352b 3330 2a78  x_7,x_0*x_5+30*x
│ │ │ │ -000714e0: 5f35 5e32 2d31 312a 785f 302a 785f 362d  _5^2-11*x_0*x_6-
│ │ │ │ -000714f0: 3338 2a78 5f31 2a78 5f36 2b31 332a 785f  38*x_1*x_6+13*x_
│ │ │ │ -00071500: 322a 785f 362d 3332 2a78 5f33 2a78 5f36  2*x_6-32*x_3*x_6
│ │ │ │ -00071510: 2d33 302a 785f 7c0a 7c2d 2d2d 2d2d 2d2d  -30*x_|.|-------
│ │ │ │ +000714c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c33  ------------|.|3
│ │ │ │ +000714d0: 362a 785f 352a 785f 372c 785f 302a 785f  6*x_5*x_7,x_0*x_
│ │ │ │ +000714e0: 352b 3330 2a78 5f35 5e32 2d31 312a 785f  5+30*x_5^2-11*x_
│ │ │ │ +000714f0: 302a 785f 362d 3338 2a78 5f31 2a78 5f36  0*x_6-38*x_1*x_6
│ │ │ │ +00071500: 2b31 332a 785f 322a 785f 362d 3332 2a78  +13*x_2*x_6-32*x
│ │ │ │ +00071510: 5f33 2a78 5f36 2d33 302a 785f 7c0a 7c2d  _3*x_6-30*x_|.|-
│ │ │ │  00071520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071560: 2d2d 2d2d 2d2d 7c0a 7c34 2a78 5f36 2b34  ------|.|4*x_6+4
│ │ │ │ -00071570: 2a78 5f35 2a78 5f36 2d32 382a 785f 365e  *x_5*x_6-28*x_6^
│ │ │ │ -00071580: 322d 3330 2a78 5f30 2a78 5f37 2d36 2a78  2-30*x_0*x_7-6*x
│ │ │ │ -00071590: 5f31 2a78 5f37 2d34 352a 785f 322a 785f  _1*x_7-45*x_2*x_
│ │ │ │ -000715a0: 372b 3334 2a78 5f33 2a78 5f37 2b32 302a  7+34*x_3*x_7+20*
│ │ │ │ -000715b0: 785f 342a 785f 7c0a 7c2d 2d2d 2d2d 2d2d  x_4*x_|.|-------
│ │ │ │ +00071560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34  ------------|.|4
│ │ │ │ +00071570: 2a78 5f36 2b34 2a78 5f35 2a78 5f36 2d32  *x_6+4*x_5*x_6-2
│ │ │ │ +00071580: 382a 785f 365e 322d 3330 2a78 5f30 2a78  8*x_6^2-30*x_0*x
│ │ │ │ +00071590: 5f37 2d36 2a78 5f31 2a78 5f37 2d34 352a  _7-6*x_1*x_7-45*
│ │ │ │ +000715a0: 785f 322a 785f 372b 3334 2a78 5f33 2a78  x_2*x_7+34*x_3*x
│ │ │ │ +000715b0: 5f37 2b32 302a 785f 342a 785f 7c0a 7c2d  _7+20*x_4*x_|.|-
│ │ │ │  000715c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000715d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000715e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000715f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071600: 2d2d 2d2d 2d2d 7c0a 7c37 2b34 382a 785f  ------|.|7+48*x_
│ │ │ │ -00071610: 352a 785f 372c 785f 332a 785f 342b 3436  5*x_7,x_3*x_4+46
│ │ │ │ -00071620: 2a78 5f35 5e32 2d33 372a 785f 302a 785f  *x_5^2-37*x_0*x_
│ │ │ │ -00071630: 362b 3237 2a78 5f31 2a78 5f36 2b33 332a  6+27*x_1*x_6+33*
│ │ │ │ -00071640: 785f 322a 785f 362b 382a 785f 332a 785f  x_2*x_6+8*x_3*x_
│ │ │ │ -00071650: 362d 3332 2a78 7c0a 7c2d 2d2d 2d2d 2d2d  6-32*x|.|-------
│ │ │ │ +00071600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c37  ------------|.|7
│ │ │ │ +00071610: 2b34 382a 785f 352a 785f 372c 785f 332a  +48*x_5*x_7,x_3*
│ │ │ │ +00071620: 785f 342b 3436 2a78 5f35 5e32 2d33 372a  x_4+46*x_5^2-37*
│ │ │ │ +00071630: 785f 302a 785f 362b 3237 2a78 5f31 2a78  x_0*x_6+27*x_1*x
│ │ │ │ +00071640: 5f36 2b33 332a 785f 322a 785f 362b 382a  _6+33*x_2*x_6+8*
│ │ │ │ +00071650: 785f 332a 785f 362d 3332 2a78 7c0a 7c2d  x_3*x_6-32*x|.|-
│ │ │ │  00071660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000716a0: 2d2d 2d2d 2d2d 7c0a 7c5f 342a 785f 362b  ------|.|_4*x_6+
│ │ │ │ -000716b0: 3432 2a78 5f35 2a78 5f36 2d33 342a 785f  42*x_5*x_6-34*x_
│ │ │ │ -000716c0: 365e 322d 3337 2a78 5f30 2a78 5f37 2d32  6^2-37*x_0*x_7-2
│ │ │ │ -000716d0: 382a 785f 312a 785f 372b 3130 2a78 5f32  8*x_1*x_7+10*x_2
│ │ │ │ -000716e0: 2a78 5f37 2d32 372a 785f 332a 785f 372d  *x_7-27*x_3*x_7-
│ │ │ │ -000716f0: 3432 2a78 5f34 7c0a 7c2d 2d2d 2d2d 2d2d  42*x_4|.|-------
│ │ │ │ +000716a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f  ------------|.|_
│ │ │ │ +000716b0: 342a 785f 362b 3432 2a78 5f35 2a78 5f36  4*x_6+42*x_5*x_6
│ │ │ │ +000716c0: 2d33 342a 785f 365e 322d 3337 2a78 5f30  -34*x_6^2-37*x_0
│ │ │ │ +000716d0: 2a78 5f37 2d32 382a 785f 312a 785f 372b  *x_7-28*x_1*x_7+
│ │ │ │ +000716e0: 3130 2a78 5f32 2a78 5f37 2d32 372a 785f  10*x_2*x_7-27*x_
│ │ │ │ +000716f0: 332a 785f 372d 3432 2a78 5f34 7c0a 7c2d  3*x_7-42*x_4|.|-
│ │ │ │  00071700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071740: 2d2d 2d2d 2d2d 7c0a 7c2a 785f 372d 382a  ------|.|*x_7-8*
│ │ │ │ -00071750: 785f 352a 785f 372c 785f 322a 785f 342d  x_5*x_7,x_2*x_4-
│ │ │ │ -00071760: 3235 2a78 5f35 5e32 2d34 2a78 5f30 2a78  25*x_5^2-4*x_0*x
│ │ │ │ -00071770: 5f36 2b32 2a78 5f31 2a78 5f36 2d33 312a  _6+2*x_1*x_6-31*
│ │ │ │ -00071780: 785f 322a 785f 362d 352a 785f 332a 785f  x_2*x_6-5*x_3*x_
│ │ │ │ -00071790: 362b 3136 2a78 7c0a 7c2d 2d2d 2d2d 2d2d  6+16*x|.|-------
│ │ │ │ +00071740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a  ------------|.|*
│ │ │ │ +00071750: 785f 372d 382a 785f 352a 785f 372c 785f  x_7-8*x_5*x_7,x_
│ │ │ │ +00071760: 322a 785f 342d 3235 2a78 5f35 5e32 2d34  2*x_4-25*x_5^2-4
│ │ │ │ +00071770: 2a78 5f30 2a78 5f36 2b32 2a78 5f31 2a78  *x_0*x_6+2*x_1*x
│ │ │ │ +00071780: 5f36 2d33 312a 785f 322a 785f 362d 352a  _6-31*x_2*x_6-5*
│ │ │ │ +00071790: 785f 332a 785f 362b 3136 2a78 7c0a 7c2d  x_3*x_6+16*x|.|-
│ │ │ │  000717a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000717b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000717c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000717d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000717e0: 2d2d 2d2d 2d2d 7c0a 7c5f 342a 785f 362d  ------|.|_4*x_6-
│ │ │ │ -000717f0: 3234 2a78 5f35 2a78 5f36 2b33 312a 785f  24*x_5*x_6+31*x_
│ │ │ │ -00071800: 365e 322d 3330 2a78 5f30 2a78 5f37 2b33  6^2-30*x_0*x_7+3
│ │ │ │ -00071810: 322a 785f 312a 785f 372b 3132 2a78 5f32  2*x_1*x_7+12*x_2
│ │ │ │ -00071820: 2a78 5f37 2d34 302a 785f 332a 785f 372b  *x_7-40*x_3*x_7+
│ │ │ │ -00071830: 332a 785f 342a 7c0a 7c2d 2d2d 2d2d 2d2d  3*x_4*|.|-------
│ │ │ │ +000717e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f  ------------|.|_
│ │ │ │ +000717f0: 342a 785f 362d 3234 2a78 5f35 2a78 5f36  4*x_6-24*x_5*x_6
│ │ │ │ +00071800: 2b33 312a 785f 365e 322d 3330 2a78 5f30  +31*x_6^2-30*x_0
│ │ │ │ +00071810: 2a78 5f37 2b33 322a 785f 312a 785f 372b  *x_7+32*x_1*x_7+
│ │ │ │ +00071820: 3132 2a78 5f32 2a78 5f37 2d34 302a 785f  12*x_2*x_7-40*x_
│ │ │ │ +00071830: 332a 785f 372b 332a 785f 342a 7c0a 7c2d  3*x_7+3*x_4*|.|-
│ │ │ │  00071840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071880: 2d2d 2d2d 2d2d 7c0a 7c78 5f37 2d32 382a  ------|.|x_7-28*
│ │ │ │ -00071890: 785f 352a 785f 372c 785f 302a 785f 342b  x_5*x_7,x_0*x_4+
│ │ │ │ -000718a0: 3135 2a78 5f35 5e32 2b34 382a 785f 302a  15*x_5^2+48*x_0*
│ │ │ │ -000718b0: 785f 362d 3530 2a78 5f31 2a78 5f36 2b34  x_6-50*x_1*x_6+4
│ │ │ │ -000718c0: 362a 785f 322a 785f 362d 3438 2a78 5f33  6*x_2*x_6-48*x_3
│ │ │ │ -000718d0: 2a78 5f36 2d20 7c0a 7c2d 2d2d 2d2d 2d2d  *x_6- |.|-------
│ │ │ │ +00071880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c78  ------------|.|x
│ │ │ │ +00071890: 5f37 2d32 382a 785f 352a 785f 372c 785f  _7-28*x_5*x_7,x_
│ │ │ │ +000718a0: 302a 785f 342b 3135 2a78 5f35 5e32 2b34  0*x_4+15*x_5^2+4
│ │ │ │ +000718b0: 382a 785f 302a 785f 362d 3530 2a78 5f31  8*x_0*x_6-50*x_1
│ │ │ │ +000718c0: 2a78 5f36 2b34 362a 785f 322a 785f 362d  *x_6+46*x_2*x_6-
│ │ │ │ +000718d0: 3438 2a78 5f33 2a78 5f36 2d20 7c0a 7c2d  48*x_3*x_6- |.|-
│ │ │ │  000718e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000718f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071920: 2d2d 2d2d 2d2d 7c0a 7c32 332a 785f 342a  ------|.|23*x_4*
│ │ │ │ -00071930: 785f 362d 3238 2a78 5f35 2a78 5f36 2b33  x_6-28*x_5*x_6+3
│ │ │ │ -00071940: 392a 785f 365e 322b 3338 2a78 5f31 2a78  9*x_6^2+38*x_1*x
│ │ │ │ -00071950: 5f37 2d35 2a78 5f33 2a78 5f37 2b35 2a78  _7-5*x_3*x_7+5*x
│ │ │ │ -00071960: 5f34 2a78 5f37 2d33 342a 785f 352a 785f  _4*x_7-34*x_5*x_
│ │ │ │ -00071970: 372c 785f 335e 7c0a 7c2d 2d2d 2d2d 2d2d  7,x_3^|.|-------
│ │ │ │ +00071920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c32  ------------|.|2
│ │ │ │ +00071930: 332a 785f 342a 785f 362d 3238 2a78 5f35  3*x_4*x_6-28*x_5
│ │ │ │ +00071940: 2a78 5f36 2b33 392a 785f 365e 322b 3338  *x_6+39*x_6^2+38
│ │ │ │ +00071950: 2a78 5f31 2a78 5f37 2d35 2a78 5f33 2a78  *x_1*x_7-5*x_3*x
│ │ │ │ +00071960: 5f37 2b35 2a78 5f34 2a78 5f37 2d33 342a  _7+5*x_4*x_7-34*
│ │ │ │ +00071970: 785f 352a 785f 372c 785f 335e 7c0a 7c2d  x_5*x_7,x_3^|.|-
│ │ │ │  00071980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000719a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000719b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000719c0: 2d2d 2d2d 2d2d 7c0a 7c32 2d33 312a 785f  ------|.|2-31*x_
│ │ │ │ -000719d0: 355e 322b 3431 2a78 5f30 2a78 5f36 2d33  5^2+41*x_0*x_6-3
│ │ │ │ -000719e0: 302a 785f 312a 785f 362d 342a 785f 322a  0*x_1*x_6-4*x_2*
│ │ │ │ -000719f0: 785f 362b 3433 2a78 5f33 2a78 5f36 2b32  x_6+43*x_3*x_6+2
│ │ │ │ -00071a00: 332a 785f 342a 785f 362b 372a 785f 352a  3*x_4*x_6+7*x_5*
│ │ │ │ -00071a10: 785f 362b 3331 7c0a 7c2d 2d2d 2d2d 2d2d  x_6+31|.|-------
│ │ │ │ +000719c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c32  ------------|.|2
│ │ │ │ +000719d0: 2d33 312a 785f 355e 322b 3431 2a78 5f30  -31*x_5^2+41*x_0
│ │ │ │ +000719e0: 2a78 5f36 2d33 302a 785f 312a 785f 362d  *x_6-30*x_1*x_6-
│ │ │ │ +000719f0: 342a 785f 322a 785f 362b 3433 2a78 5f33  4*x_2*x_6+43*x_3
│ │ │ │ +00071a00: 2a78 5f36 2b32 332a 785f 342a 785f 362b  *x_6+23*x_4*x_6+
│ │ │ │ +00071a10: 372a 785f 352a 785f 362b 3331 7c0a 7c2d  7*x_5*x_6+31|.|-
│ │ │ │  00071a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071a60: 2d2d 2d2d 2d2d 7c0a 7c2a 785f 365e 322d  ------|.|*x_6^2-
│ │ │ │ -00071a70: 3139 2a78 5f30 2a78 5f37 2b32 352a 785f  19*x_0*x_7+25*x_
│ │ │ │ -00071a80: 312a 785f 372d 3439 2a78 5f32 2a78 5f37  1*x_7-49*x_2*x_7
│ │ │ │ -00071a90: 2d31 362a 785f 332a 785f 372d 3435 2a78  -16*x_3*x_7-45*x
│ │ │ │ -00071aa0: 5f34 2a78 5f37 2b32 352a 785f 352a 785f  _4*x_7+25*x_5*x_
│ │ │ │ -00071ab0: 372c 785f 322a 7c0a 7c2d 2d2d 2d2d 2d2d  7,x_2*|.|-------
│ │ │ │ +00071a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a  ------------|.|*
│ │ │ │ +00071a70: 785f 365e 322d 3139 2a78 5f30 2a78 5f37  x_6^2-19*x_0*x_7
│ │ │ │ +00071a80: 2b32 352a 785f 312a 785f 372d 3439 2a78  +25*x_1*x_7-49*x
│ │ │ │ +00071a90: 5f32 2a78 5f37 2d31 362a 785f 332a 785f  _2*x_7-16*x_3*x_
│ │ │ │ +00071aa0: 372d 3435 2a78 5f34 2a78 5f37 2b32 352a  7-45*x_4*x_7+25*
│ │ │ │ +00071ab0: 785f 352a 785f 372c 785f 322a 7c0a 7c2d  x_5*x_7,x_2*|.|-
│ │ │ │  00071ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071b00: 2d2d 2d2d 2d2d 7c0a 7c78 5f33 2b31 332a  ------|.|x_3+13*
│ │ │ │ -00071b10: 785f 355e 322d 3435 2a78 5f30 2a78 5f36  x_5^2-45*x_0*x_6
│ │ │ │ -00071b20: 2d32 322a 785f 312a 785f 362b 3333 2a78  -22*x_1*x_6+33*x
│ │ │ │ -00071b30: 5f32 2a78 5f36 2d32 362a 785f 332a 785f  _2*x_6-26*x_3*x_
│ │ │ │ -00071b40: 362d 3231 2a78 5f34 2a78 5f36 2b33 342a  6-21*x_4*x_6+34*
│ │ │ │ -00071b50: 785f 352a 785f 7c0a 7c2d 2d2d 2d2d 2d2d  x_5*x_|.|-------
│ │ │ │ +00071b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c78  ------------|.|x
│ │ │ │ +00071b10: 5f33 2b31 332a 785f 355e 322d 3435 2a78  _3+13*x_5^2-45*x
│ │ │ │ +00071b20: 5f30 2a78 5f36 2d32 322a 785f 312a 785f  _0*x_6-22*x_1*x_
│ │ │ │ +00071b30: 362b 3333 2a78 5f32 2a78 5f36 2d32 362a  6+33*x_2*x_6-26*
│ │ │ │ +00071b40: 785f 332a 785f 362d 3231 2a78 5f34 2a78  x_3*x_6-21*x_4*x
│ │ │ │ +00071b50: 5f36 2b33 342a 785f 352a 785f 7c0a 7c2d  _6+34*x_5*x_|.|-
│ │ │ │  00071b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071ba0: 2d2d 2d2d 2d2d 7c0a 7c36 2d32 312a 785f  ------|.|6-21*x_
│ │ │ │ -00071bb0: 365e 322d 3437 2a78 5f30 2a78 5f37 2d31  6^2-47*x_0*x_7-1
│ │ │ │ -00071bc0: 302a 785f 312a 785f 372b 3239 2a78 5f32  0*x_1*x_7+29*x_2
│ │ │ │ -00071bd0: 2a78 5f37 2d34 362a 785f 332a 785f 372d  *x_7-46*x_3*x_7-
│ │ │ │ -00071be0: 785f 342a 785f 372b 3230 2a78 5f35 2a78  x_4*x_7+20*x_5*x
│ │ │ │ -00071bf0: 5f37 2c78 5f31 7c0a 7c2d 2d2d 2d2d 2d2d  _7,x_1|.|-------
│ │ │ │ +00071ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c36  ------------|.|6
│ │ │ │ +00071bb0: 2d32 312a 785f 365e 322d 3437 2a78 5f30  -21*x_6^2-47*x_0
│ │ │ │ +00071bc0: 2a78 5f37 2d31 302a 785f 312a 785f 372b  *x_7-10*x_1*x_7+
│ │ │ │ +00071bd0: 3239 2a78 5f32 2a78 5f37 2d34 362a 785f  29*x_2*x_7-46*x_
│ │ │ │ +00071be0: 332a 785f 372d 785f 342a 785f 372b 3230  3*x_7-x_4*x_7+20
│ │ │ │ +00071bf0: 2a78 5f35 2a78 5f37 2c78 5f31 7c0a 7c2d  *x_5*x_7,x_1|.|-
│ │ │ │  00071c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071c40: 2d2d 2d2d 2d2d 7c0a 7c2a 785f 332b 3232  ------|.|*x_3+22
│ │ │ │ -00071c50: 2a78 5f35 5e32 2b34 2a78 5f30 2a78 5f36  *x_5^2+4*x_0*x_6
│ │ │ │ -00071c60: 2b33 2a78 5f31 2a78 5f36 2b34 352a 785f  +3*x_1*x_6+45*x_
│ │ │ │ -00071c70: 322a 785f 362b 3337 2a78 5f33 2a78 5f36  2*x_6+37*x_3*x_6
│ │ │ │ -00071c80: 2b31 372a 785f 342a 785f 362b 3336 2a78  +17*x_4*x_6+36*x
│ │ │ │ -00071c90: 5f35 2a78 5f36 7c0a 7c2d 2d2d 2d2d 2d2d  _5*x_6|.|-------
│ │ │ │ +00071c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a  ------------|.|*
│ │ │ │ +00071c50: 785f 332b 3232 2a78 5f35 5e32 2b34 2a78  x_3+22*x_5^2+4*x
│ │ │ │ +00071c60: 5f30 2a78 5f36 2b33 2a78 5f31 2a78 5f36  _0*x_6+3*x_1*x_6
│ │ │ │ +00071c70: 2b34 352a 785f 322a 785f 362b 3337 2a78  +45*x_2*x_6+37*x
│ │ │ │ +00071c80: 5f33 2a78 5f36 2b31 372a 785f 342a 785f  _3*x_6+17*x_4*x_
│ │ │ │ +00071c90: 362b 3336 2a78 5f35 2a78 5f36 7c0a 7c2d  6+36*x_5*x_6|.|-
│ │ │ │  00071ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071ce0: 2d2d 2d2d 2d2d 7c0a 7c2d 322a 785f 365e  ------|.|-2*x_6^
│ │ │ │ -00071cf0: 322d 3331 2a78 5f30 2a78 5f37 2b33 2a78  2-31*x_0*x_7+3*x
│ │ │ │ -00071d00: 5f31 2a78 5f37 2d31 322a 785f 322a 785f  _1*x_7-12*x_2*x_
│ │ │ │ -00071d10: 372b 3139 2a78 5f33 2a78 5f37 2b32 382a  7+19*x_3*x_7+28*
│ │ │ │ -00071d20: 785f 342a 785f 372b 3330 2a78 5f35 2a78  x_4*x_7+30*x_5*x
│ │ │ │ -00071d30: 5f37 2c78 5f30 7c0a 7c2d 2d2d 2d2d 2d2d  _7,x_0|.|-------
│ │ │ │ +00071ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d  ------------|.|-
│ │ │ │ +00071cf0: 322a 785f 365e 322d 3331 2a78 5f30 2a78  2*x_6^2-31*x_0*x
│ │ │ │ +00071d00: 5f37 2b33 2a78 5f31 2a78 5f37 2d31 322a  _7+3*x_1*x_7-12*
│ │ │ │ +00071d10: 785f 322a 785f 372b 3139 2a78 5f33 2a78  x_2*x_7+19*x_3*x
│ │ │ │ +00071d20: 5f37 2b32 382a 785f 342a 785f 372b 3330  _7+28*x_4*x_7+30
│ │ │ │ +00071d30: 2a78 5f35 2a78 5f37 2c78 5f30 7c0a 7c2d  *x_5*x_7,x_0|.|-
│ │ │ │  00071d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071d80: 2d2d 2d2d 2d2d 7c0a 7c2a 785f 332d 3437  ------|.|*x_3-47
│ │ │ │ -00071d90: 2a78 5f35 5e32 2d34 332a 785f 302a 785f  *x_5^2-43*x_0*x_
│ │ │ │ -00071da0: 362b 362a 785f 312a 785f 362d 3430 2a78  6+6*x_1*x_6-40*x
│ │ │ │ -00071db0: 5f32 2a78 5f36 2b32 312a 785f 332a 785f  _2*x_6+21*x_3*x_
│ │ │ │ -00071dc0: 362b 3236 2a78 5f34 2a78 5f36 2d35 2a78  6+26*x_4*x_6-5*x
│ │ │ │ -00071dd0: 5f35 2a78 5f36 7c0a 7c2d 2d2d 2d2d 2d2d  _5*x_6|.|-------
│ │ │ │ +00071d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a  ------------|.|*
│ │ │ │ +00071d90: 785f 332d 3437 2a78 5f35 5e32 2d34 332a  x_3-47*x_5^2-43*
│ │ │ │ +00071da0: 785f 302a 785f 362b 362a 785f 312a 785f  x_0*x_6+6*x_1*x_
│ │ │ │ +00071db0: 362d 3430 2a78 5f32 2a78 5f36 2b32 312a  6-40*x_2*x_6+21*
│ │ │ │ +00071dc0: 785f 332a 785f 362b 3236 2a78 5f34 2a78  x_3*x_6+26*x_4*x
│ │ │ │ +00071dd0: 5f36 2d35 2a78 5f35 2a78 5f36 7c0a 7c2d  _6-5*x_5*x_6|.|-
│ │ │ │  00071de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071e20: 2d2d 2d2d 2d2d 7c0a 7c2d 352a 785f 365e  ------|.|-5*x_6^
│ │ │ │ -00071e30: 322b 342a 785f 302a 785f 372d 3135 2a78  2+4*x_0*x_7-15*x
│ │ │ │ -00071e40: 5f31 2a78 5f37 2b31 382a 785f 322a 785f  _1*x_7+18*x_2*x_
│ │ │ │ -00071e50: 372d 3331 2a78 5f33 2a78 5f37 2b35 302a  7-31*x_3*x_7+50*
│ │ │ │ -00071e60: 785f 342a 785f 372d 3436 2a78 5f35 2a78  x_4*x_7-46*x_5*x
│ │ │ │ -00071e70: 5f37 2c78 5f32 7c0a 7c2d 2d2d 2d2d 2d2d  _7,x_2|.|-------
│ │ │ │ +00071e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d  ------------|.|-
│ │ │ │ +00071e30: 352a 785f 365e 322b 342a 785f 302a 785f  5*x_6^2+4*x_0*x_
│ │ │ │ +00071e40: 372d 3135 2a78 5f31 2a78 5f37 2b31 382a  7-15*x_1*x_7+18*
│ │ │ │ +00071e50: 785f 322a 785f 372d 3331 2a78 5f33 2a78  x_2*x_7-31*x_3*x
│ │ │ │ +00071e60: 5f37 2b35 302a 785f 342a 785f 372d 3436  _7+50*x_4*x_7-46
│ │ │ │ +00071e70: 2a78 5f35 2a78 5f37 2c78 5f32 7c0a 7c2d  *x_5*x_7,x_2|.|-
│ │ │ │  00071e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071ec0: 2d2d 2d2d 2d2d 7c0a 7c5e 322b 342a 785f  ------|.|^2+4*x_
│ │ │ │ -00071ed0: 355e 322b 3331 2a78 5f30 2a78 5f36 2b34  5^2+31*x_0*x_6+4
│ │ │ │ -00071ee0: 312a 785f 312a 785f 362b 3331 2a78 5f32  1*x_1*x_6+31*x_2
│ │ │ │ -00071ef0: 2a78 5f36 2b32 382a 785f 332a 785f 362b  *x_6+28*x_3*x_6+
│ │ │ │ -00071f00: 3432 2a78 5f34 2a78 5f36 2d32 382a 785f  42*x_4*x_6-28*x_
│ │ │ │ -00071f10: 352a 785f 362d 7c0a 7c2d 2d2d 2d2d 2d2d  5*x_6-|.|-------
│ │ │ │ +00071ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5e  ------------|.|^
│ │ │ │ +00071ed0: 322b 342a 785f 355e 322b 3331 2a78 5f30  2+4*x_5^2+31*x_0
│ │ │ │ +00071ee0: 2a78 5f36 2b34 312a 785f 312a 785f 362b  *x_6+41*x_1*x_6+
│ │ │ │ +00071ef0: 3331 2a78 5f32 2a78 5f36 2b32 382a 785f  31*x_2*x_6+28*x_
│ │ │ │ +00071f00: 332a 785f 362b 3432 2a78 5f34 2a78 5f36  3*x_6+42*x_4*x_6
│ │ │ │ +00071f10: 2d32 382a 785f 352a 785f 362d 7c0a 7c2d  -28*x_5*x_6-|.|-
│ │ │ │  00071f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071f60: 2d2d 2d2d 2d2d 7c0a 7c34 2a78 5f36 5e32  ------|.|4*x_6^2
│ │ │ │ -00071f70: 2d37 2a78 5f30 2a78 5f37 2b31 352a 785f  -7*x_0*x_7+15*x_
│ │ │ │ -00071f80: 312a 785f 372d 392a 785f 322a 785f 372b  1*x_7-9*x_2*x_7+
│ │ │ │ -00071f90: 3331 2a78 5f33 2a78 5f37 2b33 2a78 5f34  31*x_3*x_7+3*x_4
│ │ │ │ -00071fa0: 2a78 5f37 2b37 2a78 5f35 2a78 5f37 2c78  *x_7+7*x_5*x_7,x
│ │ │ │ -00071fb0: 5f31 2a78 5f32 7c0a 7c2d 2d2d 2d2d 2d2d  _1*x_2|.|-------
│ │ │ │ +00071f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34  ------------|.|4
│ │ │ │ +00071f70: 2a78 5f36 5e32 2d37 2a78 5f30 2a78 5f37  *x_6^2-7*x_0*x_7
│ │ │ │ +00071f80: 2b31 352a 785f 312a 785f 372d 392a 785f  +15*x_1*x_7-9*x_
│ │ │ │ +00071f90: 322a 785f 372b 3331 2a78 5f33 2a78 5f37  2*x_7+31*x_3*x_7
│ │ │ │ +00071fa0: 2b33 2a78 5f34 2a78 5f37 2b37 2a78 5f35  +3*x_4*x_7+7*x_5
│ │ │ │ +00071fb0: 2a78 5f37 2c78 5f31 2a78 5f32 7c0a 7c2d  *x_7,x_1*x_2|.|-
│ │ │ │  00071fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072000: 2d2d 2d2d 2d2d 7c0a 7c2d 3436 2a78 5f35  ------|.|-46*x_5
│ │ │ │ -00072010: 5e32 2d36 2a78 5f30 2a78 5f36 2d35 302a  ^2-6*x_0*x_6-50*
│ │ │ │ -00072020: 785f 312a 785f 362b 3332 2a78 5f32 2a78  x_1*x_6+32*x_2*x
│ │ │ │ -00072030: 5f36 2d31 302a 785f 332a 785f 362b 3432  _6-10*x_3*x_6+42
│ │ │ │ -00072040: 2a78 5f34 2a78 5f36 2b33 332a 785f 352a  *x_4*x_6+33*x_5*
│ │ │ │ -00072050: 785f 362b 3138 7c0a 7c2d 2d2d 2d2d 2d2d  x_6+18|.|-------
│ │ │ │ +00072000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d  ------------|.|-
│ │ │ │ +00072010: 3436 2a78 5f35 5e32 2d36 2a78 5f30 2a78  46*x_5^2-6*x_0*x
│ │ │ │ +00072020: 5f36 2d35 302a 785f 312a 785f 362b 3332  _6-50*x_1*x_6+32
│ │ │ │ +00072030: 2a78 5f32 2a78 5f36 2d31 302a 785f 332a  *x_2*x_6-10*x_3*
│ │ │ │ +00072040: 785f 362b 3432 2a78 5f34 2a78 5f36 2b33  x_6+42*x_4*x_6+3
│ │ │ │ +00072050: 332a 785f 352a 785f 362b 3138 7c0a 7c2d  3*x_5*x_6+18|.|-
│ │ │ │  00072060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000720a0: 2d2d 2d2d 2d2d 7c0a 7c2a 785f 365e 322d  ------|.|*x_6^2-
│ │ │ │ -000720b0: 392a 785f 302a 785f 372d 3230 2a78 5f31  9*x_0*x_7-20*x_1
│ │ │ │ -000720c0: 2a78 5f37 2b34 352a 785f 322a 785f 372d  *x_7+45*x_2*x_7-
│ │ │ │ -000720d0: 392a 785f 332a 785f 372b 3130 2a78 5f34  9*x_3*x_7+10*x_4
│ │ │ │ -000720e0: 2a78 5f37 2d38 2a78 5f35 2a78 5f37 2c78  *x_7-8*x_5*x_7,x
│ │ │ │ -000720f0: 5f30 2a78 5f32 7c0a 7c2d 2d2d 2d2d 2d2d  _0*x_2|.|-------
│ │ │ │ +000720a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a  ------------|.|*
│ │ │ │ +000720b0: 785f 365e 322d 392a 785f 302a 785f 372d  x_6^2-9*x_0*x_7-
│ │ │ │ +000720c0: 3230 2a78 5f31 2a78 5f37 2b34 352a 785f  20*x_1*x_7+45*x_
│ │ │ │ +000720d0: 322a 785f 372d 392a 785f 332a 785f 372b  2*x_7-9*x_3*x_7+
│ │ │ │ +000720e0: 3130 2a78 5f34 2a78 5f37 2d38 2a78 5f35  10*x_4*x_7-8*x_5
│ │ │ │ +000720f0: 2a78 5f37 2c78 5f30 2a78 5f32 7c0a 7c2d  *x_7,x_0*x_2|.|-
│ │ │ │  00072100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072140: 2d2d 2d2d 2d2d 7c0a 7c2d 392a 785f 355e  ------|.|-9*x_5^
│ │ │ │ -00072150: 322b 3334 2a78 5f30 2a78 5f36 2d34 352a  2+34*x_0*x_6-45*
│ │ │ │ -00072160: 785f 312a 785f 362b 3139 2a78 5f32 2a78  x_1*x_6+19*x_2*x
│ │ │ │ -00072170: 5f36 2b32 342a 785f 332a 785f 362b 3233  _6+24*x_3*x_6+23
│ │ │ │ -00072180: 2a78 5f34 2a78 5f36 2d33 372a 785f 352a  *x_4*x_6-37*x_5*
│ │ │ │ -00072190: 785f 362d 3434 7c0a 7c2d 2d2d 2d2d 2d2d  x_6-44|.|-------
│ │ │ │ +00072140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d  ------------|.|-
│ │ │ │ +00072150: 392a 785f 355e 322b 3334 2a78 5f30 2a78  9*x_5^2+34*x_0*x
│ │ │ │ +00072160: 5f36 2d34 352a 785f 312a 785f 362b 3139  _6-45*x_1*x_6+19
│ │ │ │ +00072170: 2a78 5f32 2a78 5f36 2b32 342a 785f 332a  *x_2*x_6+24*x_3*
│ │ │ │ +00072180: 785f 362b 3233 2a78 5f34 2a78 5f36 2d33  x_6+23*x_4*x_6-3
│ │ │ │ +00072190: 372a 785f 352a 785f 362d 3434 7c0a 7c2d  7*x_5*x_6-44|.|-
│ │ │ │  000721a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000721b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000721c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000721d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000721e0: 2d2d 2d2d 2d2d 7c0a 7c2a 785f 365e 322b  ------|.|*x_6^2+
│ │ │ │ -000721f0: 3234 2a78 5f30 2a78 5f37 2d33 332a 785f  24*x_0*x_7-33*x_
│ │ │ │ -00072200: 322a 785f 372b 3431 2a78 5f33 2a78 5f37  2*x_7+41*x_3*x_7
│ │ │ │ -00072210: 2d34 302a 785f 342a 785f 372b 342a 785f  -40*x_4*x_7+4*x_
│ │ │ │ -00072220: 352a 785f 372c 785f 315e 322b 785f 312a  5*x_7,x_1^2+x_1*
│ │ │ │ -00072230: 785f 342b 785f 7c0a 7c2d 2d2d 2d2d 2d2d  x_4+x_|.|-------
│ │ │ │ +000721e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a  ------------|.|*
│ │ │ │ +000721f0: 785f 365e 322b 3234 2a78 5f30 2a78 5f37  x_6^2+24*x_0*x_7
│ │ │ │ +00072200: 2d33 332a 785f 322a 785f 372b 3431 2a78  -33*x_2*x_7+41*x
│ │ │ │ +00072210: 5f33 2a78 5f37 2d34 302a 785f 342a 785f  _3*x_7-40*x_4*x_
│ │ │ │ +00072220: 372b 342a 785f 352a 785f 372c 785f 315e  7+4*x_5*x_7,x_1^
│ │ │ │ +00072230: 322b 785f 312a 785f 342b 785f 7c0a 7c2d  2+x_1*x_4+x_|.|-
│ │ │ │  00072240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072280: 2d2d 2d2d 2d2d 7c0a 7c34 5e32 2d32 382a  ------|.|4^2-28*
│ │ │ │ -00072290: 785f 355e 322d 3333 2a78 5f30 2a78 5f36  x_5^2-33*x_0*x_6
│ │ │ │ -000722a0: 2d31 372a 785f 312a 785f 362b 3131 2a78  -17*x_1*x_6+11*x
│ │ │ │ -000722b0: 5f33 2a78 5f36 2b32 302a 785f 342a 785f  _3*x_6+20*x_4*x_
│ │ │ │ -000722c0: 362b 3235 2a78 5f35 2a78 5f36 2d32 312a  6+25*x_5*x_6-21*
│ │ │ │ -000722d0: 785f 365e 322d 7c0a 7c2d 2d2d 2d2d 2d2d  x_6^2-|.|-------
│ │ │ │ +00072280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34  ------------|.|4
│ │ │ │ +00072290: 5e32 2d32 382a 785f 355e 322d 3333 2a78  ^2-28*x_5^2-33*x
│ │ │ │ +000722a0: 5f30 2a78 5f36 2d31 372a 785f 312a 785f  _0*x_6-17*x_1*x_
│ │ │ │ +000722b0: 362b 3131 2a78 5f33 2a78 5f36 2b32 302a  6+11*x_3*x_6+20*
│ │ │ │ +000722c0: 785f 342a 785f 362b 3235 2a78 5f35 2a78  x_4*x_6+25*x_5*x
│ │ │ │ +000722d0: 5f36 2d32 312a 785f 365e 322d 7c0a 7c2d  _6-21*x_6^2-|.|-
│ │ │ │  000722e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000722f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072320: 2d2d 2d2d 2d2d 7c0a 7c32 322a 785f 302a  ------|.|22*x_0*
│ │ │ │ -00072330: 785f 372b 3234 2a78 5f31 2a78 5f37 2d31  x_7+24*x_1*x_7-1
│ │ │ │ -00072340: 342a 785f 322a 785f 372b 352a 785f 332a  4*x_2*x_7+5*x_3*
│ │ │ │ -00072350: 785f 372d 3339 2a78 5f34 2a78 5f37 2d31  x_7-39*x_4*x_7-1
│ │ │ │ -00072360: 382a 785f 352a 785f 372c 785f 302a 785f  8*x_5*x_7,x_0*x_
│ │ │ │ -00072370: 312d 3437 2a78 7c0a 7c2d 2d2d 2d2d 2d2d  1-47*x|.|-------
│ │ │ │ +00072320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c32  ------------|.|2
│ │ │ │ +00072330: 322a 785f 302a 785f 372b 3234 2a78 5f31  2*x_0*x_7+24*x_1
│ │ │ │ +00072340: 2a78 5f37 2d31 342a 785f 322a 785f 372b  *x_7-14*x_2*x_7+
│ │ │ │ +00072350: 352a 785f 332a 785f 372d 3339 2a78 5f34  5*x_3*x_7-39*x_4
│ │ │ │ +00072360: 2a78 5f37 2d31 382a 785f 352a 785f 372c  *x_7-18*x_5*x_7,
│ │ │ │ +00072370: 785f 302a 785f 312d 3437 2a78 7c0a 7c2d  x_0*x_1-47*x|.|-
│ │ │ │  00072380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000723a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000723b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000723c0: 2d2d 2d2d 2d2d 7c0a 7c5f 355e 322d 352a  ------|.|_5^2-5*
│ │ │ │ -000723d0: 785f 302a 785f 362d 392a 785f 312a 785f  x_0*x_6-9*x_1*x_
│ │ │ │ -000723e0: 362d 3435 2a78 5f32 2a78 5f36 2b34 382a  6-45*x_2*x_6+48*
│ │ │ │ -000723f0: 785f 332a 785f 362b 3435 2a78 5f34 2a78  x_3*x_6+45*x_4*x
│ │ │ │ -00072400: 5f36 2d32 392a 785f 352a 785f 362b 332a  _6-29*x_5*x_6+3*
│ │ │ │ -00072410: 785f 365e 322b 7c0a 7c2d 2d2d 2d2d 2d2d  x_6^2+|.|-------
│ │ │ │ +000723c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f  ------------|.|_
│ │ │ │ +000723d0: 355e 322d 352a 785f 302a 785f 362d 392a  5^2-5*x_0*x_6-9*
│ │ │ │ +000723e0: 785f 312a 785f 362d 3435 2a78 5f32 2a78  x_1*x_6-45*x_2*x
│ │ │ │ +000723f0: 5f36 2b34 382a 785f 332a 785f 362b 3435  _6+48*x_3*x_6+45
│ │ │ │ +00072400: 2a78 5f34 2a78 5f36 2d32 392a 785f 352a  *x_4*x_6-29*x_5*
│ │ │ │ +00072410: 785f 362b 332a 785f 365e 322b 7c0a 7c2d  x_6+3*x_6^2+|.|-
│ │ │ │  00072420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072460: 2d2d 2d2d 2d2d 7c0a 7c32 392a 785f 302a  ------|.|29*x_0*
│ │ │ │ -00072470: 785f 372b 3430 2a78 5f31 2a78 5f37 2b34  x_7+40*x_1*x_7+4
│ │ │ │ -00072480: 362a 785f 322a 785f 372b 3237 2a78 5f33  6*x_2*x_7+27*x_3
│ │ │ │ -00072490: 2a78 5f37 2d33 362a 785f 342a 785f 372d  *x_7-36*x_4*x_7-
│ │ │ │ -000724a0: 3339 2a78 5f35 2a78 5f37 2c78 5f30 5e32  39*x_5*x_7,x_0^2
│ │ │ │ -000724b0: 2d33 312a 785f 7c0a 7c2d 2d2d 2d2d 2d2d  -31*x_|.|-------
│ │ │ │ +00072460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c32  ------------|.|2
│ │ │ │ +00072470: 392a 785f 302a 785f 372b 3430 2a78 5f31  9*x_0*x_7+40*x_1
│ │ │ │ +00072480: 2a78 5f37 2b34 362a 785f 322a 785f 372b  *x_7+46*x_2*x_7+
│ │ │ │ +00072490: 3237 2a78 5f33 2a78 5f37 2d33 362a 785f  27*x_3*x_7-36*x_
│ │ │ │ +000724a0: 342a 785f 372d 3339 2a78 5f35 2a78 5f37  4*x_7-39*x_5*x_7
│ │ │ │ +000724b0: 2c78 5f30 5e32 2d33 312a 785f 7c0a 7c2d  ,x_0^2-31*x_|.|-
│ │ │ │  000724c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000724d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000724e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000724f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072500: 2d2d 2d2d 2d2d 7c0a 7c35 5e32 2b33 362a  ------|.|5^2+36*
│ │ │ │ -00072510: 785f 302a 785f 362d 3330 2a78 5f31 2a78  x_0*x_6-30*x_1*x
│ │ │ │ -00072520: 5f36 2d31 302a 785f 322a 785f 362b 3432  _6-10*x_2*x_6+42
│ │ │ │ -00072530: 2a78 5f33 2a78 5f36 2b39 2a78 5f34 2a78  *x_3*x_6+9*x_4*x
│ │ │ │ -00072540: 5f36 2b33 342a 785f 352a 785f 362d 362a  _6+34*x_5*x_6-6*
│ │ │ │ -00072550: 785f 365e 322b 7c0a 7c2d 2d2d 2d2d 2d2d  x_6^2+|.|-------
│ │ │ │ +00072500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c35  ------------|.|5
│ │ │ │ +00072510: 5e32 2b33 362a 785f 302a 785f 362d 3330  ^2+36*x_0*x_6-30
│ │ │ │ +00072520: 2a78 5f31 2a78 5f36 2d31 302a 785f 322a  *x_1*x_6-10*x_2*
│ │ │ │ +00072530: 785f 362b 3432 2a78 5f33 2a78 5f36 2b39  x_6+42*x_3*x_6+9
│ │ │ │ +00072540: 2a78 5f34 2a78 5f36 2b33 342a 785f 352a  *x_4*x_6+34*x_5*
│ │ │ │ +00072550: 785f 362d 362a 785f 365e 322b 7c0a 7c2d  x_6-6*x_6^2+|.|-
│ │ │ │  00072560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000725a0: 2d2d 2d2d 2d2d 7c0a 7c34 382a 785f 302a  ------|.|48*x_0*
│ │ │ │ -000725b0: 785f 372d 3437 2a78 5f31 2a78 5f37 2d31  x_7-47*x_1*x_7-1
│ │ │ │ -000725c0: 392a 785f 322a 785f 372b 3235 2a78 5f33  9*x_2*x_7+25*x_3
│ │ │ │ -000725d0: 2a78 5f37 2b32 382a 785f 342a 785f 372b  *x_7+28*x_4*x_7+
│ │ │ │ -000725e0: 3334 2a78 5f35 2a78 5f37 293b 2020 2020  34*x_5*x_7);    
│ │ │ │ -000725f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000725a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34  ------------|.|4
│ │ │ │ +000725b0: 382a 785f 302a 785f 372d 3437 2a78 5f31  8*x_0*x_7-47*x_1
│ │ │ │ +000725c0: 2a78 5f37 2d31 392a 785f 322a 785f 372b  *x_7-19*x_2*x_7+
│ │ │ │ +000725d0: 3235 2a78 5f33 2a78 5f37 2b32 382a 785f  25*x_3*x_7+28*x_
│ │ │ │ +000725e0: 342a 785f 372b 3334 2a78 5f35 2a78 5f37  4*x_7+34*x_5*x_7
│ │ │ │ +000725f0: 293b 2020 2020 2020 2020 2020 7c0a 2b2d  );          |.+-
│ │ │ │  00072600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072640: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 7068  ------+.|i6 : ph
│ │ │ │ -00072650: 6920 3d20 7261 7469 6f6e 616c 4d61 7028  i = rationalMap(
│ │ │ │ -00072660: 432c 332c 3229 3b20 2020 2020 2020 2020  C,3,2);         
│ │ │ │ +00072640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00072650: 3620 3a20 7068 6920 3d20 7261 7469 6f6e  6 : phi = ration
│ │ │ │ +00072660: 616c 4d61 7028 432c 332c 3229 3b20 2020  alMap(C,3,2);   
│ │ │ │  00072670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072690: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00072690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000726a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000726b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000726c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000726d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000726e0: 2020 2020 2020 7c0a 7c6f 3620 3a20 5261        |.|o6 : Ra
│ │ │ │ -000726f0: 7469 6f6e 616c 4d61 7020 2863 7562 6963  tionalMap (cubic
│ │ │ │ -00072700: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ -00072710: 6f6d 2050 505e 3720 746f 2050 505e 3729  om PP^7 to PP^7)
│ │ │ │ -00072720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072730: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000726e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000726f0: 3620 3a20 5261 7469 6f6e 616c 4d61 7020  6 : RationalMap 
│ │ │ │ +00072700: 2863 7562 6963 2072 6174 696f 6e61 6c20  (cubic rational 
│ │ │ │ +00072710: 6d61 7020 6672 6f6d 2050 505e 3720 746f  map from PP^7 to
│ │ │ │ +00072720: 2050 505e 3729 2020 2020 2020 2020 2020   PP^7)          
│ │ │ │ +00072730: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00072740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072780: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 7469  ------+.|i7 : ti
│ │ │ │ -00072790: 6d65 2069 7344 6f6d 696e 616e 7428 7068  me isDominant(ph
│ │ │ │ -000727a0: 692c 4365 7274 6966 793d 3e74 7275 6529  i,Certify=>true)
│ │ │ │ -000727b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00072780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00072790: 3720 3a20 7469 6d65 2069 7344 6f6d 696e  7 : time isDomin
│ │ │ │ +000727a0: 616e 7428 7068 692c 4365 7274 6966 793d  ant(phi,Certify=
│ │ │ │ +000727b0: 3e74 7275 6529 2020 2020 2020 2020 2020  >true)          
│ │ │ │  000727c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000727d0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -000727e0: 6420 342e 3037 3336 3373 2028 6370 7529  d 4.07363s (cpu)
│ │ │ │ -000727f0: 3b20 322e 3735 3330 3373 2028 7468 7265  ; 2.75303s (thre
│ │ │ │ -00072800: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ -00072810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072820: 2020 2020 2020 7c0a 7c43 6572 7469 6679        |.|Certify
│ │ │ │ -00072830: 3a20 6f75 7470 7574 2063 6572 7469 6669  : output certifi
│ │ │ │ -00072840: 6564 2120 2020 2020 2020 2020 2020 2020  ed!             
│ │ │ │ +000727d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000727e0: 2d2d 2075 7365 6420 332e 3230 3630 3673  -- used 3.20606s
│ │ │ │ +000727f0: 2028 6370 7529 3b20 322e 3437 3336 3973   (cpu); 2.47369s
│ │ │ │ +00072800: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +00072810: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +00072820: 2020 2020 2020 2020 2020 2020 7c0a 7c43              |.|C
│ │ │ │ +00072830: 6572 7469 6679 3a20 6f75 7470 7574 2063  ertify: output c
│ │ │ │ +00072840: 6572 7469 6669 6564 2120 2020 2020 2020  ertified!       
│ │ │ │  00072850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072870: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00072870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00072880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000728a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000728b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000728c0: 2020 2020 2020 7c0a 7c6f 3720 3d20 6661        |.|o7 = fa
│ │ │ │ -000728d0: 6c73 6520 2020 2020 2020 2020 2020 2020  lse             
│ │ │ │ +000728c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000728d0: 3720 3d20 6661 6c73 6520 2020 2020 2020  7 = false       
│ │ │ │  000728e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000728f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072910: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00072910: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00072920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072960: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ -00072970: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ -00072980: 2a6e 6f74 6520 6973 4269 7261 7469 6f6e  *note isBiration
│ │ │ │ -00072990: 616c 3a20 6973 4269 7261 7469 6f6e 616c  al: isBirational
│ │ │ │ -000729a0: 2c20 2d2d 2077 6865 7468 6572 2061 2072  , -- whether a r
│ │ │ │ -000729b0: 6174 696f 6e61 6c20 6d61 7020 6973 2062  ational map is b
│ │ │ │ -000729c0: 6972 6174 696f 6e61 6c0a 0a57 6179 7320  irational..Ways 
│ │ │ │ -000729d0: 746f 2075 7365 2069 7344 6f6d 696e 616e  to use isDominan
│ │ │ │ -000729e0: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  t:.=============
│ │ │ │ -000729f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00072a00: 2269 7344 6f6d 696e 616e 7428 5261 7469  "isDominant(Rati
│ │ │ │ -00072a10: 6f6e 616c 4d61 7029 220a 2020 2a20 2269  onalMap)".  * "i
│ │ │ │ -00072a20: 7344 6f6d 696e 616e 7428 5269 6e67 4d61  sDominant(RingMa
│ │ │ │ -00072a30: 7029 220a 0a46 6f72 2074 6865 2070 726f  p)"..For the pro
│ │ │ │ -00072a40: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -00072a50: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -00072a60: 6f62 6a65 6374 202a 6e6f 7465 2069 7344  object *note isD
│ │ │ │ -00072a70: 6f6d 696e 616e 743a 2069 7344 6f6d 696e  ominant: isDomin
│ │ │ │ -00072a80: 616e 742c 2069 7320 6120 2a6e 6f74 6520  ant, is a *note 
│ │ │ │ -00072a90: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ -00072aa0: 7769 7468 0a6f 7074 696f 6e73 3a20 284d  with.options: (M
│ │ │ │ -00072ab0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -00072ac0: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
│ │ │ │ -00072ad0: 7469 6f6e 732c 2e0a 1f0a 4669 6c65 3a20  tions,....File: 
│ │ │ │ -00072ae0: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ -00072af0: 6465 3a20 6973 496e 7665 7273 654d 6170  de: isInverseMap
│ │ │ │ -00072b00: 2c20 4e65 7874 3a20 6973 496e 7665 7273  , Next: isInvers
│ │ │ │ -00072b10: 654d 6170 5f6c 7052 6174 696f 6e61 6c4d  eMap_lpRationalM
│ │ │ │ -00072b20: 6170 5f63 6d52 6174 696f 6e61 6c4d 6170  ap_cmRationalMap
│ │ │ │ -00072b30: 5f72 702c 2050 7265 763a 2069 7344 6f6d  _rp, Prev: isDom
│ │ │ │ -00072b40: 696e 616e 742c 2055 703a 2054 6f70 0a0a  inant, Up: Top..
│ │ │ │ -00072b50: 6973 496e 7665 7273 654d 6170 202d 2d20  isInverseMap -- 
│ │ │ │ -00072b60: 6368 6563 6b73 2077 6865 7468 6572 2061  checks whether a
│ │ │ │ -00072b70: 2072 6174 696f 6e61 6c20 6d61 7020 6973   rational map is
│ │ │ │ -00072b80: 2074 6865 2069 6e76 6572 7365 206f 6620   the inverse of 
│ │ │ │ -00072b90: 616e 6f74 6865 720a 2a2a 2a2a 2a2a 2a2a  another.********
│ │ │ │ +00072960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +00072970: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +00072980: 0a0a 2020 2a20 2a6e 6f74 6520 6973 4269  ..  * *note isBi
│ │ │ │ +00072990: 7261 7469 6f6e 616c 3a20 6973 4269 7261  rational: isBira
│ │ │ │ +000729a0: 7469 6f6e 616c 2c20 2d2d 2077 6865 7468  tional, -- wheth
│ │ │ │ +000729b0: 6572 2061 2072 6174 696f 6e61 6c20 6d61  er a rational ma
│ │ │ │ +000729c0: 7020 6973 2062 6972 6174 696f 6e61 6c0a  p is birational.
│ │ │ │ +000729d0: 0a57 6179 7320 746f 2075 7365 2069 7344  .Ways to use isD
│ │ │ │ +000729e0: 6f6d 696e 616e 743a 0a3d 3d3d 3d3d 3d3d  ominant:.=======
│ │ │ │ +000729f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00072a00: 0a0a 2020 2a20 2269 7344 6f6d 696e 616e  ..  * "isDominan
│ │ │ │ +00072a10: 7428 5261 7469 6f6e 616c 4d61 7029 220a  t(RationalMap)".
│ │ │ │ +00072a20: 2020 2a20 2269 7344 6f6d 696e 616e 7428    * "isDominant(
│ │ │ │ +00072a30: 5269 6e67 4d61 7029 220a 0a46 6f72 2074  RingMap)"..For t
│ │ │ │ +00072a40: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +00072a50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00072a60: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +00072a70: 7465 2069 7344 6f6d 696e 616e 743a 2069  te isDominant: i
│ │ │ │ +00072a80: 7344 6f6d 696e 616e 742c 2069 7320 6120  sDominant, is a 
│ │ │ │ +00072a90: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +00072aa0: 6374 696f 6e20 7769 7468 0a6f 7074 696f  ction with.optio
│ │ │ │ +00072ab0: 6e73 3a20 284d 6163 6175 6c61 7932 446f  ns: (Macaulay2Do
│ │ │ │ +00072ac0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00072ad0: 5769 7468 4f70 7469 6f6e 732c 2e0a 1f0a  WithOptions,....
│ │ │ │ +00072ae0: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ +00072af0: 666f 2c20 4e6f 6465 3a20 6973 496e 7665  fo, Node: isInve
│ │ │ │ +00072b00: 7273 654d 6170 2c20 4e65 7874 3a20 6973  rseMap, Next: is
│ │ │ │ +00072b10: 496e 7665 7273 654d 6170 5f6c 7052 6174  InverseMap_lpRat
│ │ │ │ +00072b20: 696f 6e61 6c4d 6170 5f63 6d52 6174 696f  ionalMap_cmRatio
│ │ │ │ +00072b30: 6e61 6c4d 6170 5f72 702c 2050 7265 763a  nalMap_rp, Prev:
│ │ │ │ +00072b40: 2069 7344 6f6d 696e 616e 742c 2055 703a   isDominant, Up:
│ │ │ │ +00072b50: 2054 6f70 0a0a 6973 496e 7665 7273 654d   Top..isInverseM
│ │ │ │ +00072b60: 6170 202d 2d20 6368 6563 6b73 2077 6865  ap -- checks whe
│ │ │ │ +00072b70: 7468 6572 2061 2072 6174 696f 6e61 6c20  ther a rational 
│ │ │ │ +00072b80: 6d61 7020 6973 2074 6865 2069 6e76 6572  map is the inver
│ │ │ │ +00072b90: 7365 206f 6620 616e 6f74 6865 720a 2a2a  se of another.**
│ │ │ │  00072ba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00072bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00072bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00072bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00072be0: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ -00072bf0: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ -00072c00: 2020 2020 2020 2020 6973 496e 7665 7273          isInvers
│ │ │ │ -00072c10: 654d 6170 2870 6869 2c70 7369 290a 2020  eMap(phi,psi).  
│ │ │ │ -00072c20: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00072c30: 2a20 7068 692c 2061 202a 6e6f 7465 2072  * phi, a *note r
│ │ │ │ -00072c40: 696e 6720 6d61 703a 2028 4d61 6361 756c  ing map: (Macaul
│ │ │ │ -00072c50: 6179 3244 6f63 2952 696e 674d 6170 2c2c  ay2Doc)RingMap,,
│ │ │ │ -00072c60: 2072 6570 7265 7365 6e74 696e 6720 6120   representing a 
│ │ │ │ -00072c70: 7261 7469 6f6e 616c 0a20 2020 2020 2020  rational.       
│ │ │ │ -00072c80: 206d 6170 2024 5c50 6869 3a58 205c 6461   map $\Phi:X \da
│ │ │ │ -00072c90: 7368 7269 6768 7461 7272 6f77 2059 240a  shrightarrow Y$.
│ │ │ │ -00072ca0: 2020 2020 2020 2a20 7073 692c 2061 202a        * psi, a *
│ │ │ │ -00072cb0: 6e6f 7465 2072 696e 6720 6d61 703a 2028  note ring map: (
│ │ │ │ -00072cc0: 4d61 6361 756c 6179 3244 6f63 2952 696e  Macaulay2Doc)Rin
│ │ │ │ -00072cd0: 674d 6170 2c2c 2072 6570 7265 7365 6e74  gMap,, represent
│ │ │ │ -00072ce0: 696e 6720 6120 7261 7469 6f6e 616c 0a20  ing a rational. 
│ │ │ │ -00072cf0: 2020 2020 2020 206d 6170 2024 5c50 7369         map $\Psi
│ │ │ │ -00072d00: 3a59 205c 6461 7368 7269 6768 7461 7272  :Y \dashrightarr
│ │ │ │ -00072d10: 6f77 2058 240a 2020 2a20 4f75 7470 7574  ow X$.  * Output
│ │ │ │ -00072d20: 733a 0a20 2020 2020 202a 2061 202a 6e6f  s:.      * a *no
│ │ │ │ -00072d30: 7465 2042 6f6f 6c65 616e 2076 616c 7565  te Boolean value
│ │ │ │ -00072d40: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00072d50: 426f 6f6c 6561 6e2c 2c20 6163 636f 7264  Boolean,, accord
│ │ │ │ -00072d60: 696e 6720 746f 2074 6865 0a20 2020 2020  ing to the.     
│ │ │ │ -00072d70: 2020 2063 6f6e 6469 7469 6f6e 2074 6861     condition tha
│ │ │ │ -00072d80: 7420 7468 6520 636f 6d70 6f73 6974 696f  t the compositio
│ │ │ │ -00072d90: 6e20 245c 5073 695c 2c5c 5068 693a 5820  n $\Psi\,\Phi:X 
│ │ │ │ -00072da0: 5c64 6173 6872 6967 6874 6172 726f 7720  \dashrightarrow 
│ │ │ │ -00072db0: 5824 0a20 2020 2020 2020 2063 6f69 6e63  X$.        coinc
│ │ │ │ -00072dc0: 6964 6573 2077 6974 6820 7468 6520 6964  ides with the id
│ │ │ │ -00072dd0: 656e 7469 7479 206f 6620 2458 2420 2861  entity of $X$ (a
│ │ │ │ -00072de0: 7320 6120 7261 7469 6f6e 616c 206d 6170  s a rational map
│ │ │ │ -00072df0: 290a 0a57 6179 7320 746f 2075 7365 2069  )..Ways to use i
│ │ │ │ -00072e00: 7349 6e76 6572 7365 4d61 703a 0a3d 3d3d  sInverseMap:.===
│ │ │ │ -00072e10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00072e20: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 7349  ======..  * "isI
│ │ │ │ -00072e30: 6e76 6572 7365 4d61 7028 5269 6e67 4d61  nverseMap(RingMa
│ │ │ │ -00072e40: 702c 5269 6e67 4d61 7029 220a 2020 2a20  p,RingMap)".  * 
│ │ │ │ -00072e50: 2a6e 6f74 6520 6973 496e 7665 7273 654d  *note isInverseM
│ │ │ │ -00072e60: 6170 2852 6174 696f 6e61 6c4d 6170 2c52  ap(RationalMap,R
│ │ │ │ -00072e70: 6174 696f 6e61 6c4d 6170 293a 0a20 2020  ationalMap):.   
│ │ │ │ -00072e80: 2069 7349 6e76 6572 7365 4d61 705f 6c70   isInverseMap_lp
│ │ │ │ -00072e90: 5261 7469 6f6e 616c 4d61 705f 636d 5261  RationalMap_cmRa
│ │ │ │ -00072ea0: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -00072eb0: 2063 6865 636b 7320 7768 6574 6865 7220   checks whether 
│ │ │ │ -00072ec0: 7477 6f20 7261 7469 6f6e 616c 0a20 2020  two rational.   
│ │ │ │ -00072ed0: 206d 6170 7320 6172 6520 6f6e 6520 7468   maps are one th
│ │ │ │ -00072ee0: 6520 696e 7665 7273 6520 6f66 2074 6865  e inverse of the
│ │ │ │ -00072ef0: 206f 7468 6572 0a0a 466f 7220 7468 6520   other..For the 
│ │ │ │ -00072f00: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00072f10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -00072f20: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -00072f30: 6973 496e 7665 7273 654d 6170 3a20 6973  isInverseMap: is
│ │ │ │ -00072f40: 496e 7665 7273 654d 6170 2c20 6973 2061  InverseMap, is a
│ │ │ │ -00072f50: 202a 6e6f 7465 206d 6574 686f 6420 6675   *note method fu
│ │ │ │ -00072f60: 6e63 7469 6f6e 3a0a 284d 6163 6175 6c61  nction:.(Macaula
│ │ │ │ -00072f70: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ -00072f80: 7469 6f6e 2c2e 0a1f 0a46 696c 653a 2043  tion,....File: C
│ │ │ │ -00072f90: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ -00072fa0: 653a 2069 7349 6e76 6572 7365 4d61 705f  e: isInverseMap_
│ │ │ │ -00072fb0: 6c70 5261 7469 6f6e 616c 4d61 705f 636d  lpRationalMap_cm
│ │ │ │ -00072fc0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -00072fd0: 4e65 7874 3a20 6973 4973 6f6d 6f72 7068  Next: isIsomorph
│ │ │ │ -00072fe0: 6973 6d5f 6c70 5261 7469 6f6e 616c 4d61  ism_lpRationalMa
│ │ │ │ -00072ff0: 705f 7270 2c20 5072 6576 3a20 6973 496e  p_rp, Prev: isIn
│ │ │ │ -00073000: 7665 7273 654d 6170 2c20 5570 3a20 546f  verseMap, Up: To
│ │ │ │ -00073010: 700a 0a69 7349 6e76 6572 7365 4d61 7028  p..isInverseMap(
│ │ │ │ -00073020: 5261 7469 6f6e 616c 4d61 702c 5261 7469  RationalMap,Rati
│ │ │ │ -00073030: 6f6e 616c 4d61 7029 202d 2d20 6368 6563  onalMap) -- chec
│ │ │ │ -00073040: 6b73 2077 6865 7468 6572 2074 776f 2072  ks whether two r
│ │ │ │ -00073050: 6174 696f 6e61 6c20 6d61 7073 2061 7265  ational maps are
│ │ │ │ -00073060: 206f 6e65 2074 6865 2069 6e76 6572 7365   one the inverse
│ │ │ │ -00073070: 206f 6620 7468 6520 6f74 6865 720a 2a2a   of the other.**
│ │ │ │ -00073080: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00072bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00072be0: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +00072bf0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ +00072c00: 6167 653a 200a 2020 2020 2020 2020 6973  age: .        is
│ │ │ │ +00072c10: 496e 7665 7273 654d 6170 2870 6869 2c70  InverseMap(phi,p
│ │ │ │ +00072c20: 7369 290a 2020 2a20 496e 7075 7473 3a0a  si).  * Inputs:.
│ │ │ │ +00072c30: 2020 2020 2020 2a20 7068 692c 2061 202a        * phi, a *
│ │ │ │ +00072c40: 6e6f 7465 2072 696e 6720 6d61 703a 2028  note ring map: (
│ │ │ │ +00072c50: 4d61 6361 756c 6179 3244 6f63 2952 696e  Macaulay2Doc)Rin
│ │ │ │ +00072c60: 674d 6170 2c2c 2072 6570 7265 7365 6e74  gMap,, represent
│ │ │ │ +00072c70: 696e 6720 6120 7261 7469 6f6e 616c 0a20  ing a rational. 
│ │ │ │ +00072c80: 2020 2020 2020 206d 6170 2024 5c50 6869         map $\Phi
│ │ │ │ +00072c90: 3a58 205c 6461 7368 7269 6768 7461 7272  :X \dashrightarr
│ │ │ │ +00072ca0: 6f77 2059 240a 2020 2020 2020 2a20 7073  ow Y$.      * ps
│ │ │ │ +00072cb0: 692c 2061 202a 6e6f 7465 2072 696e 6720  i, a *note ring 
│ │ │ │ +00072cc0: 6d61 703a 2028 4d61 6361 756c 6179 3244  map: (Macaulay2D
│ │ │ │ +00072cd0: 6f63 2952 696e 674d 6170 2c2c 2072 6570  oc)RingMap,, rep
│ │ │ │ +00072ce0: 7265 7365 6e74 696e 6720 6120 7261 7469  resenting a rati
│ │ │ │ +00072cf0: 6f6e 616c 0a20 2020 2020 2020 206d 6170  onal.        map
│ │ │ │ +00072d00: 2024 5c50 7369 3a59 205c 6461 7368 7269   $\Psi:Y \dashri
│ │ │ │ +00072d10: 6768 7461 7272 6f77 2058 240a 2020 2a20  ghtarrow X$.  * 
│ │ │ │ +00072d20: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +00072d30: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e   a *note Boolean
│ │ │ │ +00072d40: 2076 616c 7565 3a20 284d 6163 6175 6c61   value: (Macaula
│ │ │ │ +00072d50: 7932 446f 6329 426f 6f6c 6561 6e2c 2c20  y2Doc)Boolean,, 
│ │ │ │ +00072d60: 6163 636f 7264 696e 6720 746f 2074 6865  according to the
│ │ │ │ +00072d70: 0a20 2020 2020 2020 2063 6f6e 6469 7469  .        conditi
│ │ │ │ +00072d80: 6f6e 2074 6861 7420 7468 6520 636f 6d70  on that the comp
│ │ │ │ +00072d90: 6f73 6974 696f 6e20 245c 5073 695c 2c5c  osition $\Psi\,\
│ │ │ │ +00072da0: 5068 693a 5820 5c64 6173 6872 6967 6874  Phi:X \dashright
│ │ │ │ +00072db0: 6172 726f 7720 5824 0a20 2020 2020 2020  arrow X$.       
│ │ │ │ +00072dc0: 2063 6f69 6e63 6964 6573 2077 6974 6820   coincides with 
│ │ │ │ +00072dd0: 7468 6520 6964 656e 7469 7479 206f 6620  the identity of 
│ │ │ │ +00072de0: 2458 2420 2861 7320 6120 7261 7469 6f6e  $X$ (as a ration
│ │ │ │ +00072df0: 616c 206d 6170 290a 0a57 6179 7320 746f  al map)..Ways to
│ │ │ │ +00072e00: 2075 7365 2069 7349 6e76 6572 7365 4d61   use isInverseMa
│ │ │ │ +00072e10: 703a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  p:.=============
│ │ │ │ +00072e20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00072e30: 2a20 2269 7349 6e76 6572 7365 4d61 7028  * "isInverseMap(
│ │ │ │ +00072e40: 5269 6e67 4d61 702c 5269 6e67 4d61 7029  RingMap,RingMap)
│ │ │ │ +00072e50: 220a 2020 2a20 2a6e 6f74 6520 6973 496e  ".  * *note isIn
│ │ │ │ +00072e60: 7665 7273 654d 6170 2852 6174 696f 6e61  verseMap(Rationa
│ │ │ │ +00072e70: 6c4d 6170 2c52 6174 696f 6e61 6c4d 6170  lMap,RationalMap
│ │ │ │ +00072e80: 293a 0a20 2020 2069 7349 6e76 6572 7365  ):.    isInverse
│ │ │ │ +00072e90: 4d61 705f 6c70 5261 7469 6f6e 616c 4d61  Map_lpRationalMa
│ │ │ │ +00072ea0: 705f 636d 5261 7469 6f6e 616c 4d61 705f  p_cmRationalMap_
│ │ │ │ +00072eb0: 7270 2c20 2d2d 2063 6865 636b 7320 7768  rp, -- checks wh
│ │ │ │ +00072ec0: 6574 6865 7220 7477 6f20 7261 7469 6f6e  ether two ration
│ │ │ │ +00072ed0: 616c 0a20 2020 206d 6170 7320 6172 6520  al.    maps are 
│ │ │ │ +00072ee0: 6f6e 6520 7468 6520 696e 7665 7273 6520  one the inverse 
│ │ │ │ +00072ef0: 6f66 2074 6865 206f 7468 6572 0a0a 466f  of the other..Fo
│ │ │ │ +00072f00: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00072f10: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00072f20: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00072f30: 2a6e 6f74 6520 6973 496e 7665 7273 654d  *note isInverseM
│ │ │ │ +00072f40: 6170 3a20 6973 496e 7665 7273 654d 6170  ap: isInverseMap
│ │ │ │ +00072f50: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ +00072f60: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ +00072f70: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +00072f80: 6f64 4675 6e63 7469 6f6e 2c2e 0a1f 0a46  odFunction,....F
│ │ │ │ +00072f90: 696c 653a 2043 7265 6d6f 6e61 2e69 6e66  ile: Cremona.inf
│ │ │ │ +00072fa0: 6f2c 204e 6f64 653a 2069 7349 6e76 6572  o, Node: isInver
│ │ │ │ +00072fb0: 7365 4d61 705f 6c70 5261 7469 6f6e 616c  seMap_lpRational
│ │ │ │ +00072fc0: 4d61 705f 636d 5261 7469 6f6e 616c 4d61  Map_cmRationalMa
│ │ │ │ +00072fd0: 705f 7270 2c20 4e65 7874 3a20 6973 4973  p_rp, Next: isIs
│ │ │ │ +00072fe0: 6f6d 6f72 7068 6973 6d5f 6c70 5261 7469  omorphism_lpRati
│ │ │ │ +00072ff0: 6f6e 616c 4d61 705f 7270 2c20 5072 6576  onalMap_rp, Prev
│ │ │ │ +00073000: 3a20 6973 496e 7665 7273 654d 6170 2c20  : isInverseMap, 
│ │ │ │ +00073010: 5570 3a20 546f 700a 0a69 7349 6e76 6572  Up: Top..isInver
│ │ │ │ +00073020: 7365 4d61 7028 5261 7469 6f6e 616c 4d61  seMap(RationalMa
│ │ │ │ +00073030: 702c 5261 7469 6f6e 616c 4d61 7029 202d  p,RationalMap) -
│ │ │ │ +00073040: 2d20 6368 6563 6b73 2077 6865 7468 6572  - checks whether
│ │ │ │ +00073050: 2074 776f 2072 6174 696f 6e61 6c20 6d61   two rational ma
│ │ │ │ +00073060: 7073 2061 7265 206f 6e65 2074 6865 2069  ps are one the i
│ │ │ │ +00073070: 6e76 6572 7365 206f 6620 7468 6520 6f74  nverse of the ot
│ │ │ │ +00073080: 6865 720a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  her.************
│ │ │ │  00073090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000730a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000730b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000730c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000730d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000730e0: 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073  ********..Synops
│ │ │ │ -000730f0: 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  is.========..  *
│ │ │ │ -00073100: 2046 756e 6374 696f 6e3a 202a 6e6f 7465   Function: *note
│ │ │ │ -00073110: 2069 7349 6e76 6572 7365 4d61 703a 2069   isInverseMap: i
│ │ │ │ -00073120: 7349 6e76 6572 7365 4d61 702c 0a20 202a  sInverseMap,.  *
│ │ │ │ -00073130: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00073140: 2069 7349 6e76 6572 7365 4d61 7028 7068   isInverseMap(ph
│ │ │ │ -00073150: 692c 7073 6929 0a20 202a 2049 6e70 7574  i,psi).  * Input
│ │ │ │ -00073160: 733a 0a20 2020 2020 202a 2070 6869 2c20  s:.      * phi, 
│ │ │ │ -00073170: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ -00073180: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ -00073190: 702c 0a20 2020 2020 202a 2070 7369 2c20  p,.      * psi, 
│ │ │ │ -000731a0: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ -000731b0: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ -000731c0: 702c 0a20 202a 204f 7574 7075 7473 3a0a  p,.  * Outputs:.
│ │ │ │ -000731d0: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ -000731e0: 426f 6f6c 6561 6e20 7661 6c75 653a 2028  Boolean value: (
│ │ │ │ -000731f0: 4d61 6361 756c 6179 3244 6f63 2942 6f6f  Macaulay2Doc)Boo
│ │ │ │ -00073200: 6c65 616e 2c2c 2077 6865 7468 6572 2070  lean,, whether p
│ │ │ │ -00073210: 6869 202a 2070 7369 203d 3d20 310a 2020  hi * psi == 1.  
│ │ │ │ -00073220: 2020 2020 2020 616e 6420 7073 6920 2a20        and psi * 
│ │ │ │ -00073230: 7068 6920 3d3d 2031 0a20 202a 2043 6f6e  phi == 1.  * Con
│ │ │ │ -00073240: 7365 7175 656e 6365 733a 0a20 2020 2020  sequences:.     
│ │ │ │ -00073250: 202a 2049 6620 7468 6520 616e 7377 6572   * If the answer
│ │ │ │ -00073260: 2069 7320 6166 6669 726d 6174 6976 652c   is affirmative,
│ │ │ │ -00073270: 2074 6865 6e20 7468 6520 7379 7374 656d   then the system
│ │ │ │ -00073280: 2077 696c 6c20 6265 2069 6e66 6f72 6d65   will be informe
│ │ │ │ -00073290: 6420 616e 6420 736f 0a20 2020 2020 2020  d and so.       
│ │ │ │ -000732a0: 2063 6f6d 6d61 6e64 7320 6c69 6b65 2027   commands like '
│ │ │ │ -000732b0: 696e 7665 7273 6520 7068 6927 2077 696c  inverse phi' wil
│ │ │ │ -000732c0: 6c20 6578 6563 7574 6520 6661 7374 2e0a  l execute fast..
│ │ │ │ -000732d0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -000732e0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 5261  ==..  * *note Ra
│ │ │ │ -000732f0: 7469 6f6e 616c 4d61 7020 3d3d 2052 6174  tionalMap == Rat
│ │ │ │ -00073300: 696f 6e61 6c4d 6170 3a20 5261 7469 6f6e  ionalMap: Ration
│ │ │ │ -00073310: 616c 4d61 7020 3d3d 2052 6174 696f 6e61  alMap == Rationa
│ │ │ │ -00073320: 6c4d 6170 2c20 2d2d 2065 7175 616c 6974  lMap, -- equalit
│ │ │ │ -00073330: 790a 2020 2020 6f66 2072 6174 696f 6e61  y.    of rationa
│ │ │ │ -00073340: 6c20 6d61 7073 0a20 202a 202a 6e6f 7465  l maps.  * *note
│ │ │ │ -00073350: 2052 6174 696f 6e61 6c4d 6170 202a 2052   RationalMap * R
│ │ │ │ -00073360: 6174 696f 6e61 6c4d 6170 3a20 5261 7469  ationalMap: Rati
│ │ │ │ -00073370: 6f6e 616c 4d61 7020 5f73 7420 5261 7469  onalMap _st Rati
│ │ │ │ -00073380: 6f6e 616c 4d61 702c 202d 2d0a 2020 2020  onalMap, --.    
│ │ │ │ -00073390: 636f 6d70 6f73 6974 696f 6e20 6f66 2072  composition of r
│ │ │ │ -000733a0: 6174 696f 6e61 6c20 6d61 7073 0a20 202a  ational maps.  *
│ │ │ │ -000733b0: 202a 6e6f 7465 2069 7349 6e76 6572 7365   *note isInverse
│ │ │ │ -000733c0: 4d61 7028 5269 6e67 4d61 702c 5269 6e67  Map(RingMap,Ring
│ │ │ │ -000733d0: 4d61 7029 3a20 6973 496e 7665 7273 654d  Map): isInverseM
│ │ │ │ -000733e0: 6170 2c20 2d2d 2063 6865 636b 7320 7768  ap, -- checks wh
│ │ │ │ -000733f0: 6574 6865 7220 610a 2020 2020 7261 7469  ether a.    rati
│ │ │ │ -00073400: 6f6e 616c 206d 6170 2069 7320 7468 6520  onal map is the 
│ │ │ │ -00073410: 696e 7665 7273 6520 6f66 2061 6e6f 7468  inverse of anoth
│ │ │ │ -00073420: 6572 0a20 202a 202a 6e6f 7465 2069 6e76  er.  * *note inv
│ │ │ │ -00073430: 6572 7365 2852 6174 696f 6e61 6c4d 6170  erse(RationalMap
│ │ │ │ -00073440: 293a 2069 6e76 6572 7365 5f6c 7052 6174  ): inverse_lpRat
│ │ │ │ -00073450: 696f 6e61 6c4d 6170 5f72 702c 202d 2d20  ionalMap_rp, -- 
│ │ │ │ -00073460: 696e 7665 7273 6520 6f66 2061 0a20 2020  inverse of a.   
│ │ │ │ -00073470: 2062 6972 6174 696f 6e61 6c20 6d61 700a   birational map.
│ │ │ │ -00073480: 2020 2a20 2a6e 6f74 6520 666f 7263 6549    * *note forceI
│ │ │ │ -00073490: 6e76 6572 7365 4d61 703a 2066 6f72 6365  nverseMap: force
│ │ │ │ -000734a0: 496e 7665 7273 654d 6170 2c20 2d2d 2064  InverseMap, -- d
│ │ │ │ -000734b0: 6563 6c61 7265 2074 6861 7420 7477 6f20  eclare that two 
│ │ │ │ -000734c0: 7261 7469 6f6e 616c 206d 6170 730a 2020  rational maps.  
│ │ │ │ -000734d0: 2020 6172 6520 6f6e 6520 7468 6520 696e    are one the in
│ │ │ │ -000734e0: 7665 7273 6520 6f66 2074 6865 206f 7468  verse of the oth
│ │ │ │ -000734f0: 6572 0a0a 5761 7973 2074 6f20 7573 6520  er..Ways to use 
│ │ │ │ -00073500: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ -00073510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00073520: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00073530: 2069 7349 6e76 6572 7365 4d61 7028 5261   isInverseMap(Ra
│ │ │ │ -00073540: 7469 6f6e 616c 4d61 702c 5261 7469 6f6e  tionalMap,Ration
│ │ │ │ -00073550: 616c 4d61 7029 3a0a 2020 2020 6973 496e  alMap):.    isIn
│ │ │ │ -00073560: 7665 7273 654d 6170 5f6c 7052 6174 696f  verseMap_lpRatio
│ │ │ │ -00073570: 6e61 6c4d 6170 5f63 6d52 6174 696f 6e61  nalMap_cmRationa
│ │ │ │ -00073580: 6c4d 6170 5f72 702c 202d 2d20 6368 6563  lMap_rp, -- chec
│ │ │ │ -00073590: 6b73 2077 6865 7468 6572 2074 776f 2072  ks whether two r
│ │ │ │ -000735a0: 6174 696f 6e61 6c0a 2020 2020 6d61 7073  ational.    maps
│ │ │ │ -000735b0: 2061 7265 206f 6e65 2074 6865 2069 6e76   are one the inv
│ │ │ │ -000735c0: 6572 7365 206f 6620 7468 6520 6f74 6865  erse of the othe
│ │ │ │ -000735d0: 720a 1f0a 4669 6c65 3a20 4372 656d 6f6e  r...File: Cremon
│ │ │ │ -000735e0: 612e 696e 666f 2c20 4e6f 6465 3a20 6973  a.info, Node: is
│ │ │ │ -000735f0: 4973 6f6d 6f72 7068 6973 6d5f 6c70 5261  Isomorphism_lpRa
│ │ │ │ -00073600: 7469 6f6e 616c 4d61 705f 7270 2c20 4e65  tionalMap_rp, Ne
│ │ │ │ -00073610: 7874 3a20 6973 4d6f 7270 6869 736d 2c20  xt: isMorphism, 
│ │ │ │ -00073620: 5072 6576 3a20 6973 496e 7665 7273 654d  Prev: isInverseM
│ │ │ │ -00073630: 6170 5f6c 7052 6174 696f 6e61 6c4d 6170  ap_lpRationalMap
│ │ │ │ -00073640: 5f63 6d52 6174 696f 6e61 6c4d 6170 5f72  _cmRationalMap_r
│ │ │ │ -00073650: 702c 2055 703a 2054 6f70 0a0a 6973 4973  p, Up: Top..isIs
│ │ │ │ -00073660: 6f6d 6f72 7068 6973 6d28 5261 7469 6f6e  omorphism(Ration
│ │ │ │ -00073670: 616c 4d61 7029 202d 2d20 7768 6574 6865  alMap) -- whethe
│ │ │ │ -00073680: 7220 6120 6269 7261 7469 6f6e 616c 206d  r a birational m
│ │ │ │ -00073690: 6170 2069 7320 616e 2069 736f 6d6f 7270  ap is an isomorp
│ │ │ │ -000736a0: 6869 736d 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  hism.***********
│ │ │ │ +000730e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +000730f0: 5379 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d  Synopsis.=======
│ │ │ │ +00073100: 3d0a 0a20 202a 2046 756e 6374 696f 6e3a  =..  * Function:
│ │ │ │ +00073110: 202a 6e6f 7465 2069 7349 6e76 6572 7365   *note isInverse
│ │ │ │ +00073120: 4d61 703a 2069 7349 6e76 6572 7365 4d61  Map: isInverseMa
│ │ │ │ +00073130: 702c 0a20 202a 2055 7361 6765 3a20 0a20  p,.  * Usage: . 
│ │ │ │ +00073140: 2020 2020 2020 2069 7349 6e76 6572 7365         isInverse
│ │ │ │ +00073150: 4d61 7028 7068 692c 7073 6929 0a20 202a  Map(phi,psi).  *
│ │ │ │ +00073160: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00073170: 2070 6869 2c20 6120 2a6e 6f74 6520 7261   phi, a *note ra
│ │ │ │ +00073180: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ +00073190: 6f6e 616c 4d61 702c 0a20 2020 2020 202a  onalMap,.      *
│ │ │ │ +000731a0: 2070 7369 2c20 6120 2a6e 6f74 6520 7261   psi, a *note ra
│ │ │ │ +000731b0: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ +000731c0: 6f6e 616c 4d61 702c 0a20 202a 204f 7574  onalMap,.  * Out
│ │ │ │ +000731d0: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ +000731e0: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661  *note Boolean va
│ │ │ │ +000731f0: 6c75 653a 2028 4d61 6361 756c 6179 3244  lue: (Macaulay2D
│ │ │ │ +00073200: 6f63 2942 6f6f 6c65 616e 2c2c 2077 6865  oc)Boolean,, whe
│ │ │ │ +00073210: 7468 6572 2070 6869 202a 2070 7369 203d  ther phi * psi =
│ │ │ │ +00073220: 3d20 310a 2020 2020 2020 2020 616e 6420  = 1.        and 
│ │ │ │ +00073230: 7073 6920 2a20 7068 6920 3d3d 2031 0a20  psi * phi == 1. 
│ │ │ │ +00073240: 202a 2043 6f6e 7365 7175 656e 6365 733a   * Consequences:
│ │ │ │ +00073250: 0a20 2020 2020 202a 2049 6620 7468 6520  .      * If the 
│ │ │ │ +00073260: 616e 7377 6572 2069 7320 6166 6669 726d  answer is affirm
│ │ │ │ +00073270: 6174 6976 652c 2074 6865 6e20 7468 6520  ative, then the 
│ │ │ │ +00073280: 7379 7374 656d 2077 696c 6c20 6265 2069  system will be i
│ │ │ │ +00073290: 6e66 6f72 6d65 6420 616e 6420 736f 0a20  nformed and so. 
│ │ │ │ +000732a0: 2020 2020 2020 2063 6f6d 6d61 6e64 7320         commands 
│ │ │ │ +000732b0: 6c69 6b65 2027 696e 7665 7273 6520 7068  like 'inverse ph
│ │ │ │ +000732c0: 6927 2077 696c 6c20 6578 6563 7574 6520  i' will execute 
│ │ │ │ +000732d0: 6661 7374 2e0a 0a53 6565 2061 6c73 6f0a  fast...See also.
│ │ │ │ +000732e0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +000732f0: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ +00073300: 3d3d 2052 6174 696f 6e61 6c4d 6170 3a20  == RationalMap: 
│ │ │ │ +00073310: 5261 7469 6f6e 616c 4d61 7020 3d3d 2052  RationalMap == R
│ │ │ │ +00073320: 6174 696f 6e61 6c4d 6170 2c20 2d2d 2065  ationalMap, -- e
│ │ │ │ +00073330: 7175 616c 6974 790a 2020 2020 6f66 2072  quality.    of r
│ │ │ │ +00073340: 6174 696f 6e61 6c20 6d61 7073 0a20 202a  ational maps.  *
│ │ │ │ +00073350: 202a 6e6f 7465 2052 6174 696f 6e61 6c4d   *note RationalM
│ │ │ │ +00073360: 6170 202a 2052 6174 696f 6e61 6c4d 6170  ap * RationalMap
│ │ │ │ +00073370: 3a20 5261 7469 6f6e 616c 4d61 7020 5f73  : RationalMap _s
│ │ │ │ +00073380: 7420 5261 7469 6f6e 616c 4d61 702c 202d  t RationalMap, -
│ │ │ │ +00073390: 2d0a 2020 2020 636f 6d70 6f73 6974 696f  -.    compositio
│ │ │ │ +000733a0: 6e20 6f66 2072 6174 696f 6e61 6c20 6d61  n of rational ma
│ │ │ │ +000733b0: 7073 0a20 202a 202a 6e6f 7465 2069 7349  ps.  * *note isI
│ │ │ │ +000733c0: 6e76 6572 7365 4d61 7028 5269 6e67 4d61  nverseMap(RingMa
│ │ │ │ +000733d0: 702c 5269 6e67 4d61 7029 3a20 6973 496e  p,RingMap): isIn
│ │ │ │ +000733e0: 7665 7273 654d 6170 2c20 2d2d 2063 6865  verseMap, -- che
│ │ │ │ +000733f0: 636b 7320 7768 6574 6865 7220 610a 2020  cks whether a.  
│ │ │ │ +00073400: 2020 7261 7469 6f6e 616c 206d 6170 2069    rational map i
│ │ │ │ +00073410: 7320 7468 6520 696e 7665 7273 6520 6f66  s the inverse of
│ │ │ │ +00073420: 2061 6e6f 7468 6572 0a20 202a 202a 6e6f   another.  * *no
│ │ │ │ +00073430: 7465 2069 6e76 6572 7365 2852 6174 696f  te inverse(Ratio
│ │ │ │ +00073440: 6e61 6c4d 6170 293a 2069 6e76 6572 7365  nalMap): inverse
│ │ │ │ +00073450: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +00073460: 702c 202d 2d20 696e 7665 7273 6520 6f66  p, -- inverse of
│ │ │ │ +00073470: 2061 0a20 2020 2062 6972 6174 696f 6e61   a.    birationa
│ │ │ │ +00073480: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ +00073490: 666f 7263 6549 6e76 6572 7365 4d61 703a  forceInverseMap:
│ │ │ │ +000734a0: 2066 6f72 6365 496e 7665 7273 654d 6170   forceInverseMap
│ │ │ │ +000734b0: 2c20 2d2d 2064 6563 6c61 7265 2074 6861  , -- declare tha
│ │ │ │ +000734c0: 7420 7477 6f20 7261 7469 6f6e 616c 206d  t two rational m
│ │ │ │ +000734d0: 6170 730a 2020 2020 6172 6520 6f6e 6520  aps.    are one 
│ │ │ │ +000734e0: 7468 6520 696e 7665 7273 6520 6f66 2074  the inverse of t
│ │ │ │ +000734f0: 6865 206f 7468 6572 0a0a 5761 7973 2074  he other..Ways t
│ │ │ │ +00073500: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ +00073510: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ +00073520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00073530: 202a 6e6f 7465 2069 7349 6e76 6572 7365   *note isInverse
│ │ │ │ +00073540: 4d61 7028 5261 7469 6f6e 616c 4d61 702c  Map(RationalMap,
│ │ │ │ +00073550: 5261 7469 6f6e 616c 4d61 7029 3a0a 2020  RationalMap):.  
│ │ │ │ +00073560: 2020 6973 496e 7665 7273 654d 6170 5f6c    isInverseMap_l
│ │ │ │ +00073570: 7052 6174 696f 6e61 6c4d 6170 5f63 6d52  pRationalMap_cmR
│ │ │ │ +00073580: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +00073590: 2d20 6368 6563 6b73 2077 6865 7468 6572  - checks whether
│ │ │ │ +000735a0: 2074 776f 2072 6174 696f 6e61 6c0a 2020   two rational.  
│ │ │ │ +000735b0: 2020 6d61 7073 2061 7265 206f 6e65 2074    maps are one t
│ │ │ │ +000735c0: 6865 2069 6e76 6572 7365 206f 6620 7468  he inverse of th
│ │ │ │ +000735d0: 6520 6f74 6865 720a 1f0a 4669 6c65 3a20  e other...File: 
│ │ │ │ +000735e0: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ +000735f0: 6465 3a20 6973 4973 6f6d 6f72 7068 6973  de: isIsomorphis
│ │ │ │ +00073600: 6d5f 6c70 5261 7469 6f6e 616c 4d61 705f  m_lpRationalMap_
│ │ │ │ +00073610: 7270 2c20 4e65 7874 3a20 6973 4d6f 7270  rp, Next: isMorp
│ │ │ │ +00073620: 6869 736d 2c20 5072 6576 3a20 6973 496e  hism, Prev: isIn
│ │ │ │ +00073630: 7665 7273 654d 6170 5f6c 7052 6174 696f  verseMap_lpRatio
│ │ │ │ +00073640: 6e61 6c4d 6170 5f63 6d52 6174 696f 6e61  nalMap_cmRationa
│ │ │ │ +00073650: 6c4d 6170 5f72 702c 2055 703a 2054 6f70  lMap_rp, Up: Top
│ │ │ │ +00073660: 0a0a 6973 4973 6f6d 6f72 7068 6973 6d28  ..isIsomorphism(
│ │ │ │ +00073670: 5261 7469 6f6e 616c 4d61 7029 202d 2d20  RationalMap) -- 
│ │ │ │ +00073680: 7768 6574 6865 7220 6120 6269 7261 7469  whether a birati
│ │ │ │ +00073690: 6f6e 616c 206d 6170 2069 7320 616e 2069  onal map is an i
│ │ │ │ +000736a0: 736f 6d6f 7270 6869 736d 0a2a 2a2a 2a2a  somorphism.*****
│ │ │ │  000736b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000736c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000736d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000736e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ -000736f0: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ -00073700: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -00073710: 2a6e 6f74 6520 6973 4973 6f6d 6f72 7068  *note isIsomorph
│ │ │ │ -00073720: 6973 6d3a 2028 4d61 6361 756c 6179 3244  ism: (Macaulay2D
│ │ │ │ -00073730: 6f63 2969 7349 736f 6d6f 7270 6869 736d  oc)isIsomorphism
│ │ │ │ -00073740: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ -00073750: 2020 2020 2020 6973 4973 6f6d 6f72 7068        isIsomorph
│ │ │ │ -00073760: 6973 6d20 7068 690a 2020 2a20 496e 7075  ism phi.  * Inpu
│ │ │ │ -00073770: 7473 3a0a 2020 2020 2020 2a20 7068 692c  ts:.      * phi,
│ │ │ │ -00073780: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ -00073790: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ -000737a0: 6170 2c0a 2020 2a20 4f75 7470 7574 733a  ap,.  * Outputs:
│ │ │ │ -000737b0: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ -000737c0: 2042 6f6f 6c65 616e 2076 616c 7565 3a20   Boolean value: 
│ │ │ │ -000737d0: 284d 6163 6175 6c61 7932 446f 6329 426f  (Macaulay2Doc)Bo
│ │ │ │ -000737e0: 6f6c 6561 6e2c 2c20 7768 6574 6865 7220  olean,, whether 
│ │ │ │ -000737f0: 7068 6920 6973 2061 6e0a 2020 2020 2020  phi is an.      
│ │ │ │ -00073800: 2020 6973 6f6d 6f72 7068 6973 6d0a 0a44    isomorphism..D
│ │ │ │ -00073810: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00073820: 3d3d 3d3d 3d3d 0a0a 5468 6973 206d 6574  ======..This met
│ │ │ │ -00073830: 686f 6420 636f 6d70 7574 6573 2074 6865  hod computes the
│ │ │ │ -00073840: 2069 6e76 6572 7365 2072 6174 696f 6e61   inverse rationa
│ │ │ │ -00073850: 6c20 6d61 7020 7573 696e 6720 2a6e 6f74  l map using *not
│ │ │ │ -00073860: 6520 696e 7665 7273 653a 0a69 6e76 6572  e inverse:.inver
│ │ │ │ -00073870: 7365 5f6c 7052 6174 696f 6e61 6c4d 6170  se_lpRationalMap
│ │ │ │ -00073880: 5f72 702c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  _rp,...+--------
│ │ │ │ +000736e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000736f0: 2a2a 2a0a 0a53 796e 6f70 7369 730a 3d3d  ***..Synopsis.==
│ │ │ │ +00073700: 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675 6e63  ======..  * Func
│ │ │ │ +00073710: 7469 6f6e 3a20 2a6e 6f74 6520 6973 4973  tion: *note isIs
│ │ │ │ +00073720: 6f6d 6f72 7068 6973 6d3a 2028 4d61 6361  omorphism: (Maca
│ │ │ │ +00073730: 756c 6179 3244 6f63 2969 7349 736f 6d6f  ulay2Doc)isIsomo
│ │ │ │ +00073740: 7270 6869 736d 2c0a 2020 2a20 5573 6167  rphism,.  * Usag
│ │ │ │ +00073750: 653a 200a 2020 2020 2020 2020 6973 4973  e: .        isIs
│ │ │ │ +00073760: 6f6d 6f72 7068 6973 6d20 7068 690a 2020  omorphism phi.  
│ │ │ │ +00073770: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ +00073780: 2a20 7068 692c 2061 202a 6e6f 7465 2072  * phi, a *note r
│ │ │ │ +00073790: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ +000737a0: 696f 6e61 6c4d 6170 2c0a 2020 2a20 4f75  ionalMap,.  * Ou
│ │ │ │ +000737b0: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ +000737c0: 202a 6e6f 7465 2042 6f6f 6c65 616e 2076   *note Boolean v
│ │ │ │ +000737d0: 616c 7565 3a20 284d 6163 6175 6c61 7932  alue: (Macaulay2
│ │ │ │ +000737e0: 446f 6329 426f 6f6c 6561 6e2c 2c20 7768  Doc)Boolean,, wh
│ │ │ │ +000737f0: 6574 6865 7220 7068 6920 6973 2061 6e0a  ether phi is an.
│ │ │ │ +00073800: 2020 2020 2020 2020 6973 6f6d 6f72 7068          isomorph
│ │ │ │ +00073810: 6973 6d0a 0a44 6573 6372 6970 7469 6f6e  ism..Description
│ │ │ │ +00073820: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ +00073830: 6973 206d 6574 686f 6420 636f 6d70 7574  is method comput
│ │ │ │ +00073840: 6573 2074 6865 2069 6e76 6572 7365 2072  es the inverse r
│ │ │ │ +00073850: 6174 696f 6e61 6c20 6d61 7020 7573 696e  ational map usin
│ │ │ │ +00073860: 6720 2a6e 6f74 6520 696e 7665 7273 653a  g *note inverse:
│ │ │ │ +00073870: 0a69 6e76 6572 7365 5f6c 7052 6174 696f  .inverse_lpRatio
│ │ │ │ +00073880: 6e61 6c4d 6170 5f72 702c 2e0a 0a2b 2d2d  nalMap_rp,...+--
│ │ │ │  00073890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000738a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000738b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000738c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000738d0: 3120 3a20 5031 203a 3d20 5151 5b61 2c62  1 : P1 := QQ[a,b
│ │ │ │ -000738e0: 5d3b 2050 3420 3a3d 2051 515b 782c 792c  ]; P4 := QQ[x,y,
│ │ │ │ -000738f0: 7a2c 775d 3b20 2020 2020 2020 2020 2020  z,w];           
│ │ │ │ +000738c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000738d0: 2d2d 2b0a 7c69 3120 3a20 5031 203a 3d20  --+.|i1 : P1 := 
│ │ │ │ +000738e0: 5151 5b61 2c62 5d3b 2050 3420 3a3d 2051  QQ[a,b]; P4 := Q
│ │ │ │ +000738f0: 515b 782c 792c 7a2c 775d 3b20 2020 2020  Q[x,y,z,w];     
│ │ │ │  00073900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073910: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00073910: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  00073920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00073930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00073940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00073950: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -00073960: 3a20 7068 6920 3d20 7261 7469 6f6e 616c  : phi = rational
│ │ │ │ -00073970: 4d61 7028 7b61 5e34 2c61 5e33 2a62 2c61  Map({a^4,a^3*b,a
│ │ │ │ -00073980: 5e32 2a62 5e32 2c61 2a62 5e33 2c62 5e34  ^2*b^2,a*b^3,b^4
│ │ │ │ -00073990: 7d2c 446f 6d69 6e61 6e74 3d3e 7472 7565  },Dominant=>true
│ │ │ │ -000739a0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ +00073950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00073960: 2b0a 7c69 3320 3a20 7068 6920 3d20 7261  +.|i3 : phi = ra
│ │ │ │ +00073970: 7469 6f6e 616c 4d61 7028 7b61 5e34 2c61  tionalMap({a^4,a
│ │ │ │ +00073980: 5e33 2a62 2c61 5e32 2a62 5e32 2c61 2a62  ^3*b,a^2*b^2,a*b
│ │ │ │ +00073990: 5e33 2c62 5e34 7d2c 446f 6d69 6e61 6e74  ^3,b^4},Dominant
│ │ │ │ +000739a0: 3d3e 7472 7565 297c 0a7c 2020 2020 2020  =>true)|.|      
│ │ │ │  000739b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000739c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000739d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000739e0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -000739f0: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -00073a00: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +000739e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000739f0: 7c6f 3320 3d20 2d2d 2072 6174 696f 6e61  |o3 = -- rationa
│ │ │ │ +00073a00: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │  00073a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00073a30: 0a7c 2020 2020 2073 6f75 7263 653a 2050  .|     source: P
│ │ │ │ -00073a40: 726f 6a28 5151 5b61 2c20 625d 2920 2020  roj(QQ[a, b])   
│ │ │ │ -00073a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073a30: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ +00073a40: 7263 653a 2050 726f 6a28 5151 5b61 2c20  rce: Proj(QQ[a, 
│ │ │ │ +00073a50: 625d 2920 2020 2020 2020 2020 2020 2020  b])             
│ │ │ │  00073a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073a70: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ -00073a80: 7267 6574 3a20 7375 6276 6172 6965 7479  rget: subvariety
│ │ │ │ -00073a90: 206f 6620 5072 6f6a 2851 515b 7420 2c20   of Proj(QQ[t , 
│ │ │ │ -00073aa0: 7420 2c20 7420 2c20 7420 2c20 7420 5d29  t , t , t , t ])
│ │ │ │ -00073ab0: 2064 6566 696e 6564 2062 7920 207c 0a7c   defined by  |.|
│ │ │ │ -00073ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073a70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00073a80: 2020 2020 7461 7267 6574 3a20 7375 6276      target: subv
│ │ │ │ +00073a90: 6172 6965 7479 206f 6620 5072 6f6a 2851  ariety of Proj(Q
│ │ │ │ +00073aa0: 515b 7420 2c20 7420 2c20 7420 2c20 7420  Q[t , t , t , t 
│ │ │ │ +00073ab0: 2c20 7420 5d29 2064 6566 696e 6564 2062  , t ]) defined b
│ │ │ │ +00073ac0: 7920 207c 0a7c 2020 2020 2020 2020 2020  y  |.|          
│ │ │ │  00073ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073ae0: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ -00073af0: 3320 2020 3420 2020 2020 2020 2020 2020  3   4           
│ │ │ │ -00073b00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00073b10: 2020 2020 7b20 2020 2020 2020 2020 2020      {           
│ │ │ │ +00073ae0: 2020 2020 2020 2020 2020 3020 2020 3120            0   1 
│ │ │ │ +00073af0: 2020 3220 2020 3320 2020 3420 2020 2020    2   3   4     
│ │ │ │ +00073b00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00073b10: 2020 2020 2020 2020 2020 7b20 2020 2020            {     
│ │ │ │  00073b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073b40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00073b50: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00073b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073b50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00073b60: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  00073b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073b90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00073ba0: 2020 2074 2020 2d20 7420 7420 2c20 2020     t  - t t ,   
│ │ │ │ -00073bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073b90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00073ba0: 2020 2020 2020 2020 2074 2020 2d20 7420           t  - t 
│ │ │ │ +00073bb0: 7420 2c20 2020 2020 2020 2020 2020 2020  t ,             
│ │ │ │  00073bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073bd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00073be0: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -00073bf0: 3220 3420 2020 2020 2020 2020 2020 2020  2 4             
│ │ │ │ +00073bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00073be0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00073bf0: 2033 2020 2020 3220 3420 2020 2020 2020   3    2 4       
│ │ │ │  00073c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073c20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00073c20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00073c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073c60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00073c70: 2020 2020 2020 2020 7420 7420 202d 2074          t t  - t
│ │ │ │ -00073c80: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │ +00073c60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00073c70: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00073c80: 7420 202d 2074 2074 202c 2020 2020 2020  t  - t t ,      
│ │ │ │  00073c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073ca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00073cb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00073cc0: 3220 3320 2020 2031 2034 2020 2020 2020  2 3    1 4      
│ │ │ │ +00073ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073cb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00073cc0: 2020 2020 2020 3220 3320 2020 2031 2034        2 3    1 4
│ │ │ │  00073cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00073cf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00073d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073d30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00073d40: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -00073d50: 2020 2d20 7420 7420 2c20 2020 2020 2020    - t t ,       
│ │ │ │ +00073d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073d40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00073d50: 2020 2074 2074 2020 2d20 7420 7420 2c20     t t  - t t , 
│ │ │ │  00073d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073d80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00073d90: 2020 2020 2031 2033 2020 2020 3020 3420       1 3    0 4 
│ │ │ │ -00073da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073d80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00073d90: 2020 2020 2020 2020 2020 2031 2033 2020             1 3  
│ │ │ │ +00073da0: 2020 3020 3420 2020 2020 2020 2020 2020    0 4           
│ │ │ │  00073db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073dc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00073dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00073de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073e10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00073e20: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00073e10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00073e20: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00073e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073e50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00073e60: 2020 2020 2020 2020 2074 2020 2d20 7420           t  - t 
│ │ │ │ -00073e70: 7420 2c20 2020 2020 2020 2020 2020 2020  t ,             
│ │ │ │ +00073e50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00073e60: 7c20 2020 2020 2020 2020 2020 2020 2074  |              t
│ │ │ │ +00073e70: 2020 2d20 7420 7420 2c20 2020 2020 2020    - t t ,       
│ │ │ │  00073e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00073ea0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00073eb0: 2032 2020 2020 3020 3420 2020 2020 2020   2    0 4       
│ │ │ │ +00073e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073ea0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00073eb0: 2020 2020 2020 2032 2020 2020 3020 3420         2    0 4 
│ │ │ │  00073ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073ee0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00073ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00073ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073f20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00073f30: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -00073f40: 7420 202d 2074 2074 202c 2020 2020 2020  t  - t t ,      
│ │ │ │ +00073f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073f30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00073f40: 2020 2020 7420 7420 202d 2074 2074 202c      t t  - t t ,
│ │ │ │  00073f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073f70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00073f80: 2020 2020 2020 3120 3220 2020 2030 2033        1 2    0 3
│ │ │ │ -00073f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073f70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00073f80: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
│ │ │ │ +00073f90: 2020 2030 2033 2020 2020 2020 2020 2020     0 3          
│ │ │ │  00073fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073fb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00073fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00073fc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00073fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074000: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00074010: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00074000: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00074010: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  00074020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074040: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00074050: 2020 2020 2020 2020 2020 7420 202d 2074            t  - t
│ │ │ │ -00074060: 2074 2020 2020 2020 2020 2020 2020 2020   t              
│ │ │ │ +00074040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00074050: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00074060: 7420 202d 2074 2074 2020 2020 2020 2020  t  - t t        
│ │ │ │  00074070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074090: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000740a0: 2020 3120 2020 2030 2032 2020 2020 2020    1    0 2      
│ │ │ │ +00074090: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000740a0: 2020 2020 2020 2020 3120 2020 2030 2032          1    0 2
│ │ │ │  000740b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000740c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000740d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000740e0: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │ +000740d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000740e0: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │  000740f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074110: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00074120: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ -00074130: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │ +00074110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074120: 2020 2020 7c0a 7c20 2020 2020 6465 6669      |.|     defi
│ │ │ │ +00074130: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │  00074140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00074170: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ -00074180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074160: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00074170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074180: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │  00074190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000741a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000741b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000741c0: 2020 2020 2061 202c 2020 2020 2020 2020       a ,        
│ │ │ │ +000741a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000741b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000741c0: 2020 2020 2020 2020 2020 2061 202c 2020             a ,  
│ │ │ │  000741d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000741e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000741f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000741f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00074200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074230: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00074240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074250: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +00074230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074240: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00074250: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │  00074260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074280: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00074290: 2020 2020 2020 2020 2020 6120 622c 2020            a b,  
│ │ │ │ -000742a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00074290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000742a0: 6120 622c 2020 2020 2020 2020 2020 2020  a b,            
│ │ │ │  000742b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000742c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000742d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000742c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000742d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000742e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000742f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00074310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00074320: 2020 2020 2020 2020 2032 2032 2020 2020           2 2    
│ │ │ │ -00074330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074310: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00074320: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00074330: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00074340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074350: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00074360: 2020 2020 2020 2020 2020 2020 2020 2061                 a
│ │ │ │ -00074370: 2062 202c 2020 2020 2020 2020 2020 2020   b ,            
│ │ │ │ +00074350: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00074360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074370: 2020 2020 2061 2062 202c 2020 2020 2020       a b ,      
│ │ │ │  00074380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074390: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000743a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000743a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000743b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000743c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000743d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000743e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000743e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000743f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074400: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00074400: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │  00074410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00074430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074440: 2020 2020 612a 6220 2c20 2020 2020 2020      a*b ,       
│ │ │ │ +00074420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074430: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00074440: 2020 2020 2020 2020 2020 612a 6220 2c20            a*b , 
│ │ │ │  00074450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074470: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00074470: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00074480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000744a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000744b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000744c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000744d0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +000744b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000744c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000744d0: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │  000744e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000744f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074500: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00074510: 2020 2020 2020 2020 2062 2020 2020 2020           b      
│ │ │ │ +00074500: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00074510: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │  00074520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074540: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00074550: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │ -00074560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00074550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074560: 2020 2020 207d 2020 2020 2020 2020 2020       }          
│ │ │ │  00074570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00074590: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00074580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074590: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000745a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000745b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000745c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000745d0: 2020 2020 207c 0a7c 6f33 203a 2052 6174       |.|o3 : Rat
│ │ │ │ -000745e0: 696f 6e61 6c4d 6170 2028 646f 6d69 6e61  ionalMap (domina
│ │ │ │ -000745f0: 6e74 2072 6174 696f 6e61 6c20 6d61 7020  nt rational map 
│ │ │ │ -00074600: 6672 6f6d 2050 505e 3120 746f 2063 7572  from PP^1 to cur
│ │ │ │ -00074610: 7665 2069 6e20 5050 5e34 2920 7c0a 2b2d  ve in PP^4) |.+-
│ │ │ │ -00074620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000745d0: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +000745e0: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +000745f0: 646f 6d69 6e61 6e74 2072 6174 696f 6e61  dominant rationa
│ │ │ │ +00074600: 6c20 6d61 7020 6672 6f6d 2050 505e 3120  l map from PP^1 
│ │ │ │ +00074610: 746f 2063 7572 7665 2069 6e20 5050 5e34  to curve in PP^4
│ │ │ │ +00074620: 2920 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ) |.+-----------
│ │ │ │  00074630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074660: 2d2d 2d2b 0a7c 6934 203a 2069 7349 736f  ---+.|i4 : isIso
│ │ │ │ -00074670: 6d6f 7270 6869 736d 2070 6869 2020 2020  morphism phi    
│ │ │ │ -00074680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074660: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +00074670: 2069 7349 736f 6d6f 7270 6869 736d 2070   isIsomorphism p
│ │ │ │ +00074680: 6869 2020 2020 2020 2020 2020 2020 2020  hi              
│ │ │ │  00074690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000746a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000746b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000746a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000746b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000746c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000746d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000746e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000746f0: 207c 0a7c 6f34 203d 2074 7275 6520 2020   |.|o4 = true   
│ │ │ │ -00074700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000746f0: 2020 2020 2020 207c 0a7c 6f34 203d 2074         |.|o4 = t
│ │ │ │ +00074700: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │  00074710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074730: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00074740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00074730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00074740: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00074750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00074780: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00074790: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2069  ===..  * *note i
│ │ │ │ -000747a0: 6465 616c 2852 6174 696f 6e61 6c4d 6170  deal(RationalMap
│ │ │ │ -000747b0: 293a 2069 6465 616c 5f6c 7052 6174 696f  ): ideal_lpRatio
│ │ │ │ -000747c0: 6e61 6c4d 6170 5f72 702c 202d 2d20 6261  nalMap_rp, -- ba
│ │ │ │ -000747d0: 7365 206c 6f63 7573 206f 6620 610a 2020  se locus of a.  
│ │ │ │ -000747e0: 2020 7261 7469 6f6e 616c 206d 6170 0a20    rational map. 
│ │ │ │ -000747f0: 202a 202a 6e6f 7465 2069 7342 6972 6174   * *note isBirat
│ │ │ │ -00074800: 696f 6e61 6c3a 2069 7342 6972 6174 696f  ional: isBiratio
│ │ │ │ -00074810: 6e61 6c2c 202d 2d20 7768 6574 6865 7220  nal, -- whether 
│ │ │ │ -00074820: 6120 7261 7469 6f6e 616c 206d 6170 2069  a rational map i
│ │ │ │ -00074830: 7320 6269 7261 7469 6f6e 616c 0a20 202a  s birational.  *
│ │ │ │ -00074840: 202a 6e6f 7465 2069 734d 6f72 7068 6973   *note isMorphis
│ │ │ │ -00074850: 6d3a 2069 734d 6f72 7068 6973 6d2c 202d  m: isMorphism, -
│ │ │ │ -00074860: 2d20 7768 6574 6865 7220 6120 7261 7469  - whether a rati
│ │ │ │ -00074870: 6f6e 616c 206d 6170 2069 7320 6120 6d6f  onal map is a mo
│ │ │ │ -00074880: 7270 6869 736d 0a0a 5761 7973 2074 6f20  rphism..Ways to 
│ │ │ │ -00074890: 7573 6520 7468 6973 206d 6574 686f 643a  use this method:
│ │ │ │ -000748a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -000748b0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  =========..  * *
│ │ │ │ -000748c0: 6e6f 7465 2069 7349 736f 6d6f 7270 6869  note isIsomorphi
│ │ │ │ -000748d0: 736d 2852 6174 696f 6e61 6c4d 6170 293a  sm(RationalMap):
│ │ │ │ -000748e0: 2069 7349 736f 6d6f 7270 6869 736d 5f6c   isIsomorphism_l
│ │ │ │ -000748f0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -00074900: 202d 2d0a 2020 2020 7768 6574 6865 7220   --.    whether 
│ │ │ │ -00074910: 6120 6269 7261 7469 6f6e 616c 206d 6170  a birational map
│ │ │ │ -00074920: 2069 7320 616e 2069 736f 6d6f 7270 6869   is an isomorphi
│ │ │ │ -00074930: 736d 0a1f 0a46 696c 653a 2043 7265 6d6f  sm...File: Cremo
│ │ │ │ -00074940: 6e61 2e69 6e66 6f2c 204e 6f64 653a 2069  na.info, Node: i
│ │ │ │ -00074950: 734d 6f72 7068 6973 6d2c 204e 6578 743a  sMorphism, Next:
│ │ │ │ -00074960: 206b 6572 6e65 6c5f 6c70 5269 6e67 4d61   kernel_lpRingMa
│ │ │ │ -00074970: 705f 636d 5a5a 5f72 702c 2050 7265 763a  p_cmZZ_rp, Prev:
│ │ │ │ -00074980: 2069 7349 736f 6d6f 7270 6869 736d 5f6c   isIsomorphism_l
│ │ │ │ -00074990: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -000749a0: 2055 703a 2054 6f70 0a0a 6973 4d6f 7270   Up: Top..isMorp
│ │ │ │ -000749b0: 6869 736d 202d 2d20 7768 6574 6865 7220  hism -- whether 
│ │ │ │ -000749c0: 6120 7261 7469 6f6e 616c 206d 6170 2069  a rational map i
│ │ │ │ -000749d0: 7320 6120 6d6f 7270 6869 736d 0a2a 2a2a  s a morphism.***
│ │ │ │ -000749e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00074770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00074780: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ +00074790: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +000747a0: 6e6f 7465 2069 6465 616c 2852 6174 696f  note ideal(Ratio
│ │ │ │ +000747b0: 6e61 6c4d 6170 293a 2069 6465 616c 5f6c  nalMap): ideal_l
│ │ │ │ +000747c0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +000747d0: 202d 2d20 6261 7365 206c 6f63 7573 206f   -- base locus o
│ │ │ │ +000747e0: 6620 610a 2020 2020 7261 7469 6f6e 616c  f a.    rational
│ │ │ │ +000747f0: 206d 6170 0a20 202a 202a 6e6f 7465 2069   map.  * *note i
│ │ │ │ +00074800: 7342 6972 6174 696f 6e61 6c3a 2069 7342  sBirational: isB
│ │ │ │ +00074810: 6972 6174 696f 6e61 6c2c 202d 2d20 7768  irational, -- wh
│ │ │ │ +00074820: 6574 6865 7220 6120 7261 7469 6f6e 616c  ether a rational
│ │ │ │ +00074830: 206d 6170 2069 7320 6269 7261 7469 6f6e   map is biration
│ │ │ │ +00074840: 616c 0a20 202a 202a 6e6f 7465 2069 734d  al.  * *note isM
│ │ │ │ +00074850: 6f72 7068 6973 6d3a 2069 734d 6f72 7068  orphism: isMorph
│ │ │ │ +00074860: 6973 6d2c 202d 2d20 7768 6574 6865 7220  ism, -- whether 
│ │ │ │ +00074870: 6120 7261 7469 6f6e 616c 206d 6170 2069  a rational map i
│ │ │ │ +00074880: 7320 6120 6d6f 7270 6869 736d 0a0a 5761  s a morphism..Wa
│ │ │ │ +00074890: 7973 2074 6f20 7573 6520 7468 6973 206d  ys to use this m
│ │ │ │ +000748a0: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d  ethod:.=========
│ │ │ │ +000748b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +000748c0: 0a20 202a 202a 6e6f 7465 2069 7349 736f  .  * *note isIso
│ │ │ │ +000748d0: 6d6f 7270 6869 736d 2852 6174 696f 6e61  morphism(Rationa
│ │ │ │ +000748e0: 6c4d 6170 293a 2069 7349 736f 6d6f 7270  lMap): isIsomorp
│ │ │ │ +000748f0: 6869 736d 5f6c 7052 6174 696f 6e61 6c4d  hism_lpRationalM
│ │ │ │ +00074900: 6170 5f72 702c 202d 2d0a 2020 2020 7768  ap_rp, --.    wh
│ │ │ │ +00074910: 6574 6865 7220 6120 6269 7261 7469 6f6e  ether a biration
│ │ │ │ +00074920: 616c 206d 6170 2069 7320 616e 2069 736f  al map is an iso
│ │ │ │ +00074930: 6d6f 7270 6869 736d 0a1f 0a46 696c 653a  morphism...File:
│ │ │ │ +00074940: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ +00074950: 6f64 653a 2069 734d 6f72 7068 6973 6d2c  ode: isMorphism,
│ │ │ │ +00074960: 204e 6578 743a 206b 6572 6e65 6c5f 6c70   Next: kernel_lp
│ │ │ │ +00074970: 5269 6e67 4d61 705f 636d 5a5a 5f72 702c  RingMap_cmZZ_rp,
│ │ │ │ +00074980: 2050 7265 763a 2069 7349 736f 6d6f 7270   Prev: isIsomorp
│ │ │ │ +00074990: 6869 736d 5f6c 7052 6174 696f 6e61 6c4d  hism_lpRationalM
│ │ │ │ +000749a0: 6170 5f72 702c 2055 703a 2054 6f70 0a0a  ap_rp, Up: Top..
│ │ │ │ +000749b0: 6973 4d6f 7270 6869 736d 202d 2d20 7768  isMorphism -- wh
│ │ │ │ +000749c0: 6574 6865 7220 6120 7261 7469 6f6e 616c  ether a rational
│ │ │ │ +000749d0: 206d 6170 2069 7320 6120 6d6f 7270 6869   map is a morphi
│ │ │ │ +000749e0: 736d 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  sm.*************
│ │ │ │  000749f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00074a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00074a10: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ -00074a20: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ -00074a30: 2020 2020 2020 2020 6973 4d6f 7270 6869          isMorphi
│ │ │ │ -00074a40: 736d 2070 6869 0a20 202a 2049 6e70 7574  sm phi.  * Input
│ │ │ │ -00074a50: 733a 0a20 2020 2020 202a 2070 6869 2c20  s:.      * phi, 
│ │ │ │ -00074a60: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ -00074a70: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ -00074a80: 702c 0a20 202a 204f 7574 7075 7473 3a0a  p,.  * Outputs:.
│ │ │ │ -00074a90: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ -00074aa0: 426f 6f6c 6561 6e20 7661 6c75 653a 2028  Boolean value: (
│ │ │ │ -00074ab0: 4d61 6361 756c 6179 3244 6f63 2942 6f6f  Macaulay2Doc)Boo
│ │ │ │ -00074ac0: 6c65 616e 2c2c 2077 6865 7468 6572 2070  lean,, whether p
│ │ │ │ -00074ad0: 6869 2069 7320 610a 2020 2020 2020 2020  hi is a.        
│ │ │ │ -00074ae0: 6d6f 7270 6869 736d 2028 692e 652e 2c20  morphism (i.e., 
│ │ │ │ -00074af0: 6576 6572 7977 6865 7265 2064 6566 696e  everywhere defin
│ │ │ │ -00074b00: 6564 290a 0a44 6573 6372 6970 7469 6f6e  ed)..Description
│ │ │ │ -00074b10: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ -00074b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00074a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00074a10: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +00074a20: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ +00074a30: 6167 653a 200a 2020 2020 2020 2020 6973  age: .        is
│ │ │ │ +00074a40: 4d6f 7270 6869 736d 2070 6869 0a20 202a  Morphism phi.  *
│ │ │ │ +00074a50: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00074a60: 2070 6869 2c20 6120 2a6e 6f74 6520 7261   phi, a *note ra
│ │ │ │ +00074a70: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ +00074a80: 6f6e 616c 4d61 702c 0a20 202a 204f 7574  onalMap,.  * Out
│ │ │ │ +00074a90: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ +00074aa0: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661  *note Boolean va
│ │ │ │ +00074ab0: 6c75 653a 2028 4d61 6361 756c 6179 3244  lue: (Macaulay2D
│ │ │ │ +00074ac0: 6f63 2942 6f6f 6c65 616e 2c2c 2077 6865  oc)Boolean,, whe
│ │ │ │ +00074ad0: 7468 6572 2070 6869 2069 7320 610a 2020  ther phi is a.  
│ │ │ │ +00074ae0: 2020 2020 2020 6d6f 7270 6869 736d 2028        morphism (
│ │ │ │ +00074af0: 692e 652e 2c20 6576 6572 7977 6865 7265  i.e., everywhere
│ │ │ │ +00074b00: 2064 6566 696e 6564 290a 0a44 6573 6372   defined)..Descr
│ │ │ │ +00074b10: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00074b20: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ==..+-----------
│ │ │ │  00074b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00074b70: 3120 3a20 7068 6920 3d20 7175 6164 726f  1 : phi = quadro
│ │ │ │ -00074b80: 5175 6164 7269 6343 7265 6d6f 6e61 5472  QuadricCremonaTr
│ │ │ │ -00074b90: 616e 7366 6f72 6d61 7469 6f6e 2835 2c31  ansformation(5,1
│ │ │ │ -00074ba0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00074bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00074b70: 2d2d 2b0a 7c69 3120 3a20 7068 6920 3d20  --+.|i1 : phi = 
│ │ │ │ +00074b80: 7175 6164 726f 5175 6164 7269 6343 7265  quadroQuadricCre
│ │ │ │ +00074b90: 6d6f 6e61 5472 616e 7366 6f72 6d61 7469  monaTransformati
│ │ │ │ +00074ba0: 6f6e 2835 2c31 2920 2020 2020 2020 2020  on(5,1)         
│ │ │ │ +00074bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074bc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00074bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074c00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00074c10: 3120 3d20 2d2d 2072 6174 696f 6e61 6c20  1 = -- rational 
│ │ │ │ -00074c20: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +00074c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074c10: 2020 7c0a 7c6f 3120 3d20 2d2d 2072 6174    |.|o1 = -- rat
│ │ │ │ +00074c20: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │  00074c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074c60: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -00074c70: 2851 515b 782c 2079 2c20 7a2c 2074 2c20  (QQ[x, y, z, t, 
│ │ │ │ -00074c80: 752c 2076 5d29 2020 2020 2020 2020 2020  u, v])          
│ │ │ │ +00074c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074c60: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +00074c70: 3a20 5072 6f6a 2851 515b 782c 2079 2c20  : Proj(QQ[x, y, 
│ │ │ │ +00074c80: 7a2c 2074 2c20 752c 2076 5d29 2020 2020  z, t, u, v])    
│ │ │ │  00074c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074cb0: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -00074cc0: 2851 515b 782c 2079 2c20 7a2c 2074 2c20  (QQ[x, y, z, t, 
│ │ │ │ -00074cd0: 752c 2076 5d29 2020 2020 2020 2020 2020  u, v])          
│ │ │ │ +00074ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074cb0: 2020 7c0a 7c20 2020 2020 7461 7267 6574    |.|     target
│ │ │ │ +00074cc0: 3a20 5072 6f6a 2851 515b 782c 2079 2c20  : Proj(QQ[x, y, 
│ │ │ │ +00074cd0: 7a2c 2074 2c20 752c 2076 5d29 2020 2020  z, t, u, v])    
│ │ │ │  00074ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074d00: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -00074d10: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +00074cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074d00: 2020 7c0a 7c20 2020 2020 6465 6669 6e69    |.|     defini
│ │ │ │ +00074d10: 6e67 2066 6f72 6d73 3a20 7b20 2020 2020  ng forms: {     
│ │ │ │  00074d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074d40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074d60: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00074d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074d50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00074d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074d70: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00074d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074d90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074db0: 2020 2020 2079 2a7a 202d 2076 202c 2020       y*z - v ,  
│ │ │ │ -00074dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074da0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00074db0: 2020 2020 2020 2020 2020 2079 2a7a 202d             y*z -
│ │ │ │ +00074dc0: 2076 202c 2020 2020 2020 2020 2020 2020   v ,            
│ │ │ │  00074dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074de0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074df0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00074e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074e30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074e50: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00074e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074e40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00074e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074e60: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00074e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074e80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074ea0: 2020 2020 2078 2a7a 202d 2075 202c 2020       x*z - u ,  
│ │ │ │ -00074eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074e90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00074ea0: 2020 2020 2020 2020 2020 2078 2a7a 202d             x*z -
│ │ │ │ +00074eb0: 2075 202c 2020 2020 2020 2020 2020 2020   u ,            
│ │ │ │  00074ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074ed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074ee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00074ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074f20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074f40: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00074f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074f30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00074f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074f50: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00074f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074f70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074f90: 2020 2020 2078 2a79 202d 2074 202c 2020       x*y - t ,  
│ │ │ │ -00074fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074f80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00074f90: 2020 2020 2020 2020 2020 2078 2a79 202d             x*y -
│ │ │ │ +00074fa0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │  00074fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00074fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074fd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00074fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075010: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00075020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075030: 2020 2020 202d 207a 2a74 202b 2075 2a76       - z*t + u*v
│ │ │ │ -00075040: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00075010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075020: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00075030: 2020 2020 2020 2020 2020 202d 207a 2a74             - z*t
│ │ │ │ +00075040: 202b 2075 2a76 2c20 2020 2020 2020 2020   + u*v,         
│ │ │ │  00075050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075060: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00075070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075070: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00075080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000750a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000750b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000750c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000750d0: 2020 2020 202d 2079 2a75 202b 2074 2a76       - y*u + t*v
│ │ │ │ -000750e0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +000750b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000750c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000750d0: 2020 2020 2020 2020 2020 202d 2079 2a75             - y*u
│ │ │ │ +000750e0: 202b 2074 2a76 2c20 2020 2020 2020 2020   + t*v,         
│ │ │ │  000750f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075100: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00075110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075110: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00075120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00075160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075170: 2020 2020 2074 2a75 202d 2078 2a76 2020       t*u - x*v  
│ │ │ │ -00075180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075160: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00075170: 2020 2020 2020 2020 2020 2074 2a75 202d             t*u -
│ │ │ │ +00075180: 2078 2a76 2020 2020 2020 2020 2020 2020   x*v            
│ │ │ │  00075190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000751a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000751b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000751c0: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +000751a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000751b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000751c0: 2020 2020 2020 2020 2020 7d20 2020 2020            }     
│ │ │ │  000751d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000751e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000751f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00075200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000751f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075200: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00075210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00075250: 3120 3a20 5261 7469 6f6e 616c 4d61 7020  1 : RationalMap 
│ │ │ │ -00075260: 2843 7265 6d6f 6e61 2074 7261 6e73 666f  (Cremona transfo
│ │ │ │ -00075270: 726d 6174 696f 6e20 6f66 2050 505e 3520  rmation of PP^5 
│ │ │ │ -00075280: 6f66 2074 7970 6520 2832 2c32 2929 2020  of type (2,2))  
│ │ │ │ -00075290: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -000752a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00075240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075250: 2020 7c0a 7c6f 3120 3a20 5261 7469 6f6e    |.|o1 : Ration
│ │ │ │ +00075260: 616c 4d61 7020 2843 7265 6d6f 6e61 2074  alMap (Cremona t
│ │ │ │ +00075270: 7261 6e73 666f 726d 6174 696f 6e20 6f66  ransformation of
│ │ │ │ +00075280: 2050 505e 3520 6f66 2074 7970 6520 2832   PP^5 of type (2
│ │ │ │ +00075290: 2c32 2929 2020 2020 2020 2020 2020 2020  ,2))            
│ │ │ │ +000752a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000752b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000752c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000752d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000752e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000752f0: 3220 3a20 6973 4d6f 7270 6869 736d 2070  2 : isMorphism p
│ │ │ │ -00075300: 6869 2020 2020 2020 2020 2020 2020 2020  hi              
│ │ │ │ +000752e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000752f0: 2d2d 2b0a 7c69 3220 3a20 6973 4d6f 7270  --+.|i2 : isMorp
│ │ │ │ +00075300: 6869 736d 2070 6869 2020 2020 2020 2020  hism phi        
│ │ │ │  00075310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075330: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00075340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075340: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00075350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075380: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00075390: 3220 3d20 6661 6c73 6520 2020 2020 2020  2 = false       
│ │ │ │ +00075380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00075390: 2020 7c0a 7c6f 3220 3d20 6661 6c73 6520    |.|o2 = false 
│ │ │ │  000753a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000753b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000753c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000753d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -000753e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000753d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000753e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000753f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00075430: 3320 3a20 7068 6927 203d 206c 6173 7420  3 : phi' = last 
│ │ │ │ -00075440: 6772 6170 6820 7068 693b 2020 2020 2020  graph phi;      
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│ │ │ │  000754b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000754c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
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│ │ │ │ -000755c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000755c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
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│ │ │ │  000755e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000755f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00075630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075650: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  00075670: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00075690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000756a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000756b0: 3420 3d20 7472 7565 2020 2020 2020 2020  4 = true        
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│ │ │ │ +000756b0: 2020 7c0a 7c6f 3420 3d20 7472 7565 2020    |.|o4 = true  
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│ │ │ │  000756d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000756e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000756f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00075700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00075720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00075770: 6c28 5261 7469 6f6e 616c 4d61 7029 3a20  l(RationalMap): 
│ │ │ │ -00075780: 6964 6561 6c5f 6c70 5261 7469 6f6e 616c  ideal_lpRational
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│ │ │ │ -000757a0: 6c6f 6375 7320 6f66 2061 0a20 2020 2072  locus of a.    r
│ │ │ │ -000757b0: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
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│ │ │ │ -000757d0: 6973 6d28 5261 7469 6f6e 616c 4d61 7029  ism(RationalMap)
│ │ │ │ -000757e0: 3a20 6973 4973 6f6d 6f72 7068 6973 6d5f  : isIsomorphism_
│ │ │ │ -000757f0: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00075800: 2c20 2d2d 0a20 2020 2077 6865 7468 6572  , --.    whether
│ │ │ │ -00075810: 2061 2062 6972 6174 696f 6e61 6c20 6d61   a birational ma
│ │ │ │ -00075820: 7020 6973 2061 6e20 6973 6f6d 6f72 7068  p is an isomorph
│ │ │ │ -00075830: 6973 6d0a 0a57 6179 7320 746f 2075 7365  ism..Ways to use
│ │ │ │ -00075840: 2069 734d 6f72 7068 6973 6d3a 0a3d 3d3d   isMorphism:.===
│ │ │ │ -00075850: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00075860: 3d3d 3d3d 0a0a 2020 2a20 2269 734d 6f72  ====..  * "isMor
│ │ │ │ -00075870: 7068 6973 6d28 5261 7469 6f6e 616c 4d61  phism(RationalMa
│ │ │ │ -00075880: 7029 220a 0a46 6f72 2074 6865 2070 726f  p)"..For the pro
│ │ │ │ -00075890: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -000758a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -000758b0: 6f62 6a65 6374 202a 6e6f 7465 2069 734d  object *note isM
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│ │ │ │ -000758d0: 6973 6d2c 2069 7320 6120 2a6e 6f74 6520  ism, is a *note 
│ │ │ │ -000758e0: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ -000758f0: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ -00075900: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ -00075910: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -00075920: 696e 666f 2c20 4e6f 6465 3a20 6b65 726e  info, Node: kern
│ │ │ │ -00075930: 656c 5f6c 7052 696e 674d 6170 5f63 6d5a  el_lpRingMap_cmZ
│ │ │ │ -00075940: 5a5f 7270 2c20 4e65 7874 3a20 6d61 705f  Z_rp, Next: map_
│ │ │ │ -00075950: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00075960: 2c20 5072 6576 3a20 6973 4d6f 7270 6869  , Prev: isMorphi
│ │ │ │ -00075970: 736d 2c20 5570 3a20 546f 700a 0a6b 6572  sm, Up: Top..ker
│ │ │ │ -00075980: 6e65 6c28 5269 6e67 4d61 702c 5a5a 2920  nel(RingMap,ZZ) 
│ │ │ │ -00075990: 2d2d 2068 6f6d 6f67 656e 656f 7573 2063  -- homogeneous c
│ │ │ │ -000759a0: 6f6d 706f 6e65 6e74 7320 6f66 2074 6865  omponents of the
│ │ │ │ -000759b0: 206b 6572 6e65 6c20 6f66 2061 2068 6f6d   kernel of a hom
│ │ │ │ -000759c0: 6f67 656e 656f 7573 2072 696e 6720 6d61  ogeneous ring ma
│ │ │ │ -000759d0: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  p.**************
│ │ │ │ +00075740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00075750: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ +00075760: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +00075770: 6520 6964 6561 6c28 5261 7469 6f6e 616c  e ideal(Rational
│ │ │ │ +00075780: 4d61 7029 3a20 6964 6561 6c5f 6c70 5261  Map): ideal_lpRa
│ │ │ │ +00075790: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ +000757a0: 2062 6173 6520 6c6f 6375 7320 6f66 2061   base locus of a
│ │ │ │ +000757b0: 0a20 2020 2072 6174 696f 6e61 6c20 6d61  .    rational ma
│ │ │ │ +000757c0: 700a 2020 2a20 2a6e 6f74 6520 6973 4973  p.  * *note isIs
│ │ │ │ +000757d0: 6f6d 6f72 7068 6973 6d28 5261 7469 6f6e  omorphism(Ration
│ │ │ │ +000757e0: 616c 4d61 7029 3a20 6973 4973 6f6d 6f72  alMap): isIsomor
│ │ │ │ +000757f0: 7068 6973 6d5f 6c70 5261 7469 6f6e 616c  phism_lpRational
│ │ │ │ +00075800: 4d61 705f 7270 2c20 2d2d 0a20 2020 2077  Map_rp, --.    w
│ │ │ │ +00075810: 6865 7468 6572 2061 2062 6972 6174 696f  hether a biratio
│ │ │ │ +00075820: 6e61 6c20 6d61 7020 6973 2061 6e20 6973  nal map is an is
│ │ │ │ +00075830: 6f6d 6f72 7068 6973 6d0a 0a57 6179 7320  omorphism..Ways 
│ │ │ │ +00075840: 746f 2075 7365 2069 734d 6f72 7068 6973  to use isMorphis
│ │ │ │ +00075850: 6d3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  m:.=============
│ │ │ │ +00075860: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00075870: 2269 734d 6f72 7068 6973 6d28 5261 7469  "isMorphism(Rati
│ │ │ │ +00075880: 6f6e 616c 4d61 7029 220a 0a46 6f72 2074  onalMap)"..For t
│ │ │ │ +00075890: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +000758a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000758b0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +000758c0: 7465 2069 734d 6f72 7068 6973 6d3a 2069  te isMorphism: i
│ │ │ │ +000758d0: 734d 6f72 7068 6973 6d2c 2069 7320 6120  sMorphism, is a 
│ │ │ │ +000758e0: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +000758f0: 6374 696f 6e3a 0a28 4d61 6361 756c 6179  ction:.(Macaulay
│ │ │ │ +00075900: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ +00075910: 696f 6e2c 2e0a 1f0a 4669 6c65 3a20 4372  ion,....File: Cr
│ │ │ │ +00075920: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ +00075930: 3a20 6b65 726e 656c 5f6c 7052 696e 674d  : kernel_lpRingM
│ │ │ │ +00075940: 6170 5f63 6d5a 5a5f 7270 2c20 4e65 7874  ap_cmZZ_rp, Next
│ │ │ │ +00075950: 3a20 6d61 705f 6c70 5261 7469 6f6e 616c  : map_lpRational
│ │ │ │ +00075960: 4d61 705f 7270 2c20 5072 6576 3a20 6973  Map_rp, Prev: is
│ │ │ │ +00075970: 4d6f 7270 6869 736d 2c20 5570 3a20 546f  Morphism, Up: To
│ │ │ │ +00075980: 700a 0a6b 6572 6e65 6c28 5269 6e67 4d61  p..kernel(RingMa
│ │ │ │ +00075990: 702c 5a5a 2920 2d2d 2068 6f6d 6f67 656e  p,ZZ) -- homogen
│ │ │ │ +000759a0: 656f 7573 2063 6f6d 706f 6e65 6e74 7320  eous components 
│ │ │ │ +000759b0: 6f66 2074 6865 206b 6572 6e65 6c20 6f66  of the kernel of
│ │ │ │ +000759c0: 2061 2068 6f6d 6f67 656e 656f 7573 2072   a homogeneous r
│ │ │ │ +000759d0: 696e 6720 6d61 700a 2a2a 2a2a 2a2a 2a2a  ing map.********
│ │ │ │  000759e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000759f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00075a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00075a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00075a20: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ -00075a30: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2046  .========..  * F
│ │ │ │ -00075a40: 756e 6374 696f 6e3a 202a 6e6f 7465 206b  unction: *note k
│ │ │ │ -00075a50: 6572 6e65 6c3a 2028 4d61 6361 756c 6179  ernel: (Macaulay
│ │ │ │ -00075a60: 3244 6f63 296b 6572 6e65 6c2c 0a20 202a  2Doc)kernel,.  *
│ │ │ │ -00075a70: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00075a80: 206b 6572 6e65 6c28 7068 692c 6429 0a20   kernel(phi,d). 
│ │ │ │ -00075a90: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -00075aa0: 202a 2070 6869 2c20 6120 2a6e 6f74 6520   * phi, a *note 
│ │ │ │ -00075ab0: 7269 6e67 206d 6170 3a20 284d 6163 6175  ring map: (Macau
│ │ │ │ -00075ac0: 6c61 7932 446f 6329 5269 6e67 4d61 702c  lay2Doc)RingMap,
│ │ │ │ -00075ad0: 2c20 244b 5b79 5f30 2c5c 6c64 6f74 732c  , $K[y_0,\ldots,
│ │ │ │ -00075ae0: 795f 6d5d 2f4a 205c 746f 0a20 2020 2020  y_m]/J \to.     
│ │ │ │ -00075af0: 2020 204b 5b78 5f30 2c5c 6c64 6f74 732c     K[x_0,\ldots,
│ │ │ │ -00075b00: 785f 6e5d 2f49 242c 2064 6566 696e 6564  x_n]/I$, defined
│ │ │ │ -00075b10: 2062 7920 686f 6d6f 6765 6e65 6f75 7320   by homogeneous 
│ │ │ │ -00075b20: 666f 726d 7320 6f66 2074 6865 2073 616d  forms of the sam
│ │ │ │ -00075b30: 6520 6465 6772 6565 0a20 2020 2020 2020  e degree.       
│ │ │ │ -00075b40: 2061 6e64 2077 6865 7265 2024 4a24 2061   and where $J$ a
│ │ │ │ -00075b50: 6e64 2024 4924 2061 7265 2068 6f6d 6f67  nd $I$ are homog
│ │ │ │ -00075b60: 656e 656f 7573 2069 6465 616c 730a 2020  eneous ideals.  
│ │ │ │ -00075b70: 2020 2020 2a20 642c 2061 6e20 2a6e 6f74      * d, an *not
│ │ │ │ -00075b80: 6520 696e 7465 6765 723a 2028 4d61 6361  e integer: (Maca
│ │ │ │ -00075b90: 756c 6179 3244 6f63 295a 5a2c 0a20 202a  ulay2Doc)ZZ,.  *
│ │ │ │ -00075ba0: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ -00075bb0: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ -00075bc0: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ -00075bd0: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ -00075be0: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ -00075bf0: 2020 202a 202a 6e6f 7465 2053 7562 7269     * *note Subri
│ │ │ │ -00075c00: 6e67 4c69 6d69 743a 2028 4d61 6361 756c  ngLimit: (Macaul
│ │ │ │ -00075c10: 6179 3244 6f63 296b 6572 6e65 6c5f 6c70  ay2Doc)kernel_lp
│ │ │ │ -00075c20: 5f70 645f 7064 5f70 645f 636d 5375 6272  _pd_pd_pd_cmSubr
│ │ │ │ -00075c30: 696e 674c 696d 6974 3d3e 5f0a 2020 2020  ingLimit=>_.    
│ │ │ │ -00075c40: 2020 2020 7064 5f70 645f 7064 5f72 702c      pd_pd_pd_rp,
│ │ │ │ -00075c50: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -00075c60: 2076 616c 7565 2069 6e66 696e 6974 792c   value infinity,
│ │ │ │ -00075c70: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00075c80: 2020 2020 2a20 7468 6520 2a6e 6f74 6520      * the *note 
│ │ │ │ -00075c90: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ -00075ca0: 3244 6f63 2949 6465 616c 2c20 6765 6e65  2Doc)Ideal, gene
│ │ │ │ -00075cb0: 7261 7465 6420 6279 2061 6c6c 2068 6f6d  rated by all hom
│ │ │ │ -00075cc0: 6f67 656e 656f 7573 0a20 2020 2020 2020  ogeneous.       
│ │ │ │ -00075cd0: 2065 6c65 6d65 6e74 7320 6f66 2064 6567   elements of deg
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│ │ │ │ -00075cf0: 746f 2074 6865 206b 6572 6e65 6c20 6f66  to the kernel of
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│ │ │ │ -00075dc0: 616c 6765 6272 612e 2046 6f72 2073 6d61  algebra. For sma
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│ │ │ │ +00075a60: 6361 756c 6179 3244 6f63 296b 6572 6e65  caulay2Doc)kerne
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│ │ │ │ +00075a90: 692c 6429 0a20 202a 2049 6e70 7574 733a  i,d).  * Inputs:
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│ │ │ │ +00075ac0: 284d 6163 6175 6c61 7932 446f 6329 5269  (Macaulay2Doc)Ri
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│ │ │ │ +00075b30: 6865 2073 616d 6520 6465 6772 6565 0a20  he same degree. 
│ │ │ │ +00075b40: 2020 2020 2020 2061 6e64 2077 6865 7265         and where
│ │ │ │ +00075b50: 2024 4a24 2061 6e64 2024 4924 2061 7265   $J$ and $I$ are
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│ │ │ │ +00075b70: 616c 730a 2020 2020 2020 2a20 642c 2061  als.      * d, a
│ │ │ │ +00075b80: 6e20 2a6e 6f74 6520 696e 7465 6765 723a  n *note integer:
│ │ │ │ +00075b90: 2028 4d61 6361 756c 6179 3244 6f63 295a   (Macaulay2Doc)Z
│ │ │ │ +00075ba0: 5a2c 0a20 202a 202a 6e6f 7465 204f 7074  Z,.  * *note Opt
│ │ │ │ +00075bb0: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ +00075bc0: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
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│ │ │ │ +00075be0: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ +00075bf0: 2c3a 0a20 2020 2020 202a 202a 6e6f 7465  ,:.      * *note
│ │ │ │ +00075c00: 2053 7562 7269 6e67 4c69 6d69 743a 2028   SubringLimit: (
│ │ │ │ +00075c10: 4d61 6361 756c 6179 3244 6f63 296b 6572  Macaulay2Doc)ker
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│ │ │ │ +00075c50: 7064 5f72 702c 203d 3e20 2e2e 2e2c 2064  pd_rp, => ..., d
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│ │ │ │ +00075c80: 7473 3a0a 2020 2020 2020 2a20 7468 6520  ts:.      * the 
│ │ │ │ +00075c90: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ +00075ca0: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
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│ │ │ │ +00075d70: 7265 6374 0a61 6c67 6f72 6974 686d 2e20  rect.algorithm. 
│ │ │ │ +00075d80: 5765 2074 616b 6520 6164 7661 6e74 6167  We take advantag
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│ │ │ │ -00077390: 7820 202b 2032 7820 7820 202b 2034 7820  x  + 2x x  + 4x 
│ │ │ │ -000773a0: 7820 202b 2032 7820 7820 202b 7c0a 7c20  x  + 2x x  +|.| 
│ │ │ │ -000773b0: 3120 3220 2020 2020 3020 3420 2020 2031  1 2     0 4    1
│ │ │ │ -000773c0: 2034 2020 2020 2020 3120 3520 2020 2032   4      1 5    2
│ │ │ │ -000773d0: 2035 2020 2020 2034 2035 2020 2020 2030   5     4 5     0
│ │ │ │ -000773e0: 2036 2020 2020 2031 2036 2020 2020 2034   6     1 6     4
│ │ │ │ -000773f0: 2036 2020 2020 2032 2037 2020 7c0a 7c2d   6     2 7  |.|-
│ │ │ │ -00077400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00077350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00077360: 2d2d 7c0a 7c78 2078 2020 2b20 3578 2078  --|.|x x  + 5x x
│ │ │ │ +00077370: 2020 2b20 7820 7820 202d 2031 3478 2078    + x x  - 14x x
│ │ │ │ +00077380: 2020 2d20 7820 7820 202d 2038 7820 7820    - x x  - 8x x 
│ │ │ │ +00077390: 202d 2038 7820 7820 202b 2032 7820 7820   - 8x x  + 2x x 
│ │ │ │ +000773a0: 202b 2034 7820 7820 202b 2032 7820 7820   + 4x x  + 2x x 
│ │ │ │ +000773b0: 202b 7c0a 7c20 3120 3220 2020 2020 3020   +|.| 1 2     0 
│ │ │ │ +000773c0: 3420 2020 2031 2034 2020 2020 2020 3120  4    1 4      1 
│ │ │ │ +000773d0: 3520 2020 2032 2035 2020 2020 2034 2035  5    2 5     4 5
│ │ │ │ +000773e0: 2020 2020 2030 2036 2020 2020 2031 2036       0 6     1 6
│ │ │ │ +000773f0: 2020 2020 2034 2036 2020 2020 2032 2037       4 6     2 7
│ │ │ │ +00077400: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  00077410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00077440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34  ------------|.|4
│ │ │ │ -00077450: 7820 7820 202b 2033 7820 7820 202d 2037  x x  + 3x x  - 7
│ │ │ │ -00077460: 7820 7820 202b 2032 7820 7820 202d 2033  x x  + 2x x  - 3
│ │ │ │ -00077470: 7820 7820 7d29 2020 2020 2020 2020 2020  x x })          
│ │ │ │ +00077440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00077450: 2d2d 7c0a 7c34 7820 7820 202b 2033 7820  --|.|4x x  + 3x 
│ │ │ │ +00077460: 7820 202d 2037 7820 7820 202b 2032 7820  x  - 7x x  + 2x 
│ │ │ │ +00077470: 7820 202d 2033 7820 7820 7d29 2020 2020  x  - 3x x })    
│ │ │ │  00077480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077490: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000774a0: 2030 2038 2020 2020 2031 2038 2020 2020   0 8     1 8    
│ │ │ │ -000774b0: 2035 2038 2020 2020 2036 2038 2020 2020   5 8     6 8    
│ │ │ │ -000774c0: 2037 2038 2020 2020 2020 2020 2020 2020   7 8            
│ │ │ │ +00077490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000774a0: 2020 7c0a 7c20 2030 2038 2020 2020 2031    |.|  0 8     1
│ │ │ │ +000774b0: 2038 2020 2020 2035 2038 2020 2020 2036   8     5 8     6
│ │ │ │ +000774c0: 2038 2020 2020 2037 2038 2020 2020 2020   8     7 8      
│ │ │ │  000774d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000774e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -000774f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000774e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000774f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00077500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00077530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00077540: 3220 3a20 7469 6d65 206b 6572 6e65 6c28  2 : time kernel(
│ │ │ │ -00077550: 7068 692c 3129 2020 2020 2020 2020 2020  phi,1)          
│ │ │ │ +00077530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00077540: 2d2d 2b0a 7c69 3220 3a20 7469 6d65 206b  --+.|i2 : time k
│ │ │ │ +00077550: 6572 6e65 6c28 7068 692c 3129 2020 2020  ernel(phi,1)    
│ │ │ │  00077560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077580: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00077590: 2d2d 2075 7365 6420 302e 3131 3533 3438  -- used 0.115348
│ │ │ │ -000775a0: 7320 2863 7075 293b 2030 2e30 3336 3531  s (cpu); 0.03651
│ │ │ │ -000775b0: 3832 7320 2874 6872 6561 6429 3b20 3073  82s (thread); 0s
│ │ │ │ -000775c0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ -000775d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000775e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077590: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +000775a0: 3031 3939 3973 2028 6370 7529 3b20 302e  01999s (cpu); 0.
│ │ │ │ +000775b0: 3031 3932 3631 3673 2028 7468 7265 6164  0192616s (thread
│ │ │ │ +000775c0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +000775d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000775e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000775f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077620: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00077630: 3220 3d20 6964 6561 6c20 2829 2020 2020  2 = ideal ()    
│ │ │ │ -00077640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077630: 2020 7c0a 7c6f 3220 3d20 6964 6561 6c20    |.|o2 = ideal 
│ │ │ │ +00077640: 2829 2020 2020 2020 2020 2020 2020 2020  ()              
│ │ │ │  00077650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077670: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00077680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077680: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00077690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000776a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000776b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000776c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000776d0: 3220 3a20 4964 6561 6c20 6f66 2051 515b  2 : Ideal of QQ[
│ │ │ │ -000776e0: 7920 2e2e 7920 205d 2020 2020 2020 2020  y ..y  ]        
│ │ │ │ +000776c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000776d0: 2020 7c0a 7c6f 3220 3a20 4964 6561 6c20    |.|o2 : Ideal 
│ │ │ │ +000776e0: 6f66 2051 515b 7920 2e2e 7920 205d 2020  of QQ[y ..y  ]  
│ │ │ │  000776f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077710: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00077720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077730: 2030 2020 2031 3120 2020 2020 2020 2020   0   11         
│ │ │ │ +00077710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077720: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00077730: 2020 2020 2020 2030 2020 2031 3120 2020         0   11   
│ │ │ │  00077740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077760: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00077770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00077760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077770: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00077780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000777a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000777b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000777c0: 3320 3a20 7469 6d65 206b 6572 6e65 6c28  3 : time kernel(
│ │ │ │ -000777d0: 7068 692c 3229 2020 2020 2020 2020 2020  phi,2)          
│ │ │ │ +000777b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000777c0: 2d2d 2b0a 7c69 3320 3a20 7469 6d65 206b  --+.|i3 : time k
│ │ │ │ +000777d0: 6572 6e65 6c28 7068 692c 3229 2020 2020  ernel(phi,2)    
│ │ │ │  000777e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000777f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00077810: 2d2d 2075 7365 6420 302e 3437 3431 3639  -- used 0.474169
│ │ │ │ -00077820: 7320 2863 7075 293b 2030 2e33 3130 3831  s (cpu); 0.31081
│ │ │ │ -00077830: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ -00077840: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -00077850: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00077860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077810: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +00077820: 3436 3839 3234 7320 2863 7075 293b 2030  468924s (cpu); 0
│ │ │ │ +00077830: 2e33 3931 3335 3373 2028 7468 7265 6164  .391353s (thread
│ │ │ │ +00077840: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +00077850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077860: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00077870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000778a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000778b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000778c0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000778d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000778a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000778b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000778c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000778d0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  000778e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000778f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00077900: 3320 3d20 6964 6561 6c20 2879 2079 2020  3 = ideal (y y  
│ │ │ │ -00077910: 2b20 7920 7920 202b 2079 2020 2b20 3579  + y y  + y  + 5y
│ │ │ │ -00077920: 2079 2020 2b20 7920 7920 202b 2035 7920   y  + y y  + 5y 
│ │ │ │ -00077930: 7920 202d 2079 2079 2020 2d20 3479 2079  y  - y y  - 4y y
│ │ │ │ -00077940: 2020 2d20 3579 2079 2020 2d20 7c0a 7c20    - 5y y  - |.| 
│ │ │ │ -00077950: 2020 2020 2020 2020 2020 2020 3220 3420              2 4 
│ │ │ │ -00077960: 2020 2033 2034 2020 2020 3420 2020 2020     3 4    4     
│ │ │ │ -00077970: 3220 3520 2020 2033 2035 2020 2020 2034  2 5    3 5     4
│ │ │ │ -00077980: 2035 2020 2020 3120 3620 2020 2020 3220   5    1 6     2 
│ │ │ │ -00077990: 3620 2020 2020 3520 3620 2020 7c0a 7c20  6     5 6   |.| 
│ │ │ │ -000779a0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │ +000778f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077900: 2020 7c0a 7c6f 3320 3d20 6964 6561 6c20    |.|o3 = ideal 
│ │ │ │ +00077910: 2879 2079 2020 2b20 7920 7920 202b 2079  (y y  + y y  + y
│ │ │ │ +00077920: 2020 2b20 3579 2079 2020 2b20 7920 7920    + 5y y  + y y 
│ │ │ │ +00077930: 202b 2035 7920 7920 202d 2079 2079 2020   + 5y y  - y y  
│ │ │ │ +00077940: 2d20 3479 2079 2020 2d20 3579 2079 2020  - 4y y  - 5y y  
│ │ │ │ +00077950: 2d20 7c0a 7c20 2020 2020 2020 2020 2020  - |.|           
│ │ │ │ +00077960: 2020 3220 3420 2020 2033 2034 2020 2020    2 4    3 4    
│ │ │ │ +00077970: 3420 2020 2020 3220 3520 2020 2033 2035  4     2 5    3 5
│ │ │ │ +00077980: 2020 2020 2034 2035 2020 2020 3120 3620       4 5    1 6 
│ │ │ │ +00077990: 2020 2020 3220 3620 2020 2020 3520 3620      2 6     5 6 
│ │ │ │ +000779a0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
│ │ │ │  000779b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000779c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000779d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000779e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -000779f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000779e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000779f0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │  00077a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00077a40: 2020 2020 3479 2079 2020 2d20 3279 2079      4y y  - 2y y
│ │ │ │ -00077a50: 2020 2d20 7920 7920 202b 2034 7920 7920    - y y  + 4y y 
│ │ │ │ -00077a60: 202d 2035 7920 7920 202d 2034 7920 7920   - 5y y  - 4y y 
│ │ │ │ -00077a70: 202b 2033 7920 7920 202d 2034 7920 7920   + 3y y  - 4y y 
│ │ │ │ -00077a80: 202d 2079 2079 2020 202d 2020 7c0a 7c20   - y y   -  |.| 
│ │ │ │ -00077a90: 2020 2020 2020 3220 3720 2020 2020 3420        2 7     4 
│ │ │ │ -00077aa0: 3720 2020 2031 2038 2020 2020 2034 2038  7    1 8     4 8
│ │ │ │ -00077ab0: 2020 2020 2035 2038 2020 2020 2035 2039       5 8     5 9
│ │ │ │ -00077ac0: 2020 2020 2037 2039 2020 2020 2038 2039       7 9     8 9
│ │ │ │ -00077ad0: 2020 2020 3320 3130 2020 2020 7c0a 7c20      3 10    |.| 
│ │ │ │ -00077ae0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │ +00077a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077a40: 2020 7c0a 7c20 2020 2020 3479 2079 2020    |.|     4y y  
│ │ │ │ +00077a50: 2d20 3279 2079 2020 2d20 7920 7920 202b  - 2y y  - y y  +
│ │ │ │ +00077a60: 2034 7920 7920 202d 2035 7920 7920 202d   4y y  - 5y y  -
│ │ │ │ +00077a70: 2034 7920 7920 202b 2033 7920 7920 202d   4y y  + 3y y  -
│ │ │ │ +00077a80: 2034 7920 7920 202d 2079 2079 2020 202d   4y y  - y y   -
│ │ │ │ +00077a90: 2020 7c0a 7c20 2020 2020 2020 3220 3720    |.|       2 7 
│ │ │ │ +00077aa0: 2020 2020 3420 3720 2020 2031 2038 2020      4 7    1 8  
│ │ │ │ +00077ab0: 2020 2034 2038 2020 2020 2035 2038 2020     4 8     5 8  
│ │ │ │ +00077ac0: 2020 2035 2039 2020 2020 2037 2039 2020     5 9     7 9  
│ │ │ │ +00077ad0: 2020 2038 2039 2020 2020 3320 3130 2020     8 9    3 10  
│ │ │ │ +00077ae0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
│ │ │ │  00077af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00077b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -00077b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00077b30: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │  00077b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077b70: 2020 2020 2020 2032 2020 2020 7c0a 7c20         2    |.| 
│ │ │ │ -00077b80: 2020 2020 3379 2079 2020 202d 2035 7920      3y y   - 5y 
│ │ │ │ -00077b90: 7920 2020 2d20 7920 7920 2020 2b20 3479  y   - y y   + 4y
│ │ │ │ -00077ba0: 2079 2020 202b 2035 7920 7920 202c 2033   y   + 5y y  , 3
│ │ │ │ -00077bb0: 7920 7920 202d 2079 2079 2020 2d20 3379  y y  - y y  - 3y
│ │ │ │ -00077bc0: 2079 2020 2d20 7920 202b 2020 7c0a 7c20   y  - y  +  |.| 
│ │ │ │ -00077bd0: 2020 2020 2020 3620 3130 2020 2020 2038        6 10     8
│ │ │ │ -00077be0: 2031 3020 2020 2034 2031 3120 2020 2020   10    4 11     
│ │ │ │ -00077bf0: 3620 3131 2020 2020 2038 2031 3120 2020  6 11     8 11   
│ │ │ │ -00077c00: 2031 2033 2020 2020 3220 3320 2020 2020   1 3    2 3     
│ │ │ │ -00077c10: 3320 3420 2020 2034 2020 2020 7c0a 7c20  3 4    4    |.| 
│ │ │ │ -00077c20: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │ +00077b70: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00077b80: 2020 7c0a 7c20 2020 2020 3379 2079 2020    |.|     3y y  
│ │ │ │ +00077b90: 202d 2035 7920 7920 2020 2d20 7920 7920   - 5y y   - y y 
│ │ │ │ +00077ba0: 2020 2b20 3479 2079 2020 202b 2035 7920    + 4y y   + 5y 
│ │ │ │ +00077bb0: 7920 202c 2033 7920 7920 202d 2079 2079  y  , 3y y  - y y
│ │ │ │ +00077bc0: 2020 2d20 3379 2079 2020 2d20 7920 202b    - 3y y  - y  +
│ │ │ │ +00077bd0: 2020 7c0a 7c20 2020 2020 2020 3620 3130    |.|       6 10
│ │ │ │ +00077be0: 2020 2020 2038 2031 3020 2020 2034 2031       8 10    4 1
│ │ │ │ +00077bf0: 3120 2020 2020 3620 3131 2020 2020 2038  1     6 11     8
│ │ │ │ +00077c00: 2031 3120 2020 2031 2033 2020 2020 3220   11    1 3    2 
│ │ │ │ +00077c10: 3320 2020 2020 3320 3420 2020 2034 2020  3     3 4    4  
│ │ │ │ +00077c20: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
│ │ │ │  00077c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00077c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -00077c70: 2020 2020 3279 2079 2020 2d20 7920 7920      2y y  - y y 
│ │ │ │ -00077c80: 202b 2079 2079 2020 2b20 3279 2079 2020   + y y  + 2y y  
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│ │ │ │ -00077cb0: 2d20 3279 2079 2020 2b20 2020 7c0a 7c20  - 2y y  +   |.| 
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│ │ │ │ +00077ef0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
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│ │ │ │ -00077f40: 2020 2020 3779 2079 2020 2b20 3879 2079      7y y  + 8y y
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│ │ │ │ -00077f90: 2020 2020 2020 3220 3620 2020 2020 3520        2 6     5 
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│ │ │ │ +00077fe0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
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│ │ │ │ -00078030: 2020 2020 7920 7920 2020 2d20 7920 7920      y y   - y y 
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│ │ │ │ -00078050: 7920 2020 2d20 3779 2079 2020 202d 2037  y   - 7y y   - 7
│ │ │ │ -00078060: 7920 7920 2029 2020 2020 2020 2020 2020  y y  )          
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│ │ │ │ -000780b0: 2036 2031 3120 2020 2020 2020 2020 2020   6 11           
│ │ │ │ -000780c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000780d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00078070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078080: 2020 7c0a 7c20 2020 2020 2030 2031 3020    |.|      0 10 
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│ │ │ │ +000780c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000780d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000780e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000780f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078110: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00078120: 3320 3a20 4964 6561 6c20 6f66 2051 515b  3 : Ideal of QQ[
│ │ │ │ -00078130: 7920 2e2e 7920 205d 2020 2020 2020 2020  y ..y  ]        
│ │ │ │ +00078110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078120: 2020 7c0a 7c6f 3320 3a20 4964 6561 6c20    |.|o3 : Ideal 
│ │ │ │ +00078130: 6f66 2051 515b 7920 2e2e 7920 205d 2020  of QQ[y ..y  ]  
│ │ │ │  00078140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078160: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00078170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078180: 2030 2020 2031 3120 2020 2020 2020 2020   0   11         
│ │ │ │ +00078160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078170: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00078180: 2020 2020 2020 2030 2020 2031 3120 2020         0   11   
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│ │ │ │  000781a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000781b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -000781c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000781b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000781c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000781d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000781e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000781f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00078200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ -00078210: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -00078220: 0a0a 2020 2a20 2a6e 6f74 6520 6b65 726e  ..  * *note kern
│ │ │ │ -00078230: 656c 2852 696e 674d 6170 293a 2028 4d61  el(RingMap): (Ma
│ │ │ │ -00078240: 6361 756c 6179 3244 6f63 296b 6572 6e65  caulay2Doc)kerne
│ │ │ │ -00078250: 6c5f 6c70 5269 6e67 4d61 705f 7270 2c20  l_lpRingMap_rp, 
│ │ │ │ -00078260: 2d2d 206b 6572 6e65 6c20 6f66 2061 0a20  -- kernel of a. 
│ │ │ │ -00078270: 2020 2072 696e 676d 6170 0a0a 5761 7973     ringmap..Ways
│ │ │ │ -00078280: 2074 6f20 7573 6520 7468 6973 206d 6574   to use this met
│ │ │ │ -00078290: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  hod:.===========
│ │ │ │ -000782a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -000782b0: 202a 202a 6e6f 7465 206b 6572 6e65 6c28   * *note kernel(
│ │ │ │ -000782c0: 5269 6e67 4d61 702c 5a5a 293a 206b 6572  RingMap,ZZ): ker
│ │ │ │ -000782d0: 6e65 6c5f 6c70 5269 6e67 4d61 705f 636d  nel_lpRingMap_cm
│ │ │ │ -000782e0: 5a5a 5f72 702c 202d 2d20 686f 6d6f 6765  ZZ_rp, -- homoge
│ │ │ │ -000782f0: 6e65 6f75 730a 2020 2020 636f 6d70 6f6e  neous.    compon
│ │ │ │ -00078300: 656e 7473 206f 6620 7468 6520 6b65 726e  ents of the kern
│ │ │ │ -00078310: 656c 206f 6620 6120 686f 6d6f 6765 6e65  el of a homogene
│ │ │ │ -00078320: 6f75 7320 7269 6e67 206d 6170 0a1f 0a46  ous ring map...F
│ │ │ │ -00078330: 696c 653a 2043 7265 6d6f 6e61 2e69 6e66  ile: Cremona.inf
│ │ │ │ -00078340: 6f2c 204e 6f64 653a 206d 6170 5f6c 7052  o, Node: map_lpR
│ │ │ │ -00078350: 6174 696f 6e61 6c4d 6170 5f72 702c 204e  ationalMap_rp, N
│ │ │ │ -00078360: 6578 743a 206d 6174 7269 785f 6c70 5261  ext: matrix_lpRa
│ │ │ │ -00078370: 7469 6f6e 616c 4d61 705f 7270 2c20 5072  tionalMap_rp, Pr
│ │ │ │ -00078380: 6576 3a20 6b65 726e 656c 5f6c 7052 696e  ev: kernel_lpRin
│ │ │ │ -00078390: 674d 6170 5f63 6d5a 5a5f 7270 2c20 5570  gMap_cmZZ_rp, Up
│ │ │ │ -000783a0: 3a20 546f 700a 0a6d 6170 2852 6174 696f  : Top..map(Ratio
│ │ │ │ -000783b0: 6e61 6c4d 6170 2920 2d2d 2067 6574 2074  nalMap) -- get t
│ │ │ │ -000783c0: 6865 2072 696e 6720 6d61 7020 6465 6669  he ring map defi
│ │ │ │ -000783d0: 6e69 6e67 2061 2072 6174 696f 6e61 6c20  ning a rational 
│ │ │ │ -000783e0: 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  map.************
│ │ │ │ +00078200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00078210: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ +00078220: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +00078230: 6520 6b65 726e 656c 2852 696e 674d 6170  e kernel(RingMap
│ │ │ │ +00078240: 293a 2028 4d61 6361 756c 6179 3244 6f63  ): (Macaulay2Doc
│ │ │ │ +00078250: 296b 6572 6e65 6c5f 6c70 5269 6e67 4d61  )kernel_lpRingMa
│ │ │ │ +00078260: 705f 7270 2c20 2d2d 206b 6572 6e65 6c20  p_rp, -- kernel 
│ │ │ │ +00078270: 6f66 2061 0a20 2020 2072 696e 676d 6170  of a.    ringmap
│ │ │ │ +00078280: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
│ │ │ │ +00078290: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d  is method:.=====
│ │ │ │ +000782a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000782b0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206b  ===..  * *note k
│ │ │ │ +000782c0: 6572 6e65 6c28 5269 6e67 4d61 702c 5a5a  ernel(RingMap,ZZ
│ │ │ │ +000782d0: 293a 206b 6572 6e65 6c5f 6c70 5269 6e67  ): kernel_lpRing
│ │ │ │ +000782e0: 4d61 705f 636d 5a5a 5f72 702c 202d 2d20  Map_cmZZ_rp, -- 
│ │ │ │ +000782f0: 686f 6d6f 6765 6e65 6f75 730a 2020 2020  homogeneous.    
│ │ │ │ +00078300: 636f 6d70 6f6e 656e 7473 206f 6620 7468  components of th
│ │ │ │ +00078310: 6520 6b65 726e 656c 206f 6620 6120 686f  e kernel of a ho
│ │ │ │ +00078320: 6d6f 6765 6e65 6f75 7320 7269 6e67 206d  mogeneous ring m
│ │ │ │ +00078330: 6170 0a1f 0a46 696c 653a 2043 7265 6d6f  ap...File: Cremo
│ │ │ │ +00078340: 6e61 2e69 6e66 6f2c 204e 6f64 653a 206d  na.info, Node: m
│ │ │ │ +00078350: 6170 5f6c 7052 6174 696f 6e61 6c4d 6170  ap_lpRationalMap
│ │ │ │ +00078360: 5f72 702c 204e 6578 743a 206d 6174 7269  _rp, Next: matri
│ │ │ │ +00078370: 785f 6c70 5261 7469 6f6e 616c 4d61 705f  x_lpRationalMap_
│ │ │ │ +00078380: 7270 2c20 5072 6576 3a20 6b65 726e 656c  rp, Prev: kernel
│ │ │ │ +00078390: 5f6c 7052 696e 674d 6170 5f63 6d5a 5a5f  _lpRingMap_cmZZ_
│ │ │ │ +000783a0: 7270 2c20 5570 3a20 546f 700a 0a6d 6170  rp, Up: Top..map
│ │ │ │ +000783b0: 2852 6174 696f 6e61 6c4d 6170 2920 2d2d  (RationalMap) --
│ │ │ │ +000783c0: 2067 6574 2074 6865 2072 696e 6720 6d61   get the ring ma
│ │ │ │ +000783d0: 7020 6465 6669 6e69 6e67 2061 2072 6174  p defining a rat
│ │ │ │ +000783e0: 696f 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a  ional map.******
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│ │ │ │ -00078480: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00078490: 2050 6869 2c20 6120 2a6e 6f74 6520 7261   Phi, a *note ra
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│ │ │ │ -00078510: 202a 6e6f 7465 2044 6567 7265 653a 2028   *note Degree: (
│ │ │ │ -00078520: 4d61 6361 756c 6179 3244 6f63 2944 6567  Macaulay2Doc)Deg
│ │ │ │ -00078530: 7265 652c 203d 3e20 2e2e 2e2c 2064 6566  ree, => ..., def
│ │ │ │ -00078540: 6175 6c74 2076 616c 7565 206e 756c 6c2c  ault value null,
│ │ │ │ -00078550: 200a 2020 2020 2020 2a20 2a6e 6f74 6520   .      * *note 
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│ │ │ │ -00078570: 6175 6c61 7932 446f 6329 6d61 705f 6c70  aulay2Doc)map_lp
│ │ │ │ -00078580: 5269 6e67 5f63 6d52 696e 675f 636d 4d61  Ring_cmRing_cmMa
│ │ │ │ -00078590: 7472 6978 5f72 702c 203d 3e20 2e2e 2e2c  trix_rp, => ...,
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│ │ │ │ -000785b0: 2076 616c 7565 206e 756c 6c2c 0a20 2020   value null,.   
│ │ │ │ -000785c0: 2020 202a 202a 6e6f 7465 2044 6567 7265     * *note Degre
│ │ │ │ -000785d0: 654d 6170 3a20 284d 6163 6175 6c61 7932  eMap: (Macaulay2
│ │ │ │ -000785e0: 446f 6329 6d61 705f 6c70 5269 6e67 5f63  Doc)map_lpRing_c
│ │ │ │ -000785f0: 6d52 696e 675f 636d 4d61 7472 6978 5f72  mRing_cmMatrix_r
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│ │ │ │ -00078610: 2020 2064 6566 6175 6c74 2076 616c 7565     default value
│ │ │ │ -00078620: 206e 756c 6c2c 0a20 202a 204f 7574 7075   null,.  * Outpu
│ │ │ │ -00078630: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ -00078640: 6f74 6520 7269 6e67 206d 6170 3a20 284d  ote ring map: (M
│ │ │ │ -00078650: 6163 6175 6c61 7932 446f 6329 5269 6e67  acaulay2Doc)Ring
│ │ │ │ -00078660: 4d61 702c 2c20 7468 6520 7269 6e67 206d  Map,, the ring m
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│ │ │ │ -00078680: 650a 2020 2020 2020 2020 7261 7469 6f6e  e.        ration
│ │ │ │ -00078690: 616c 206d 6170 2050 6869 2077 6173 2064  al map Phi was d
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│ │ │ │ -000786c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00078420: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +00078430: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2046  .========..  * F
│ │ │ │ +00078440: 756e 6374 696f 6e3a 202a 6e6f 7465 206d  unction: *note m
│ │ │ │ +00078450: 6170 3a20 284d 6163 6175 6c61 7932 446f  ap: (Macaulay2Do
│ │ │ │ +00078460: 6329 6d61 702c 0a20 202a 2055 7361 6765  c)map,.  * Usage
│ │ │ │ +00078470: 3a20 0a20 2020 2020 2020 206d 6170 2050  : .        map P
│ │ │ │ +00078480: 6869 0a20 202a 2049 6e70 7574 733a 0a20  hi.  * Inputs:. 
│ │ │ │ +00078490: 2020 2020 202a 2050 6869 2c20 6120 2a6e       * Phi, a *n
│ │ │ │ +000784a0: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ +000784b0: 3a20 5261 7469 6f6e 616c 4d61 702c 0a20  : RationalMap,. 
│ │ │ │ +000784c0: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
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│ │ │ │ +000784e0: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ +000784f0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +00078500: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ +00078510: 2020 2020 202a 202a 6e6f 7465 2044 6567       * *note Deg
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│ │ │ │ +00078560: 2a6e 6f74 6520 4465 6772 6565 4c69 6674  *note DegreeLift
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│ │ │ │ +00078580: 6d61 705f 6c70 5269 6e67 5f63 6d52 696e  map_lpRing_cmRin
│ │ │ │ +00078590: 675f 636d 4d61 7472 6978 5f72 702c 203d  g_cmMatrix_rp, =
│ │ │ │ +000785a0: 3e20 2e2e 2e2c 0a20 2020 2020 2020 2064  > ...,.        d
│ │ │ │ +000785b0: 6566 6175 6c74 2076 616c 7565 206e 756c  efault value nul
│ │ │ │ +000785c0: 6c2c 0a20 2020 2020 202a 202a 6e6f 7465  l,.      * *note
│ │ │ │ +000785d0: 2044 6567 7265 654d 6170 3a20 284d 6163   DegreeMap: (Mac
│ │ │ │ +000785e0: 6175 6c61 7932 446f 6329 6d61 705f 6c70  aulay2Doc)map_lp
│ │ │ │ +000785f0: 5269 6e67 5f63 6d52 696e 675f 636d 4d61  Ring_cmRing_cmMa
│ │ │ │ +00078600: 7472 6978 5f72 702c 203d 3e20 2e2e 2e2c  trix_rp, => ...,
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│ │ │ │ +00078620: 2076 616c 7565 206e 756c 6c2c 0a20 202a   value null,.  *
│ │ │ │ +00078630: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +00078640: 2a20 6120 2a6e 6f74 6520 7269 6e67 206d  * a *note ring m
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│ │ │ │ +00078680: 6963 6820 7468 650a 2020 2020 2020 2020  ich the.        
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│ │ │ │ +000786b0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +000786c0: 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d  =====..+--------
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│ │ │ │  000786e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000786f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00078700: 2d2d 2d2d 2d2b 0a7c 6931 203a 2051 515b  -----+.|i1 : QQ[
│ │ │ │ -00078710: 745f 302e 2e74 5f33 5d20 2020 2020 2020  t_0..t_3]       
│ │ │ │ +00078700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00078710: 203a 2051 515b 745f 302e 2e74 5f33 5d20   : QQ[t_0..t_3] 
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│ │ │ │  00078730: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00078760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078790: 207c 0a7c 6f31 203d 2051 515b 7420 2e2e   |.|o1 = QQ[t ..
│ │ │ │ -000787a0: 7420 5d20 2020 2020 2020 2020 2020 2020  t ]             
│ │ │ │ +00078790: 2020 2020 2020 207c 0a7c 6f31 203d 2051         |.|o1 = Q
│ │ │ │ +000787a0: 515b 7420 2e2e 7420 5d20 2020 2020 2020  Q[t ..t ]       
│ │ │ │  000787b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000787c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000787d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000787e0: 2020 2030 2020 2033 2020 2020 2020 2020     0   3        
│ │ │ │ +000787d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000787e0: 2020 2020 2020 2020 2030 2020 2033 2020           0   3  
│ │ │ │  000787f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078800: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00078890: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000788e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000788f0: 0a7c 6932 203a 2050 6869 203d 2072 6174  .|i2 : Phi = rat
│ │ │ │ -00078900: 696f 6e61 6c4d 6170 207b 745f 315e 322b  ionalMap {t_1^2+
│ │ │ │ -00078910: 745f 325e 322b 745f 335e 322c 745f 302a  t_2^2+t_3^2,t_0*
│ │ │ │ -00078920: 745f 312c 745f 302a 745f 322c 745f 302a  t_1,t_0*t_2,t_0*
│ │ │ │ -00078930: 745f 337d 207c 0a7c 2020 2020 2020 2020  t_3} |.|        
│ │ │ │ +000788e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000788f0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2050 6869  -----+.|i2 : Phi
│ │ │ │ +00078900: 203d 2072 6174 696f 6e61 6c4d 6170 207b   = rationalMap {
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│ │ │ │ +00078920: 322c 745f 302a 745f 312c 745f 302a 745f  2,t_0*t_1,t_0*t_
│ │ │ │ +00078930: 322c 745f 302a 745f 337d 207c 0a7c 2020  2,t_0*t_3} |.|  
│ │ │ │  00078940: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00078960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078970: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
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│ │ │ │  000789b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000789c0: 207c 0a7c 2020 2020 2073 6f75 7263 653a   |.|     source:
│ │ │ │ -000789d0: 2050 726f 6a28 5151 5b74 202c 2074 202c   Proj(QQ[t , t ,
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│ │ │ │ +000789c0: 2020 2020 2020 207c 0a7c 2020 2020 2073         |.|     s
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│ │ │ │  000789f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00078a00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00078a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00078a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078a40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00078a50: 2020 2020 2074 6172 6765 743a 2050 726f       target: Pro
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│ │ │ │ -00078a70: 2074 205d 2920 2020 2020 2020 2020 2020   t ])           
│ │ │ │ +00078a20: 2020 2020 2020 3020 2020 3120 2020 3220        0   1   2 
│ │ │ │ +00078a30: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +00078a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00078a60: 743a 2050 726f 6a28 5151 5b74 202c 2074  t: Proj(QQ[t , t
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│ │ │ │  00078a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078a90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00078aa0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -00078ab0: 3120 2020 3220 2020 3320 2020 2020 2020  1   2   3       
│ │ │ │ +00078a90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00078aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078ab0: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │  00078ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078ad0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00078ae0: 2064 6566 696e 696e 6720 666f 726d 733a   defining forms:
│ │ │ │ -00078af0: 207b 2020 2020 2020 2020 2020 2020 2020   {              
│ │ │ │ +00078ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00078ae0: 0a7c 2020 2020 2064 6566 696e 696e 6720  .|     defining 
│ │ │ │ +00078af0: 666f 726d 733a 207b 2020 2020 2020 2020  forms: {        
│ │ │ │  00078b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00078b20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00078b30: 2020 2020 2020 2020 2032 2020 2020 3220           2    2 
│ │ │ │ -00078b40: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00078b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078b20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00078b30: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00078b40: 2020 2020 3220 2020 2032 2020 2020 2020      2    2      
│ │ │ │  00078b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078b60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00078b70: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -00078b80: 202b 2074 2020 2b20 7420 2c20 2020 2020   + t  + t ,     
│ │ │ │ -00078b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078ba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00078bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078bc0: 2020 2020 2031 2020 2020 3220 2020 2033       1    2    3
│ │ │ │ -00078bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00078b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078b80: 2020 2020 7420 202b 2074 2020 2b20 7420      t  + t  + t 
│ │ │ │ +00078b90: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00078ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00078bc0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +00078bd0: 3220 2020 2033 2020 2020 2020 2020 2020  2    3          
│ │ │ │  00078be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078bf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00078bf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00078c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078c30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00078c30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00078c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078c50: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │ +00078c50: 2020 2020 2020 7420 7420 2c20 2020 2020        t t ,     
│ │ │ │  00078c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00078c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078c90: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +00078c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078c80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00078c90: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │  00078ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078cc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00078cc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00078cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078d00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00078d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078d20: 2020 7420 7420 2c20 2020 2020 2020 2020    t t ,         
│ │ │ │ +00078d00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00078d10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00078d20: 2020 2020 2020 2020 7420 7420 2c20 2020          t t ,   
│ │ │ │  00078d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078d40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00078d50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00078d60: 2020 2020 2020 2020 2030 2032 2020 2020           0 2    
│ │ │ │ -00078d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078d50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00078d60: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00078d70: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00078d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078d90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00078d90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00078da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00078de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078df0: 2020 2020 7420 7420 2020 2020 2020 2020      t t         
│ │ │ │ +00078dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078de0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00078df0: 2020 2020 2020 2020 2020 7420 7420 2020            t t   
│ │ │ │  00078e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078e20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00078e30: 2020 2020 2020 2020 2020 2030 2033 2020             0 3  
│ │ │ │ -00078e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078e20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00078e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078e40: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
│ │ │ │  00078e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078e60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00078e70: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │ -00078e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078e60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00078e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078e80: 2020 2020 207d 2020 2020 2020 2020 2020       }          
│ │ │ │  00078e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078ea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00078eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078eb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00078ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00078f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00078ef0: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
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│ │ │ │ +00078f30: 6f20 5050 5e33 2920 2020 2020 2020 207c  o PP^3)        |
│ │ │ │ +00078f40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00078f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00078f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00078f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00078f80: 0a7c 6933 203a 206d 6170 2050 6869 2020  .|i3 : map Phi  
│ │ │ │ -00078f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00078f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00078f90: 2050 6869 2020 2020 2020 2020 2020 2020   Phi            
│ │ │ │  00078fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00078fc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00078fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00079000: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00079010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00079000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00079010: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00079020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00079030: 2020 3220 2020 2032 2020 2020 3220 2020    2    2    2   
│ │ │ │ -00079040: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00079080: 2074 202c 2074 2074 202c 2074 2074 202c   t , t t , t t ,
│ │ │ │ -00079090: 2074 2074 207d 297c 0a7c 2020 2020 2020   t t })|.|      
│ │ │ │ -000790a0: 2020 2020 2020 2020 3020 2020 3320 2020          0   3   
│ │ │ │ -000790b0: 2020 2020 3020 2020 3320 2020 2020 3120      0   3     1 
│ │ │ │ -000790c0: 2020 2032 2020 2020 3320 2020 3020 3120     2    3   0 1 
│ │ │ │ -000790d0: 2020 3020 3220 2020 3020 3320 207c 0a7c    0 2   0 3  |.|
│ │ │ │ -000790e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00079030: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +00079040: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00079050: 2020 2020 2020 207c 0a7c 6f33 203d 206d         |.|o3 = m
│ │ │ │ +00079060: 6170 2028 5151 5b74 202e 2e74 205d 2c20  ap (QQ[t ..t ], 
│ │ │ │ +00079070: 5151 5b74 202e 2e74 205d 2c20 7b74 2020  QQ[t ..t ], {t  
│ │ │ │ +00079080: 2b20 7420 202b 2074 202c 2074 2074 202c  + t  + t , t t ,
│ │ │ │ +00079090: 2074 2074 202c 2074 2074 207d 297c 0a7c   t t , t t })|.|
│ │ │ │ +000790a0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +000790b0: 2020 3320 2020 2020 2020 3020 2020 3320    3       0   3 
│ │ │ │ +000790c0: 2020 2020 3120 2020 2032 2020 2020 3320      1    2    3 
│ │ │ │ +000790d0: 2020 3020 3120 2020 3020 3220 2020 3020    0 1   0 2   0 
│ │ │ │ +000790e0: 3320 207c 0a7c 2020 2020 2020 2020 2020  3  |.|          
│ │ │ │  000790f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00079100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00079110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00079120: 2020 207c 0a7c 6f33 203a 2052 696e 674d     |.|o3 : RingM
│ │ │ │ -00079130: 6170 2051 515b 7420 2e2e 7420 5d20 3c2d  ap QQ[t ..t ] <-
│ │ │ │ -00079140: 2d20 5151 5b74 202e 2e74 205d 2020 2020  - QQ[t ..t ]    
│ │ │ │ -00079150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00079160: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00079170: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00079180: 2033 2020 2020 2020 2020 2020 3020 2020   3          0   
│ │ │ │ -00079190: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -000791a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000791b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00079120: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +00079130: 2052 696e 674d 6170 2051 515b 7420 2e2e   RingMap QQ[t ..
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│ │ │ │ +00079150: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ +00079160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00079170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00079180: 2020 2030 2020 2033 2020 2020 2020 2020     0   3        
│ │ │ │ +00079190: 2020 3020 2020 3320 2020 2020 2020 2020    0   3         
│ │ │ │ +000791a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000791b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000791c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000791d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000791e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00079240: 6d61 6e64 0a6d 6170 2869 2c50 6869 2920  mand.map(i,Phi) 
│ │ │ │ -00079250: 7265 7475 726e 7320 7468 6520 692d 7468  returns the i-th
│ │ │ │ -00079260: 2072 6570 7265 7365 6e74 6174 6976 6520   representative 
│ │ │ │ -00079270: 6f66 2074 6865 206d 6170 2050 6869 2e0a  of the map Phi..
│ │ │ │ -00079280: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
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│ │ │ │ -000792a0: 7472 6978 2852 6174 696f 6e61 6c4d 6170  trix(RationalMap
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│ │ │ │ -000792e0: 736f 6369 6174 6564 2074 6f20 6120 7261  sociated to a ra
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│ │ │ │ -00079300: 2074 6f20 7573 6520 7468 6973 206d 6574   to use this met
│ │ │ │ -00079310: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  hod:.===========
│ │ │ │ -00079320: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -00079330: 202a 202a 6e6f 7465 206d 6170 2852 6174   * *note map(Rat
│ │ │ │ -00079340: 696f 6e61 6c4d 6170 293a 206d 6170 5f6c  ionalMap): map_l
│ │ │ │ -00079350: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -00079360: 202d 2d20 6765 7420 7468 6520 7269 6e67   -- get the ring
│ │ │ │ -00079370: 206d 6170 2064 6566 696e 696e 670a 2020   map defining.  
│ │ │ │ -00079380: 2020 6120 7261 7469 6f6e 616c 206d 6170    a rational map
│ │ │ │ -00079390: 0a20 202a 2022 6d61 7028 5a5a 2c52 6174  .  * "map(ZZ,Rat
│ │ │ │ -000793a0: 696f 6e61 6c4d 6170 2922 0a1f 0a46 696c  ionalMap)"...Fil
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│ │ │ │ -000793c0: 204e 6f64 653a 206d 6174 7269 785f 6c70   Node: matrix_lp
│ │ │ │ -000793d0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -000793e0: 4e65 7874 3a20 4e75 6d44 6567 7265 6573  Next: NumDegrees
│ │ │ │ -000793f0: 2c20 5072 6576 3a20 6d61 705f 6c70 5261  , Prev: map_lpRa
│ │ │ │ -00079400: 7469 6f6e 616c 4d61 705f 7270 2c20 5570  tionalMap_rp, Up
│ │ │ │ -00079410: 3a20 546f 700a 0a6d 6174 7269 7828 5261  : Top..matrix(Ra
│ │ │ │ -00079420: 7469 6f6e 616c 4d61 7029 202d 2d20 7468  tionalMap) -- th
│ │ │ │ -00079430: 6520 6d61 7472 6978 2061 7373 6f63 6961  e matrix associa
│ │ │ │ -00079440: 7465 6420 746f 2061 2072 6174 696f 6e61  ted to a rationa
│ │ │ │ -00079450: 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a  l map.**********
│ │ │ │ +000791f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ +00079200: 6520 636f 6d6d 616e 6420 6d61 7020 5068  e command map Ph
│ │ │ │ +00079210: 6920 6973 2065 7175 6976 616c 656e 7420  i is equivalent 
│ │ │ │ +00079220: 746f 206d 6170 2830 2c50 6869 292e 204d  to map(0,Phi). M
│ │ │ │ +00079230: 6f72 6520 6765 6e65 7261 6c6c 792c 2074  ore generally, t
│ │ │ │ +00079240: 6865 2063 6f6d 6d61 6e64 0a6d 6170 2869  he command.map(i
│ │ │ │ +00079250: 2c50 6869 2920 7265 7475 726e 7320 7468  ,Phi) returns th
│ │ │ │ +00079260: 6520 692d 7468 2072 6570 7265 7365 6e74  e i-th represent
│ │ │ │ +00079270: 6174 6976 6520 6f66 2074 6865 206d 6170  ative of the map
│ │ │ │ +00079280: 2050 6869 2e0a 0a53 6565 2061 6c73 6f0a   Phi...See also.
│ │ │ │ +00079290: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +000792a0: 6f74 6520 6d61 7472 6978 2852 6174 696f  ote matrix(Ratio
│ │ │ │ +000792b0: 6e61 6c4d 6170 293a 206d 6174 7269 785f  nalMap): matrix_
│ │ │ │ +000792c0: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ +000792d0: 2c20 2d2d 2074 6865 206d 6174 7269 780a  , -- the matrix.
│ │ │ │ +000792e0: 2020 2020 6173 736f 6369 6174 6564 2074      associated t
│ │ │ │ +000792f0: 6f20 6120 7261 7469 6f6e 616c 206d 6170  o a rational map
│ │ │ │ +00079300: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
│ │ │ │ +00079310: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d  is method:.=====
│ │ │ │ +00079320: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00079330: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ +00079340: 6170 2852 6174 696f 6e61 6c4d 6170 293a  ap(RationalMap):
│ │ │ │ +00079350: 206d 6170 5f6c 7052 6174 696f 6e61 6c4d   map_lpRationalM
│ │ │ │ +00079360: 6170 5f72 702c 202d 2d20 6765 7420 7468  ap_rp, -- get th
│ │ │ │ +00079370: 6520 7269 6e67 206d 6170 2064 6566 696e  e ring map defin
│ │ │ │ +00079380: 696e 670a 2020 2020 6120 7261 7469 6f6e  ing.    a ration
│ │ │ │ +00079390: 616c 206d 6170 0a20 202a 2022 6d61 7028  al map.  * "map(
│ │ │ │ +000793a0: 5a5a 2c52 6174 696f 6e61 6c4d 6170 2922  ZZ,RationalMap)"
│ │ │ │ +000793b0: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ +000793c0: 2e69 6e66 6f2c 204e 6f64 653a 206d 6174  .info, Node: mat
│ │ │ │ +000793d0: 7269 785f 6c70 5261 7469 6f6e 616c 4d61  rix_lpRationalMa
│ │ │ │ +000793e0: 705f 7270 2c20 4e65 7874 3a20 4e75 6d44  p_rp, Next: NumD
│ │ │ │ +000793f0: 6567 7265 6573 2c20 5072 6576 3a20 6d61  egrees, Prev: ma
│ │ │ │ +00079400: 705f 6c70 5261 7469 6f6e 616c 4d61 705f  p_lpRationalMap_
│ │ │ │ +00079410: 7270 2c20 5570 3a20 546f 700a 0a6d 6174  rp, Up: Top..mat
│ │ │ │ +00079420: 7269 7828 5261 7469 6f6e 616c 4d61 7029  rix(RationalMap)
│ │ │ │ +00079430: 202d 2d20 7468 6520 6d61 7472 6978 2061   -- the matrix a
│ │ │ │ +00079440: 7373 6f63 6961 7465 6420 746f 2061 2072  ssociated to a r
│ │ │ │ +00079450: 6174 696f 6e61 6c20 6d61 700a 2a2a 2a2a  ational map.****
│ │ │ │  00079460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00079470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00079480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00079490: 2a2a 2a2a 0a0a 5379 6e6f 7073 6973 0a3d  ****..Synopsis.=
│ │ │ │ -000794a0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2046 756e  =======..  * Fun
│ │ │ │ -000794b0: 6374 696f 6e3a 202a 6e6f 7465 206d 6174  ction: *note mat
│ │ │ │ -000794c0: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ -000794d0: 6f63 296d 6174 7269 782c 0a20 202a 2055  oc)matrix,.  * U
│ │ │ │ -000794e0: 7361 6765 3a20 0a20 2020 2020 2020 206d  sage: .        m
│ │ │ │ -000794f0: 6174 7269 7820 5068 690a 2020 2a20 496e  atrix Phi.  * In
│ │ │ │ -00079500: 7075 7473 3a0a 2020 2020 2020 2a20 5068  puts:.      * Ph
│ │ │ │ -00079510: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ -00079520: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ -00079530: 6c4d 6170 2c0a 2020 2a20 2a6e 6f74 6520  lMap,.  * *note 
│ │ │ │ -00079540: 4f70 7469 6f6e 616c 2069 6e70 7574 733a  Optional inputs:
│ │ │ │ -00079550: 2028 4d61 6361 756c 6179 3244 6f63 2975   (Macaulay2Doc)u
│ │ │ │ -00079560: 7369 6e67 2066 756e 6374 696f 6e73 2077  sing functions w
│ │ │ │ -00079570: 6974 6820 6f70 7469 6f6e 616c 2069 6e70  ith optional inp
│ │ │ │ -00079580: 7574 732c 3a0a 2020 2020 2020 2a20 2a6e  uts,:.      * *n
│ │ │ │ -00079590: 6f74 6520 4465 6772 6565 3a20 284d 6163  ote Degree: (Mac
│ │ │ │ -000795a0: 6175 6c61 7932 446f 6329 6d61 7472 6978  aulay2Doc)matrix
│ │ │ │ -000795b0: 5f6c 704c 6973 745f 7270 2c20 3d3e 202e  _lpList_rp, => .
│ │ │ │ -000795c0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -000795d0: 650a 2020 2020 2020 2020 6e75 6c6c 2c0a  e.        null,.
│ │ │ │ -000795e0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -000795f0: 2020 202a 2061 202a 6e6f 7465 206d 6174     * a *note mat
│ │ │ │ -00079600: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ -00079610: 6f63 294d 6174 7269 782c 2c20 7468 6520  oc)Matrix,, the 
│ │ │ │ -00079620: 6d61 7472 6978 2061 7373 6f63 6961 7465  matrix associate
│ │ │ │ -00079630: 6420 746f 2074 6865 0a20 2020 2020 2020  d to the.       
│ │ │ │ -00079640: 2072 696e 6720 6d61 7020 6465 6669 6e69   ring map defini
│ │ │ │ -00079650: 6e67 2074 6865 2072 6174 696f 6e61 6c20  ng the rational 
│ │ │ │ -00079660: 6d61 7020 5068 690a 0a44 6573 6372 6970  map Phi..Descrip
│ │ │ │ -00079670: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00079680: 0a0a 5468 6973 2069 7320 6571 7569 7661  ..This is equiva
│ │ │ │ -00079690: 6c65 6e74 2074 6f20 6d61 7472 6978 206d  lent to matrix m
│ │ │ │ -000796a0: 6170 2050 6869 2e20 4d6f 7265 6f76 6572  ap Phi. Moreover
│ │ │ │ -000796b0: 2c20 7468 6520 636f 6d6d 616e 6420 6d61  , the command ma
│ │ │ │ -000796c0: 7472 6978 2050 6869 2069 730a 6571 7569  trix Phi is.equi
│ │ │ │ -000796d0: 7661 6c65 6e74 2074 6f20 6d61 7472 6978  valent to matrix
│ │ │ │ -000796e0: 2830 2c50 6869 292c 2061 6e64 206d 6f72  (0,Phi), and mor
│ │ │ │ -000796f0: 6520 6765 6e65 7261 6c6c 7920 7468 6520  e generally the 
│ │ │ │ -00079700: 636f 6d6d 616e 6420 6d61 7472 6978 2869  command matrix(i
│ │ │ │ -00079710: 2c50 6869 290a 7265 7475 726e 7320 7468  ,Phi).returns th
│ │ │ │ -00079720: 6520 6d61 7472 6978 206f 6620 7468 6520  e matrix of the 
│ │ │ │ -00079730: 692d 7468 2072 6570 7265 7365 6e74 6174  i-th representat
│ │ │ │ -00079740: 6976 6520 6f66 2050 6869 2e0a 0a53 6565  ive of Phi...See
│ │ │ │ -00079750: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -00079760: 2020 2a20 2a6e 6f74 6520 6d61 7028 5261    * *note map(Ra
│ │ │ │ -00079770: 7469 6f6e 616c 4d61 7029 3a20 6d61 705f  tionalMap): map_
│ │ │ │ -00079780: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00079790: 2c20 2d2d 2067 6574 2074 6865 2072 696e  , -- get the rin
│ │ │ │ -000797a0: 6720 6d61 7020 6465 6669 6e69 6e67 0a20  g map defining. 
│ │ │ │ -000797b0: 2020 2061 2072 6174 696f 6e61 6c20 6d61     a rational ma
│ │ │ │ -000797c0: 700a 2020 2a20 2a6e 6f74 6520 6d61 7472  p.  * *note matr
│ │ │ │ -000797d0: 6978 2852 696e 674d 6170 293a 2028 4d61  ix(RingMap): (Ma
│ │ │ │ -000797e0: 6361 756c 6179 3244 6f63 296d 6174 7269  caulay2Doc)matri
│ │ │ │ -000797f0: 785f 6c70 5269 6e67 4d61 705f 7270 2c20  x_lpRingMap_rp, 
│ │ │ │ -00079800: 2d2d 2074 6865 206d 6174 7269 780a 2020  -- the matrix.  
│ │ │ │ -00079810: 2020 6173 736f 6369 6174 6564 2074 6f20    associated to 
│ │ │ │ -00079820: 6120 7269 6e67 206d 6170 0a0a 5761 7973  a ring map..Ways
│ │ │ │ -00079830: 2074 6f20 7573 6520 7468 6973 206d 6574   to use this met
│ │ │ │ -00079840: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  hod:.===========
│ │ │ │ -00079850: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -00079860: 202a 202a 6e6f 7465 206d 6174 7269 7828   * *note matrix(
│ │ │ │ -00079870: 5261 7469 6f6e 616c 4d61 7029 3a20 6d61  RationalMap): ma
│ │ │ │ -00079880: 7472 6978 5f6c 7052 6174 696f 6e61 6c4d  trix_lpRationalM
│ │ │ │ -00079890: 6170 5f72 702c 202d 2d20 7468 6520 6d61  ap_rp, -- the ma
│ │ │ │ -000798a0: 7472 6978 0a20 2020 2061 7373 6f63 6961  trix.    associa
│ │ │ │ -000798b0: 7465 6420 746f 2061 2072 6174 696f 6e61  ted to a rationa
│ │ │ │ -000798c0: 6c20 6d61 700a 2020 2a20 226d 6174 7269  l map.  * "matri
│ │ │ │ -000798d0: 7828 5a5a 2c52 6174 696f 6e61 6c4d 6170  x(ZZ,RationalMap
│ │ │ │ -000798e0: 2922 0a1f 0a46 696c 653a 2043 7265 6d6f  )"...File: Cremo
│ │ │ │ -000798f0: 6e61 2e69 6e66 6f2c 204e 6f64 653a 204e  na.info, Node: N
│ │ │ │ -00079900: 756d 4465 6772 6565 732c 204e 6578 743a  umDegrees, Next:
│ │ │ │ -00079910: 2070 6172 616d 6574 7269 7a65 2c20 5072   parametrize, Pr
│ │ │ │ -00079920: 6576 3a20 6d61 7472 6978 5f6c 7052 6174  ev: matrix_lpRat
│ │ │ │ -00079930: 696f 6e61 6c4d 6170 5f72 702c 2055 703a  ionalMap_rp, Up:
│ │ │ │ -00079940: 2054 6f70 0a0a 4e75 6d44 6567 7265 6573   Top..NumDegrees
│ │ │ │ -00079950: 0a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573  .**********..Des
│ │ │ │ -00079960: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00079970: 3d3d 3d3d 0a0a 5468 6973 2069 7320 616e  ====..This is an
│ │ │ │ -00079980: 206f 7074 696f 6e61 6c20 6172 6775 6d65   optional argume
│ │ │ │ -00079990: 6e74 2066 6f72 202a 6e6f 7465 2070 726f  nt for *note pro
│ │ │ │ -000799a0: 6a65 6374 6976 6544 6567 7265 6573 3a20  jectiveDegrees: 
│ │ │ │ -000799b0: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -000799c0: 732c 0a61 6e64 2061 6363 6570 7473 2061  s,.and accepts a
│ │ │ │ -000799d0: 206e 6f6e 2d6e 6567 6174 6976 6520 696e   non-negative in
│ │ │ │ -000799e0: 7465 6765 722c 2031 206c 6573 7320 7468  teger, 1 less th
│ │ │ │ -000799f0: 616e 2074 6865 206e 756d 6265 7220 6f66  an the number of
│ │ │ │ -00079a00: 2070 726f 6a65 6374 6976 650a 6465 6772   projective.degr
│ │ │ │ -00079a10: 6565 7320 746f 2062 6520 636f 6d70 7574  ees to be comput
│ │ │ │ -00079a20: 6564 2e0a 0a46 756e 6374 696f 6e73 2077  ed...Functions w
│ │ │ │ -00079a30: 6974 6820 6f70 7469 6f6e 616c 2061 7267  ith optional arg
│ │ │ │ -00079a40: 756d 656e 7420 6e61 6d65 6420 4e75 6d44  ument named NumD
│ │ │ │ -00079a50: 6567 7265 6573 3a0a 3d3d 3d3d 3d3d 3d3d  egrees:.========
│ │ │ │ +00079490: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ +000794a0: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ +000794b0: 202a 2046 756e 6374 696f 6e3a 202a 6e6f   * Function: *no
│ │ │ │ +000794c0: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ +000794d0: 756c 6179 3244 6f63 296d 6174 7269 782c  ulay2Doc)matrix,
│ │ │ │ +000794e0: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +000794f0: 2020 2020 206d 6174 7269 7820 5068 690a       matrix Phi.
│ │ │ │ +00079500: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00079510: 2020 2a20 5068 692c 2061 202a 6e6f 7465    * Phi, a *note
│ │ │ │ +00079520: 2072 6174 696f 6e61 6c20 6d61 703a 2052   rational map: R
│ │ │ │ +00079530: 6174 696f 6e61 6c4d 6170 2c0a 2020 2a20  ationalMap,.  * 
│ │ │ │ +00079540: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
│ │ │ │ +00079550: 6e70 7574 733a 2028 4d61 6361 756c 6179  nputs: (Macaulay
│ │ │ │ +00079560: 3244 6f63 2975 7369 6e67 2066 756e 6374  2Doc)using funct
│ │ │ │ +00079570: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +00079580: 616c 2069 6e70 7574 732c 3a0a 2020 2020  al inputs,:.    
│ │ │ │ +00079590: 2020 2a20 2a6e 6f74 6520 4465 6772 6565    * *note Degree
│ │ │ │ +000795a0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000795b0: 6d61 7472 6978 5f6c 704c 6973 745f 7270  matrix_lpList_rp
│ │ │ │ +000795c0: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +000795d0: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ +000795e0: 6e75 6c6c 2c0a 2020 2a20 4f75 7470 7574  null,.  * Output
│ │ │ │ +000795f0: 733a 0a20 2020 2020 202a 2061 202a 6e6f  s:.      * a *no
│ │ │ │ +00079600: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ +00079610: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ +00079620: 2c20 7468 6520 6d61 7472 6978 2061 7373  , the matrix ass
│ │ │ │ +00079630: 6f63 6961 7465 6420 746f 2074 6865 0a20  ociated to the. 
│ │ │ │ +00079640: 2020 2020 2020 2072 696e 6720 6d61 7020         ring map 
│ │ │ │ +00079650: 6465 6669 6e69 6e67 2074 6865 2072 6174  defining the rat
│ │ │ │ +00079660: 696f 6e61 6c20 6d61 7020 5068 690a 0a44  ional map Phi..D
│ │ │ │ +00079670: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00079680: 3d3d 3d3d 3d3d 0a0a 5468 6973 2069 7320  ======..This is 
│ │ │ │ +00079690: 6571 7569 7661 6c65 6e74 2074 6f20 6d61  equivalent to ma
│ │ │ │ +000796a0: 7472 6978 206d 6170 2050 6869 2e20 4d6f  trix map Phi. Mo
│ │ │ │ +000796b0: 7265 6f76 6572 2c20 7468 6520 636f 6d6d  reover, the comm
│ │ │ │ +000796c0: 616e 6420 6d61 7472 6978 2050 6869 2069  and matrix Phi i
│ │ │ │ +000796d0: 730a 6571 7569 7661 6c65 6e74 2074 6f20  s.equivalent to 
│ │ │ │ +000796e0: 6d61 7472 6978 2830 2c50 6869 292c 2061  matrix(0,Phi), a
│ │ │ │ +000796f0: 6e64 206d 6f72 6520 6765 6e65 7261 6c6c  nd more generall
│ │ │ │ +00079700: 7920 7468 6520 636f 6d6d 616e 6420 6d61  y the command ma
│ │ │ │ +00079710: 7472 6978 2869 2c50 6869 290a 7265 7475  trix(i,Phi).retu
│ │ │ │ +00079720: 726e 7320 7468 6520 6d61 7472 6978 206f  rns the matrix o
│ │ │ │ +00079730: 6620 7468 6520 692d 7468 2072 6570 7265  f the i-th repre
│ │ │ │ +00079740: 7365 6e74 6174 6976 6520 6f66 2050 6869  sentative of Phi
│ │ │ │ +00079750: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  ...See also.====
│ │ │ │ +00079760: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +00079770: 6d61 7028 5261 7469 6f6e 616c 4d61 7029  map(RationalMap)
│ │ │ │ +00079780: 3a20 6d61 705f 6c70 5261 7469 6f6e 616c  : map_lpRational
│ │ │ │ +00079790: 4d61 705f 7270 2c20 2d2d 2067 6574 2074  Map_rp, -- get t
│ │ │ │ +000797a0: 6865 2072 696e 6720 6d61 7020 6465 6669  he ring map defi
│ │ │ │ +000797b0: 6e69 6e67 0a20 2020 2061 2072 6174 696f  ning.    a ratio
│ │ │ │ +000797c0: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74  nal map.  * *not
│ │ │ │ +000797d0: 6520 6d61 7472 6978 2852 696e 674d 6170  e matrix(RingMap
│ │ │ │ +000797e0: 293a 2028 4d61 6361 756c 6179 3244 6f63  ): (Macaulay2Doc
│ │ │ │ +000797f0: 296d 6174 7269 785f 6c70 5269 6e67 4d61  )matrix_lpRingMa
│ │ │ │ +00079800: 705f 7270 2c20 2d2d 2074 6865 206d 6174  p_rp, -- the mat
│ │ │ │ +00079810: 7269 780a 2020 2020 6173 736f 6369 6174  rix.    associat
│ │ │ │ +00079820: 6564 2074 6f20 6120 7269 6e67 206d 6170  ed to a ring map
│ │ │ │ +00079830: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
│ │ │ │ +00079840: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d  is method:.=====
│ │ │ │ +00079850: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00079860: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ +00079870: 6174 7269 7828 5261 7469 6f6e 616c 4d61  atrix(RationalMa
│ │ │ │ +00079880: 7029 3a20 6d61 7472 6978 5f6c 7052 6174  p): matrix_lpRat
│ │ │ │ +00079890: 696f 6e61 6c4d 6170 5f72 702c 202d 2d20  ionalMap_rp, -- 
│ │ │ │ +000798a0: 7468 6520 6d61 7472 6978 0a20 2020 2061  the matrix.    a
│ │ │ │ +000798b0: 7373 6f63 6961 7465 6420 746f 2061 2072  ssociated to a r
│ │ │ │ +000798c0: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ +000798d0: 226d 6174 7269 7828 5a5a 2c52 6174 696f  "matrix(ZZ,Ratio
│ │ │ │ +000798e0: 6e61 6c4d 6170 2922 0a1f 0a46 696c 653a  nalMap)"...File:
│ │ │ │ +000798f0: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ +00079900: 6f64 653a 204e 756d 4465 6772 6565 732c  ode: NumDegrees,
│ │ │ │ +00079910: 204e 6578 743a 2070 6172 616d 6574 7269   Next: parametri
│ │ │ │ +00079920: 7a65 2c20 5072 6576 3a20 6d61 7472 6978  ze, Prev: matrix
│ │ │ │ +00079930: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +00079940: 702c 2055 703a 2054 6f70 0a0a 4e75 6d44  p, Up: Top..NumD
│ │ │ │ +00079950: 6567 7265 6573 0a2a 2a2a 2a2a 2a2a 2a2a  egrees.*********
│ │ │ │ +00079960: 2a0a 0a44 6573 6372 6970 7469 6f6e 0a3d  *..Description.=
│ │ │ │ +00079970: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ +00079980: 2069 7320 616e 206f 7074 696f 6e61 6c20   is an optional 
│ │ │ │ +00079990: 6172 6775 6d65 6e74 2066 6f72 202a 6e6f  argument for *no
│ │ │ │ +000799a0: 7465 2070 726f 6a65 6374 6976 6544 6567  te projectiveDeg
│ │ │ │ +000799b0: 7265 6573 3a20 7072 6f6a 6563 7469 7665  rees: projective
│ │ │ │ +000799c0: 4465 6772 6565 732c 0a61 6e64 2061 6363  Degrees,.and acc
│ │ │ │ +000799d0: 6570 7473 2061 206e 6f6e 2d6e 6567 6174  epts a non-negat
│ │ │ │ +000799e0: 6976 6520 696e 7465 6765 722c 2031 206c  ive integer, 1 l
│ │ │ │ +000799f0: 6573 7320 7468 616e 2074 6865 206e 756d  ess than the num
│ │ │ │ +00079a00: 6265 7220 6f66 2070 726f 6a65 6374 6976  ber of projectiv
│ │ │ │ +00079a10: 650a 6465 6772 6565 7320 746f 2062 6520  e.degrees to be 
│ │ │ │ +00079a20: 636f 6d70 7574 6564 2e0a 0a46 756e 6374  computed...Funct
│ │ │ │ +00079a30: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +00079a40: 616c 2061 7267 756d 656e 7420 6e61 6d65  al argument name
│ │ │ │ +00079a50: 6420 4e75 6d44 6567 7265 6573 3a0a 3d3d  d NumDegrees:.==
│ │ │ │  00079a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00079a70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00079a80: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00079a90: 2270 726f 6a65 6374 6976 6544 6567 7265  "projectiveDegre
│ │ │ │ -00079aa0: 6573 282e 2e2e 2c4e 756d 4465 6772 6565  es(...,NumDegree
│ │ │ │ -00079ab0: 733d 3e2e 2e2e 2922 0a0a 466f 7220 7468  s=>...)"..For th
│ │ │ │ -00079ac0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00079ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00079ae0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00079af0: 6520 4e75 6d44 6567 7265 6573 3a20 4e75  e NumDegrees: Nu
│ │ │ │ -00079b00: 6d44 6567 7265 6573 2c20 6973 2061 202a  mDegrees, is a *
│ │ │ │ -00079b10: 6e6f 7465 2073 796d 626f 6c3a 0a28 4d61  note symbol:.(Ma
│ │ │ │ -00079b20: 6361 756c 6179 3244 6f63 2953 796d 626f  caulay2Doc)Symbo
│ │ │ │ -00079b30: 6c2c 2e0a 1f0a 4669 6c65 3a20 4372 656d  l,....File: Crem
│ │ │ │ -00079b40: 6f6e 612e 696e 666f 2c20 4e6f 6465 3a20  ona.info, Node: 
│ │ │ │ -00079b50: 7061 7261 6d65 7472 697a 652c 204e 6578  parametrize, Nex
│ │ │ │ -00079b60: 743a 2070 6172 616d 6574 7269 7a65 5f6c  t: parametrize_l
│ │ │ │ -00079b70: 7049 6465 616c 5f72 702c 2050 7265 763a  pIdeal_rp, Prev:
│ │ │ │ -00079b80: 204e 756d 4465 6772 6565 732c 2055 703a   NumDegrees, Up:
│ │ │ │ -00079b90: 2054 6f70 0a0a 7061 7261 6d65 7472 697a   Top..parametriz
│ │ │ │ -00079ba0: 6520 2d2d 2070 6172 616d 6574 7269 7a61  e -- parametriza
│ │ │ │ -00079bb0: 7469 6f6e 206f 6620 6120 7261 7469 6f6e  tion of a ration
│ │ │ │ -00079bc0: 616c 2070 726f 6a65 6374 6976 6520 7661  al projective va
│ │ │ │ -00079bd0: 7269 6574 790a 2a2a 2a2a 2a2a 2a2a 2a2a  riety.**********
│ │ │ │ +00079a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00079a90: 0a0a 2020 2a20 2270 726f 6a65 6374 6976  ..  * "projectiv
│ │ │ │ +00079aa0: 6544 6567 7265 6573 282e 2e2e 2c4e 756d  eDegrees(...,Num
│ │ │ │ +00079ab0: 4465 6772 6565 733d 3e2e 2e2e 2922 0a0a  Degrees=>...)"..
│ │ │ │ +00079ac0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +00079ad0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +00079ae0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +00079af0: 7420 2a6e 6f74 6520 4e75 6d44 6567 7265  t *note NumDegre
│ │ │ │ +00079b00: 6573 3a20 4e75 6d44 6567 7265 6573 2c20  es: NumDegrees, 
│ │ │ │ +00079b10: 6973 2061 202a 6e6f 7465 2073 796d 626f  is a *note symbo
│ │ │ │ +00079b20: 6c3a 0a28 4d61 6361 756c 6179 3244 6f63  l:.(Macaulay2Doc
│ │ │ │ +00079b30: 2953 796d 626f 6c2c 2e0a 1f0a 4669 6c65  )Symbol,....File
│ │ │ │ +00079b40: 3a20 4372 656d 6f6e 612e 696e 666f 2c20  : Cremona.info, 
│ │ │ │ +00079b50: 4e6f 6465 3a20 7061 7261 6d65 7472 697a  Node: parametriz
│ │ │ │ +00079b60: 652c 204e 6578 743a 2070 6172 616d 6574  e, Next: paramet
│ │ │ │ +00079b70: 7269 7a65 5f6c 7049 6465 616c 5f72 702c  rize_lpIdeal_rp,
│ │ │ │ +00079b80: 2050 7265 763a 204e 756d 4465 6772 6565   Prev: NumDegree
│ │ │ │ +00079b90: 732c 2055 703a 2054 6f70 0a0a 7061 7261  s, Up: Top..para
│ │ │ │ +00079ba0: 6d65 7472 697a 6520 2d2d 2070 6172 616d  metrize -- param
│ │ │ │ +00079bb0: 6574 7269 7a61 7469 6f6e 206f 6620 6120  etrization of a 
│ │ │ │ +00079bc0: 7261 7469 6f6e 616c 2070 726f 6a65 6374  rational project
│ │ │ │ +00079bd0: 6976 6520 7661 7269 6574 790a 2a2a 2a2a  ive variety.****
│ │ │ │  00079be0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00079bf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00079c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00079c10: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469  *****..Descripti
│ │ │ │ -00079c20: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00079c30: 5365 6520 2a6e 6f74 6520 7061 7261 6d65  See *note parame
│ │ │ │ -00079c40: 7472 697a 6528 4d75 6c74 6970 726f 6a65  trize(Multiproje
│ │ │ │ -00079c50: 6374 6976 6556 6172 6965 7479 293a 0a28  ctiveVariety):.(
│ │ │ │ -00079c60: 4d75 6c74 6970 726f 6a65 6374 6976 6556  MultiprojectiveV
│ │ │ │ -00079c70: 6172 6965 7469 6573 2970 6172 616d 6574  arieties)paramet
│ │ │ │ -00079c80: 7269 7a65 5f6c 704d 756c 7469 7072 6f6a  rize_lpMultiproj
│ │ │ │ -00079c90: 6563 7469 7665 5661 7269 6574 795f 7270  ectiveVariety_rp
│ │ │ │ -00079ca0: 2c20 616e 6420 2a6e 6f74 650a 7061 7261  , and *note.para
│ │ │ │ -00079cb0: 6d65 7472 697a 6528 5175 6f74 6965 6e74  metrize(Quotient
│ │ │ │ -00079cc0: 5269 6e67 293a 2070 6172 616d 6574 7269  Ring): parametri
│ │ │ │ -00079cd0: 7a65 5f6c 7049 6465 616c 5f72 702c 2e0a  ze_lpIdeal_rp,..
│ │ │ │ -00079ce0: 0a57 6179 7320 746f 2075 7365 2070 6172  .Ways to use par
│ │ │ │ -00079cf0: 616d 6574 7269 7a65 3a0a 3d3d 3d3d 3d3d  ametrize:.======
│ │ │ │ +00079c10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573  ***********..Des
│ │ │ │ +00079c20: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00079c30: 3d3d 3d3d 0a0a 5365 6520 2a6e 6f74 6520  ====..See *note 
│ │ │ │ +00079c40: 7061 7261 6d65 7472 697a 6528 4d75 6c74  parametrize(Mult
│ │ │ │ +00079c50: 6970 726f 6a65 6374 6976 6556 6172 6965  iprojectiveVarie
│ │ │ │ +00079c60: 7479 293a 0a28 4d75 6c74 6970 726f 6a65  ty):.(Multiproje
│ │ │ │ +00079c70: 6374 6976 6556 6172 6965 7469 6573 2970  ctiveVarieties)p
│ │ │ │ +00079c80: 6172 616d 6574 7269 7a65 5f6c 704d 756c  arametrize_lpMul
│ │ │ │ +00079c90: 7469 7072 6f6a 6563 7469 7665 5661 7269  tiprojectiveVari
│ │ │ │ +00079ca0: 6574 795f 7270 2c20 616e 6420 2a6e 6f74  ety_rp, and *not
│ │ │ │ +00079cb0: 650a 7061 7261 6d65 7472 697a 6528 5175  e.parametrize(Qu
│ │ │ │ +00079cc0: 6f74 6965 6e74 5269 6e67 293a 2070 6172  otientRing): par
│ │ │ │ +00079cd0: 616d 6574 7269 7a65 5f6c 7049 6465 616c  ametrize_lpIdeal
│ │ │ │ +00079ce0: 5f72 702c 2e0a 0a57 6179 7320 746f 2075  _rp,...Ways to u
│ │ │ │ +00079cf0: 7365 2070 6172 616d 6574 7269 7a65 3a0a  se parametrize:.
│ │ │ │  00079d00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00079d10: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7061  ==..  * *note pa
│ │ │ │ -00079d20: 7261 6d65 7472 697a 6528 4964 6561 6c29  rametrize(Ideal)
│ │ │ │ -00079d30: 3a20 7061 7261 6d65 7472 697a 655f 6c70  : parametrize_lp
│ │ │ │ -00079d40: 4964 6561 6c5f 7270 2c20 2d2d 2070 6172  Ideal_rp, -- par
│ │ │ │ -00079d50: 616d 6574 7269 7a61 7469 6f6e 206f 660a  ametrization of.
│ │ │ │ -00079d60: 2020 2020 6c69 6e65 6172 2076 6172 6965      linear varie
│ │ │ │ -00079d70: 7469 6573 2061 6e64 2068 7970 6572 7175  ties and hyperqu
│ │ │ │ -00079d80: 6164 7269 6373 0a20 202a 2022 7061 7261  adrics.  * "para
│ │ │ │ -00079d90: 6d65 7472 697a 6528 506f 6c79 6e6f 6d69  metrize(Polynomi
│ │ │ │ -00079da0: 616c 5269 6e67 2922 202d 2d20 7365 6520  alRing)" -- see 
│ │ │ │ -00079db0: 2a6e 6f74 6520 7061 7261 6d65 7472 697a  *note parametriz
│ │ │ │ -00079dc0: 6528 4964 6561 6c29 3a0a 2020 2020 7061  e(Ideal):.    pa
│ │ │ │ -00079dd0: 7261 6d65 7472 697a 655f 6c70 4964 6561  rametrize_lpIdea
│ │ │ │ -00079de0: 6c5f 7270 2c20 2d2d 2070 6172 616d 6574  l_rp, -- paramet
│ │ │ │ -00079df0: 7269 7a61 7469 6f6e 206f 6620 6c69 6e65  rization of line
│ │ │ │ -00079e00: 6172 2076 6172 6965 7469 6573 2061 6e64  ar varieties and
│ │ │ │ -00079e10: 0a20 2020 2068 7970 6572 7175 6164 7269  .    hyperquadri
│ │ │ │ -00079e20: 6373 0a20 202a 2022 7061 7261 6d65 7472  cs.  * "parametr
│ │ │ │ -00079e30: 697a 6528 5175 6f74 6965 6e74 5269 6e67  ize(QuotientRing
│ │ │ │ -00079e40: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -00079e50: 7061 7261 6d65 7472 697a 6528 4964 6561  parametrize(Idea
│ │ │ │ -00079e60: 6c29 3a0a 2020 2020 7061 7261 6d65 7472  l):.    parametr
│ │ │ │ -00079e70: 697a 655f 6c70 4964 6561 6c5f 7270 2c20  ize_lpIdeal_rp, 
│ │ │ │ -00079e80: 2d2d 2070 6172 616d 6574 7269 7a61 7469  -- parametrizati
│ │ │ │ -00079e90: 6f6e 206f 6620 6c69 6e65 6172 2076 6172  on of linear var
│ │ │ │ -00079ea0: 6965 7469 6573 2061 6e64 0a20 2020 2068  ieties and.    h
│ │ │ │ -00079eb0: 7970 6572 7175 6164 7269 6373 0a0a 466f  yperquadrics..Fo
│ │ │ │ -00079ec0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00079ed0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00079ee0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00079ef0: 2a6e 6f74 6520 7061 7261 6d65 7472 697a  *note parametriz
│ │ │ │ -00079f00: 653a 2070 6172 616d 6574 7269 7a65 2c20  e: parametrize, 
│ │ │ │ -00079f10: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -00079f20: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -00079f30: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -00079f40: 4675 6e63 7469 6f6e 2c2e 0a1f 0a46 696c  Function,....Fil
│ │ │ │ -00079f50: 653a 2043 7265 6d6f 6e61 2e69 6e66 6f2c  e: Cremona.info,
│ │ │ │ -00079f60: 204e 6f64 653a 2070 6172 616d 6574 7269   Node: parametri
│ │ │ │ -00079f70: 7a65 5f6c 7049 6465 616c 5f72 702c 204e  ze_lpIdeal_rp, N
│ │ │ │ -00079f80: 6578 743a 2070 6f69 6e74 2c20 5072 6576  ext: point, Prev
│ │ │ │ -00079f90: 3a20 7061 7261 6d65 7472 697a 652c 2055  : parametrize, U
│ │ │ │ -00079fa0: 703a 2054 6f70 0a0a 7061 7261 6d65 7472  p: Top..parametr
│ │ │ │ -00079fb0: 697a 6528 4964 6561 6c29 202d 2d20 7061  ize(Ideal) -- pa
│ │ │ │ -00079fc0: 7261 6d65 7472 697a 6174 696f 6e20 6f66  rametrization of
│ │ │ │ -00079fd0: 206c 696e 6561 7220 7661 7269 6574 6965   linear varietie
│ │ │ │ -00079fe0: 7320 616e 6420 6879 7065 7271 7561 6472  s and hyperquadr
│ │ │ │ -00079ff0: 6963 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ics.************
│ │ │ │ +00079d10: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00079d20: 6f74 6520 7061 7261 6d65 7472 697a 6528  ote parametrize(
│ │ │ │ +00079d30: 4964 6561 6c29 3a20 7061 7261 6d65 7472  Ideal): parametr
│ │ │ │ +00079d40: 697a 655f 6c70 4964 6561 6c5f 7270 2c20  ize_lpIdeal_rp, 
│ │ │ │ +00079d50: 2d2d 2070 6172 616d 6574 7269 7a61 7469  -- parametrizati
│ │ │ │ +00079d60: 6f6e 206f 660a 2020 2020 6c69 6e65 6172  on of.    linear
│ │ │ │ +00079d70: 2076 6172 6965 7469 6573 2061 6e64 2068   varieties and h
│ │ │ │ +00079d80: 7970 6572 7175 6164 7269 6373 0a20 202a  yperquadrics.  *
│ │ │ │ +00079d90: 2022 7061 7261 6d65 7472 697a 6528 506f   "parametrize(Po
│ │ │ │ +00079da0: 6c79 6e6f 6d69 616c 5269 6e67 2922 202d  lynomialRing)" -
│ │ │ │ +00079db0: 2d20 7365 6520 2a6e 6f74 6520 7061 7261  - see *note para
│ │ │ │ +00079dc0: 6d65 7472 697a 6528 4964 6561 6c29 3a0a  metrize(Ideal):.
│ │ │ │ +00079dd0: 2020 2020 7061 7261 6d65 7472 697a 655f      parametrize_
│ │ │ │ +00079de0: 6c70 4964 6561 6c5f 7270 2c20 2d2d 2070  lpIdeal_rp, -- p
│ │ │ │ +00079df0: 6172 616d 6574 7269 7a61 7469 6f6e 206f  arametrization o
│ │ │ │ +00079e00: 6620 6c69 6e65 6172 2076 6172 6965 7469  f linear varieti
│ │ │ │ +00079e10: 6573 2061 6e64 0a20 2020 2068 7970 6572  es and.    hyper
│ │ │ │ +00079e20: 7175 6164 7269 6373 0a20 202a 2022 7061  quadrics.  * "pa
│ │ │ │ +00079e30: 7261 6d65 7472 697a 6528 5175 6f74 6965  rametrize(Quotie
│ │ │ │ +00079e40: 6e74 5269 6e67 2922 202d 2d20 7365 6520  ntRing)" -- see 
│ │ │ │ +00079e50: 2a6e 6f74 6520 7061 7261 6d65 7472 697a  *note parametriz
│ │ │ │ +00079e60: 6528 4964 6561 6c29 3a0a 2020 2020 7061  e(Ideal):.    pa
│ │ │ │ +00079e70: 7261 6d65 7472 697a 655f 6c70 4964 6561  rametrize_lpIdea
│ │ │ │ +00079e80: 6c5f 7270 2c20 2d2d 2070 6172 616d 6574  l_rp, -- paramet
│ │ │ │ +00079e90: 7269 7a61 7469 6f6e 206f 6620 6c69 6e65  rization of line
│ │ │ │ +00079ea0: 6172 2076 6172 6965 7469 6573 2061 6e64  ar varieties and
│ │ │ │ +00079eb0: 0a20 2020 2068 7970 6572 7175 6164 7269  .    hyperquadri
│ │ │ │ +00079ec0: 6373 0a0a 466f 7220 7468 6520 7072 6f67  cs..For the prog
│ │ │ │ +00079ed0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00079ee0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00079ef0: 626a 6563 7420 2a6e 6f74 6520 7061 7261  bject *note para
│ │ │ │ +00079f00: 6d65 7472 697a 653a 2070 6172 616d 6574  metrize: paramet
│ │ │ │ +00079f10: 7269 7a65 2c20 6973 2061 202a 6e6f 7465  rize, is a *note
│ │ │ │ +00079f20: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ +00079f30: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ +00079f40: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ +00079f50: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ +00079f60: 2e69 6e66 6f2c 204e 6f64 653a 2070 6172  .info, Node: par
│ │ │ │ +00079f70: 616d 6574 7269 7a65 5f6c 7049 6465 616c  ametrize_lpIdeal
│ │ │ │ +00079f80: 5f72 702c 204e 6578 743a 2070 6f69 6e74  _rp, Next: point
│ │ │ │ +00079f90: 2c20 5072 6576 3a20 7061 7261 6d65 7472  , Prev: parametr
│ │ │ │ +00079fa0: 697a 652c 2055 703a 2054 6f70 0a0a 7061  ize, Up: Top..pa
│ │ │ │ +00079fb0: 7261 6d65 7472 697a 6528 4964 6561 6c29  rametrize(Ideal)
│ │ │ │ +00079fc0: 202d 2d20 7061 7261 6d65 7472 697a 6174   -- parametrizat
│ │ │ │ +00079fd0: 696f 6e20 6f66 206c 696e 6561 7220 7661  ion of linear va
│ │ │ │ +00079fe0: 7269 6574 6965 7320 616e 6420 6879 7065  rieties and hype
│ │ │ │ +00079ff0: 7271 7561 6472 6963 730a 2a2a 2a2a 2a2a  rquadrics.******
│ │ │ │  0007a000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007a010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007a020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0007a030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -0007a040: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ -0007a050: 3d3d 0a0a 2020 2a20 4675 6e63 7469 6f6e  ==..  * Function
│ │ │ │ -0007a060: 3a20 2a6e 6f74 6520 7061 7261 6d65 7472  : *note parametr
│ │ │ │ -0007a070: 697a 653a 2070 6172 616d 6574 7269 7a65  ize: parametrize
│ │ │ │ -0007a080: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ -0007a090: 2020 2020 2020 7061 7261 6d65 7472 697a        parametriz
│ │ │ │ -0007a0a0: 6520 490a 2020 2a20 496e 7075 7473 3a0a  e I.  * Inputs:.
│ │ │ │ -0007a0b0: 2020 2020 2020 2a20 492c 2061 6e20 2a6e        * I, an *n
│ │ │ │ -0007a0c0: 6f74 6520 6964 6561 6c3a 2028 4d61 6361  ote ideal: (Maca
│ │ │ │ -0007a0d0: 756c 6179 3244 6f63 2949 6465 616c 2c2c  ulay2Doc)Ideal,,
│ │ │ │ -0007a0e0: 2074 6865 2069 6465 616c 206f 6620 6120   the ideal of a 
│ │ │ │ -0007a0f0: 6c69 6e65 6172 2076 6172 6965 7479 0a20  linear variety. 
│ │ │ │ -0007a100: 2020 2020 2020 206f 7220 6f66 2061 2068         or of a h
│ │ │ │ -0007a110: 7970 6572 7175 6164 7269 630a 2020 2a20  yperquadric.  * 
│ │ │ │ -0007a120: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -0007a130: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ -0007a140: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ -0007a150: 6170 2c2c 2061 2062 6972 6174 696f 6e61  ap,, a birationa
│ │ │ │ -0007a160: 6c20 6d61 7020 7068 6920 7375 6368 2074  l map phi such t
│ │ │ │ -0007a170: 6861 7420 4920 3d3d 0a20 2020 2020 2020  hat I ==.       
│ │ │ │ -0007a180: 2069 6d61 6765 2070 6869 0a0a 4465 7363   image phi..Desc
│ │ │ │ -0007a190: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -0007a1a0: 3d3d 3d0a 0a54 6869 7320 6675 6e63 7469  ===..This functi
│ │ │ │ -0007a1b0: 6f6e 2068 6173 2062 6565 6e20 696d 7072  on has been impr
│ │ │ │ -0007a1c0: 6f76 6564 2061 6e64 2065 7874 656e 6465  oved and extende
│ │ │ │ -0007a1d0: 6420 696e 2074 6865 2070 6163 6b61 6765  d in the package
│ │ │ │ -0007a1e0: 0a4d 756c 7469 7072 6f6a 6563 7469 7665  .Multiprojective
│ │ │ │ -0007a1f0: 5661 7269 6574 6965 7320 286d 6973 7369  Varieties (missi
│ │ │ │ -0007a200: 6e67 2064 6f63 756d 656e 7461 7469 6f6e  ng documentation
│ │ │ │ -0007a210: 292c 2073 6565 202a 6e6f 7465 0a70 6172  ), see *note.par
│ │ │ │ -0007a220: 616d 6574 7269 7a65 284d 756c 7469 7072  ametrize(Multipr
│ │ │ │ -0007a230: 6f6a 6563 7469 7665 5661 7269 6574 7929  ojectiveVariety)
│ │ │ │ -0007a240: 3a0a 284d 756c 7469 7072 6f6a 6563 7469  :.(Multiprojecti
│ │ │ │ -0007a250: 7665 5661 7269 6574 6965 7329 7061 7261  veVarieties)para
│ │ │ │ -0007a260: 6d65 7472 697a 655f 6c70 4d75 6c74 6970  metrize_lpMultip
│ │ │ │ -0007a270: 726f 6a65 6374 6976 6556 6172 6965 7479  rojectiveVariety
│ │ │ │ -0007a280: 5f72 702c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  _rp,...+--------
│ │ │ │ +0007a030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0007a040: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +0007a050: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 4675  ========..  * Fu
│ │ │ │ +0007a060: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7061  nction: *note pa
│ │ │ │ +0007a070: 7261 6d65 7472 697a 653a 2070 6172 616d  rametrize: param
│ │ │ │ +0007a080: 6574 7269 7a65 2c0a 2020 2a20 5573 6167  etrize,.  * Usag
│ │ │ │ +0007a090: 653a 200a 2020 2020 2020 2020 7061 7261  e: .        para
│ │ │ │ +0007a0a0: 6d65 7472 697a 6520 490a 2020 2a20 496e  metrize I.  * In
│ │ │ │ +0007a0b0: 7075 7473 3a0a 2020 2020 2020 2a20 492c  puts:.      * I,
│ │ │ │ +0007a0c0: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a   an *note ideal:
│ │ │ │ +0007a0d0: 2028 4d61 6361 756c 6179 3244 6f63 2949   (Macaulay2Doc)I
│ │ │ │ +0007a0e0: 6465 616c 2c2c 2074 6865 2069 6465 616c  deal,, the ideal
│ │ │ │ +0007a0f0: 206f 6620 6120 6c69 6e65 6172 2076 6172   of a linear var
│ │ │ │ +0007a100: 6965 7479 0a20 2020 2020 2020 206f 7220  iety.        or 
│ │ │ │ +0007a110: 6f66 2061 2068 7970 6572 7175 6164 7269  of a hyperquadri
│ │ │ │ +0007a120: 630a 2020 2a20 4f75 7470 7574 733a 0a20  c.  * Outputs:. 
│ │ │ │ +0007a130: 2020 2020 202a 2061 202a 6e6f 7465 2072       * a *note r
│ │ │ │ +0007a140: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ +0007a150: 696f 6e61 6c4d 6170 2c2c 2061 2062 6972  ionalMap,, a bir
│ │ │ │ +0007a160: 6174 696f 6e61 6c20 6d61 7020 7068 6920  ational map phi 
│ │ │ │ +0007a170: 7375 6368 2074 6861 7420 4920 3d3d 0a20  such that I ==. 
│ │ │ │ +0007a180: 2020 2020 2020 2069 6d61 6765 2070 6869         image phi
│ │ │ │ +0007a190: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +0007a1a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320  =========..This 
│ │ │ │ +0007a1b0: 6675 6e63 7469 6f6e 2068 6173 2062 6565  function has bee
│ │ │ │ +0007a1c0: 6e20 696d 7072 6f76 6564 2061 6e64 2065  n improved and e
│ │ │ │ +0007a1d0: 7874 656e 6465 6420 696e 2074 6865 2070  xtended in the p
│ │ │ │ +0007a1e0: 6163 6b61 6765 0a4d 756c 7469 7072 6f6a  ackage.Multiproj
│ │ │ │ +0007a1f0: 6563 7469 7665 5661 7269 6574 6965 7320  ectiveVarieties 
│ │ │ │ +0007a200: 286d 6973 7369 6e67 2064 6f63 756d 656e  (missing documen
│ │ │ │ +0007a210: 7461 7469 6f6e 292c 2073 6565 202a 6e6f  tation), see *no
│ │ │ │ +0007a220: 7465 0a70 6172 616d 6574 7269 7a65 284d  te.parametrize(M
│ │ │ │ +0007a230: 756c 7469 7072 6f6a 6563 7469 7665 5661  ultiprojectiveVa
│ │ │ │ +0007a240: 7269 6574 7929 3a0a 284d 756c 7469 7072  riety):.(Multipr
│ │ │ │ +0007a250: 6f6a 6563 7469 7665 5661 7269 6574 6965  ojectiveVarietie
│ │ │ │ +0007a260: 7329 7061 7261 6d65 7472 697a 655f 6c70  s)parametrize_lp
│ │ │ │ +0007a270: 4d75 6c74 6970 726f 6a65 6374 6976 6556  MultiprojectiveV
│ │ │ │ +0007a280: 6172 6965 7479 5f72 702c 2e0a 0a2b 2d2d  ariety_rp,...+--
│ │ │ │  0007a290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a2d0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2050 3920  -----+.|i1 : P9 
│ │ │ │ -0007a2e0: 3a3d 205a 5a2f 3130 3030 3030 3139 5b78  := ZZ/10000019[x
│ │ │ │ -0007a2f0: 5f30 2e2e 785f 395d 2020 2020 2020 2020  _0..x_9]        
│ │ │ │ +0007a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0007a2e0: 203a 2050 3920 3a3d 205a 5a2f 3130 3030   : P9 := ZZ/1000
│ │ │ │ +0007a2f0: 3030 3139 5b78 5f30 2e2e 785f 395d 2020  0019[x_0..x_9]  
│ │ │ │  0007a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a320: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007a320: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a370: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007a380: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ +0007a370: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007a380: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │  0007a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a3c0: 2020 2020 207c 0a7c 6f31 203d 202d 2d2d       |.|o1 = ---
│ │ │ │ -0007a3d0: 2d2d 2d2d 2d5b 7820 2e2e 7820 5d20 2020  -----[x ..x ]   
│ │ │ │ -0007a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007a3c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0007a3d0: 203d 202d 2d2d 2d2d 2d2d 2d5b 7820 2e2e   = --------[x ..
│ │ │ │ +0007a3e0: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │  0007a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a410: 2020 2020 207c 0a7c 2020 2020 2031 3030       |.|     100
│ │ │ │ -0007a420: 3030 3031 3920 2030 2020 2039 2020 2020  00019  0   9    
│ │ │ │ -0007a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007a410: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007a420: 2020 2031 3030 3030 3031 3920 2030 2020     10000019  0  
│ │ │ │ +0007a430: 2039 2020 2020 2020 2020 2020 2020 2020   9              
│ │ │ │  0007a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007a460: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a4b0: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c       |.|o1 : Pol
│ │ │ │ -0007a4c0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ -0007a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007a4b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0007a4c0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +0007a4d0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  0007a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a500: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0007a500: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0007a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a550: 2d2d 2d2d 2d2b 0a7c 6932 203a 204c 203d  -----+.|i2 : L =
│ │ │ │ -0007a560: 2074 7269 6d20 6964 6561 6c28 7261 6e64   trim ideal(rand
│ │ │ │ -0007a570: 6f6d 2831 2c50 3929 2c72 616e 646f 6d28  om(1,P9),random(
│ │ │ │ -0007a580: 312c 5039 292c 7261 6e64 6f6d 2831 2c50  1,P9),random(1,P
│ │ │ │ -0007a590: 3929 2c72 616e 646f 6d28 312c 5039 2929  9),random(1,P9))
│ │ │ │ -0007a5a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0007a560: 203a 204c 203d 2074 7269 6d20 6964 6561   : L = trim idea
│ │ │ │ +0007a570: 6c28 7261 6e64 6f6d 2831 2c50 3929 2c72  l(random(1,P9),r
│ │ │ │ +0007a580: 616e 646f 6d28 312c 5039 292c 7261 6e64  andom(1,P9),rand
│ │ │ │ +0007a590: 6f6d 2831 2c50 3929 2c72 616e 646f 6d28  om(1,P9),random(
│ │ │ │ +0007a5a0: 312c 5039 2929 2020 2020 207c 0a7c 2020  1,P9))     |.|  
│ │ │ │  0007a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a5f0: 2020 2020 207c 0a7c 6f32 203d 2069 6465       |.|o2 = ide
│ │ │ │ -0007a600: 616c 2028 7820 202d 2031 3131 3230 3136  al (x  - 1112016
│ │ │ │ -0007a610: 7820 202d 2033 3930 3133 3631 7820 202d  x  - 3901361x  -
│ │ │ │ -0007a620: 2033 3139 3338 3633 7820 202b 2034 3134   3193863x  + 414
│ │ │ │ -0007a630: 3330 3430 7820 202d 2031 3936 3434 3137  3040x  - 1964417
│ │ │ │ -0007a640: 7820 202b 207c 0a7c 2020 2020 2020 2020  x  + |.|        
│ │ │ │ -0007a650: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ -0007a660: 2034 2020 2020 2020 2020 2020 2035 2020   4           5  
│ │ │ │ -0007a670: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ -0007a680: 2020 2020 2037 2020 2020 2020 2020 2020       7          
│ │ │ │ -0007a690: 2038 2020 207c 0a7c 2020 2020 202d 2d2d   8   |.|     ---
│ │ │ │ -0007a6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007a5f0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0007a600: 203d 2069 6465 616c 2028 7820 202d 2031   = ideal (x  - 1
│ │ │ │ +0007a610: 3131 3230 3136 7820 202d 2033 3930 3133  112016x  - 39013
│ │ │ │ +0007a620: 3631 7820 202d 2033 3139 3338 3633 7820  61x  - 3193863x 
│ │ │ │ +0007a630: 202b 2034 3134 3330 3430 7820 202d 2031   + 4143040x  - 1
│ │ │ │ +0007a640: 3936 3434 3137 7820 202b 207c 0a7c 2020  964417x  + |.|  
│ │ │ │ +0007a650: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +0007a660: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +0007a670: 2020 2035 2020 2020 2020 2020 2020 2036     5           6
│ │ │ │ +0007a680: 2020 2020 2020 2020 2020 2037 2020 2020             7    
│ │ │ │ +0007a690: 2020 2020 2020 2038 2020 207c 0a7c 2020         8   |.|  
│ │ │ │ +0007a6a0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007a6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a6e0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2031 3037  -----|.|     107
│ │ │ │ -0007a6f0: 3439 3538 7820 2c20 7820 202b 2036 3332  4958x , x  + 632
│ │ │ │ -0007a700: 3238 3478 2020 2b20 3439 3234 3538 7820  284x  + 492458x 
│ │ │ │ -0007a710: 202b 2033 3836 3932 3534 7820 202b 2032   + 3869254x  + 2
│ │ │ │ -0007a720: 3834 3032 3636 7820 202b 2034 3838 3339  840266x  + 48839
│ │ │ │ -0007a730: 3734 7820 207c 0a7c 2020 2020 2020 2020  74x  |.|        
│ │ │ │ -0007a740: 2020 2020 2039 2020 2032 2020 2020 2020       9   2      
│ │ │ │ -0007a750: 2020 2020 3420 2020 2020 2020 2020 2035      4          5
│ │ │ │ -0007a760: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ -0007a770: 2020 2020 2020 2037 2020 2020 2020 2020         7        
│ │ │ │ -0007a780: 2020 2038 207c 0a7c 2020 2020 202d 2d2d     8 |.|     ---
│ │ │ │ -0007a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007a6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0007a6f0: 2020 2031 3037 3439 3538 7820 2c20 7820     1074958x , x 
│ │ │ │ +0007a700: 202b 2036 3332 3238 3478 2020 2b20 3439   + 632284x  + 49
│ │ │ │ +0007a710: 3234 3538 7820 202b 2033 3836 3932 3534  2458x  + 3869254
│ │ │ │ +0007a720: 7820 202b 2032 3834 3032 3636 7820 202b  x  + 2840266x  +
│ │ │ │ +0007a730: 2034 3838 3339 3734 7820 207c 0a7c 2020   4883974x  |.|  
│ │ │ │ +0007a740: 2020 2020 2020 2020 2020 2039 2020 2032             9   2
│ │ │ │ +0007a750: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0007a760: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │ +0007a770: 2036 2020 2020 2020 2020 2020 2037 2020   6           7  
│ │ │ │ +0007a780: 2020 2020 2020 2020 2038 207c 0a7c 2020           8 |.|  
│ │ │ │ +0007a790: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a7d0: 2d2d 2d2d 2d7c 0a7c 2020 2020 202b 2033  -----|.|     + 3
│ │ │ │ -0007a7e0: 3334 3039 3631 7820 2c20 7820 202b 2034  340961x , x  + 4
│ │ │ │ -0007a7f0: 3732 3437 3039 7820 202d 2033 3530 3533  724709x  - 35053
│ │ │ │ -0007a800: 3836 7820 202b 2032 3436 3932 3036 7820  86x  + 2469206x 
│ │ │ │ -0007a810: 202d 2031 3338 3135 3135 7820 202b 2020   - 1381515x  +  
│ │ │ │ -0007a820: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007a830: 2020 2020 2020 2039 2020 2031 2020 2020         9   1    
│ │ │ │ -0007a840: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ -0007a850: 2020 2035 2020 2020 2020 2020 2020 2036     5           6
│ │ │ │ -0007a860: 2020 2020 2020 2020 2020 2037 2020 2020             7    
│ │ │ │ -0007a870: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │ -0007a880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0007a7e0: 2020 202b 2033 3334 3039 3631 7820 2c20     + 3340961x , 
│ │ │ │ +0007a7f0: 7820 202b 2034 3732 3437 3039 7820 202d  x  + 4724709x  -
│ │ │ │ +0007a800: 2033 3530 3533 3836 7820 202b 2032 3436   3505386x  + 246
│ │ │ │ +0007a810: 3932 3036 7820 202d 2031 3338 3135 3135  9206x  - 1381515
│ │ │ │ +0007a820: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ +0007a830: 2020 2020 2020 2020 2020 2020 2039 2020               9  
│ │ │ │ +0007a840: 2031 2020 2020 2020 2020 2020 2034 2020   1           4  
│ │ │ │ +0007a850: 2020 2020 2020 2020 2035 2020 2020 2020           5      
│ │ │ │ +0007a860: 2020 2020 2036 2020 2020 2020 2020 2020       6          
│ │ │ │ +0007a870: 2037 2020 2020 2020 2020 207c 0a7c 2020   7         |.|  
│ │ │ │ +0007a880: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007a890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a8c0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2032 3333  -----|.|     233
│ │ │ │ -0007a8d0: 3132 3830 7820 202d 2034 3933 3632 3239  1280x  - 4936229
│ │ │ │ -0007a8e0: 7820 2c20 7820 202d 2032 3039 3434 3536  x , x  - 2094456
│ │ │ │ -0007a8f0: 7820 202d 2033 3933 3634 3938 7820 202d  x  - 3936498x  -
│ │ │ │ -0007a900: 2034 3636 3534 3034 7820 202d 2037 3336   4665404x  - 736
│ │ │ │ -0007a910: 3934 3378 207c 0a7c 2020 2020 2020 2020  943x |.|        
│ │ │ │ -0007a920: 2020 2020 2038 2020 2020 2020 2020 2020       8          
│ │ │ │ -0007a930: 2039 2020 2030 2020 2020 2020 2020 2020   9   0          
│ │ │ │ -0007a940: 2034 2020 2020 2020 2020 2020 2035 2020   4           5  
│ │ │ │ -0007a950: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ -0007a960: 2020 2020 377c 0a7c 2020 2020 202d 2d2d      7|.|     ---
│ │ │ │ -0007a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0007a8d0: 2020 2032 3333 3132 3830 7820 202d 2034     2331280x  - 4
│ │ │ │ +0007a8e0: 3933 3632 3239 7820 2c20 7820 202d 2032  936229x , x  - 2
│ │ │ │ +0007a8f0: 3039 3434 3536 7820 202d 2033 3933 3634  094456x  - 39364
│ │ │ │ +0007a900: 3938 7820 202d 2034 3636 3534 3034 7820  98x  - 4665404x 
│ │ │ │ +0007a910: 202d 2037 3336 3934 3378 207c 0a7c 2020   - 736943x |.|  
│ │ │ │ +0007a920: 2020 2020 2020 2020 2020 2038 2020 2020             8    
│ │ │ │ +0007a930: 2020 2020 2020 2039 2020 2030 2020 2020         9   0    
│ │ │ │ +0007a940: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +0007a950: 2020 2035 2020 2020 2020 2020 2020 2036     5           6
│ │ │ │ +0007a960: 2020 2020 2020 2020 2020 377c 0a7c 2020            7|.|  
│ │ │ │ +0007a970: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007a980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a9b0: 2d2d 2d2d 2d7c 0a7c 2020 2020 202d 2038  -----|.|     - 8
│ │ │ │ -0007a9c0: 3439 3637 3178 2020 2b20 3330 3334 3133  49671x  + 303413
│ │ │ │ -0007a9d0: 3778 2029 2020 2020 2020 2020 2020 2020  7x )            
│ │ │ │ +0007a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0007a9c0: 2020 202d 2038 3439 3637 3178 2020 2b20     - 849671x  + 
│ │ │ │ +0007a9d0: 3330 3334 3133 3778 2029 2020 2020 2020  3034137x )      
│ │ │ │  0007a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007aa00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007aa10: 2020 2020 2020 3820 2020 2020 2020 2020        8         
│ │ │ │ -0007aa20: 2020 3920 2020 2020 2020 2020 2020 2020    9             
│ │ │ │ +0007aa00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007aa10: 2020 2020 2020 2020 2020 2020 3820 2020              8   
│ │ │ │ +0007aa20: 2020 2020 2020 2020 3920 2020 2020 2020          9       
│ │ │ │  0007aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007aa50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007aa50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007aaa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007aab0: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ -0007aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007aaa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007aab0: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ +0007aac0: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │  0007aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007aaf0: 2020 2020 207c 0a7c 6f32 203a 2049 6465       |.|o2 : Ide
│ │ │ │ -0007ab00: 616c 206f 6620 2d2d 2d2d 2d2d 2d2d 5b78  al of --------[x
│ │ │ │ -0007ab10: 202e 2e78 205d 2020 2020 2020 2020 2020   ..x ]          
│ │ │ │ +0007aaf0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0007ab00: 203a 2049 6465 616c 206f 6620 2d2d 2d2d   : Ideal of ----
│ │ │ │ +0007ab10: 2d2d 2d2d 5b78 202e 2e78 205d 2020 2020  ----[x ..x ]    
│ │ │ │  0007ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ab40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007ab50: 2020 2020 2020 3130 3030 3030 3139 2020        10000019  
│ │ │ │ -0007ab60: 3020 2020 3920 2020 2020 2020 2020 2020  0   9           
│ │ │ │ +0007ab40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007ab50: 2020 2020 2020 2020 2020 2020 3130 3030              1000
│ │ │ │ +0007ab60: 3030 3139 2020 3020 2020 3920 2020 2020  0019  0   9     
│ │ │ │  0007ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ab90: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0007ab90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0007aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0007b730: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ -0007b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b720: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b740: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │  0007b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b770: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007b770: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b7c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007b7d0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -0007b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b7c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b7e0: 2020 2020 7420 2020 2020 2020 2020 2020      t           
│ │ │ │  0007b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007b820: 2020 2020 2020 2020 2020 2020 2020 2035                 5
│ │ │ │ -0007b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b830: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │  0007b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b860: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007b870: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ -0007b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b860: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b880: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  0007b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b8b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007b8b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b900: 2020 2020 207c 0a7c 6f33 203a 2052 6174       |.|o3 : Rat
│ │ │ │ -0007b910: 696f 6e61 6c4d 6170 2028 6c69 6e65 6172  ionalMap (linear
│ │ │ │ -0007b920: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ -0007b930: 6f6d 2050 505e 3520 746f 2050 505e 3929  om PP^5 to PP^9)
│ │ │ │ -0007b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b950: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0007b900: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0007b910: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +0007b920: 6c69 6e65 6172 2072 6174 696f 6e61 6c20  linear rational 
│ │ │ │ +0007b930: 6d61 7020 6672 6f6d 2050 505e 3520 746f  map from PP^5 to
│ │ │ │ +0007b940: 2050 505e 3929 2020 2020 2020 2020 2020   PP^9)          
│ │ │ │ +0007b950: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  0007b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007b9a0: 2d2d 2d2d 2d7c 0a7c 3834 3936 3731 7420  -----|.|849671t 
│ │ │ │ -0007b9b0: 202d 2033 3033 3431 3337 7420 2c20 2020   - 3034137t ,   
│ │ │ │ -0007b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3834  -----------|.|84
│ │ │ │ +0007b9b0: 3936 3731 7420 202d 2033 3033 3431 3337  9671t  - 3034137
│ │ │ │ +0007b9c0: 7420 2c20 2020 2020 2020 2020 2020 2020  t ,             
│ │ │ │  0007b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b9f0: 2020 2020 207c 0a7c 2020 2020 2020 2034       |.|       4
│ │ │ │ -0007ba00: 2020 2020 2020 2020 2020 2035 2020 2020             5    
│ │ │ │ -0007ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b9f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007ba00: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ +0007ba10: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │  0007ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ba40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007ba40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ba90: 2020 2020 207c 0a7c 202d 2032 3333 3132       |.| - 23312
│ │ │ │ -0007baa0: 3830 7420 202b 2034 3933 3632 3239 7420  80t  + 4936229t 
│ │ │ │ -0007bab0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0007ba90: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +0007baa0: 2032 3333 3132 3830 7420 202b 2034 3933   2331280t  + 493
│ │ │ │ +0007bab0: 3632 3239 7420 2c20 2020 2020 2020 2020  6229t ,         
│ │ │ │  0007bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007baf0: 2020 2034 2020 2020 2020 2020 2020 2035     4           5
│ │ │ │ -0007bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007bae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007baf0: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ +0007bb00: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │  0007bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bb30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007bb30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bb80: 2020 2020 207c 0a7c 2034 3838 3339 3734       |.| 4883974
│ │ │ │ -0007bb90: 7420 202d 2033 3334 3039 3631 7420 2c20  t  - 3340961t , 
│ │ │ │ -0007bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007bb80: 2020 2020 2020 2020 2020 207c 0a7c 2034             |.| 4
│ │ │ │ +0007bb90: 3838 3339 3734 7420 202d 2033 3334 3039  883974t  - 33409
│ │ │ │ +0007bba0: 3631 7420 2c20 2020 2020 2020 2020 2020  61t ,           
│ │ │ │  0007bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bbd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007bbe0: 2034 2020 2020 2020 2020 2020 2035 2020   4           5  
│ │ │ │ -0007bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007bbd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007bbe0: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +0007bbf0: 2020 2035 2020 2020 2020 2020 2020 2020     5            
│ │ │ │  0007bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bc20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007bc20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bc70: 2020 2020 207c 0a7c 2031 3936 3434 3137       |.| 1964417
│ │ │ │ -0007bc80: 7420 202d 2031 3037 3439 3538 7420 2c20  t  - 1074958t , 
│ │ │ │ -0007bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007bc70: 2020 2020 2020 2020 2020 207c 0a7c 2031             |.| 1
│ │ │ │ +0007bc80: 3936 3434 3137 7420 202d 2031 3037 3439  964417t  - 10749
│ │ │ │ +0007bc90: 3538 7420 2c20 2020 2020 2020 2020 2020  58t ,           
│ │ │ │  0007bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bcc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007bcd0: 2034 2020 2020 2020 2020 2020 2035 2020   4           5  
│ │ │ │ -0007bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007bcc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007bcd0: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +0007bce0: 2020 2035 2020 2020 2020 2020 2020 2020     5            
│ │ │ │  0007bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bd10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0007bd10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0007bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007bd60: 2d2d 2d2d 2d2b 0a7c 6934 203a 2051 203d  -----+.|i4 : Q =
│ │ │ │ -0007bd70: 2074 7269 6d20 6964 6561 6c28 7261 6e64   trim ideal(rand
│ │ │ │ -0007bd80: 6f6d 2832 2c50 3929 2c72 616e 646f 6d28  om(2,P9),random(
│ │ │ │ -0007bd90: 312c 5039 292c 7261 6e64 6f6d 2831 2c50  1,P9),random(1,P
│ │ │ │ -0007bda0: 3929 2920 2020 2020 2020 2020 2020 2020  9))             
│ │ │ │ -0007bdb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +0007bd70: 203a 2051 203d 2074 7269 6d20 6964 6561   : Q = trim idea
│ │ │ │ +0007bd80: 6c28 7261 6e64 6f6d 2832 2c50 3929 2c72  l(random(2,P9),r
│ │ │ │ +0007bd90: 616e 646f 6d28 312c 5039 292c 7261 6e64  andom(1,P9),rand
│ │ │ │ +0007bda0: 6f6d 2831 2c50 3929 2920 2020 2020 2020  om(1,P9))       
│ │ │ │ +0007bdb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007be00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007be00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007be50: 2020 2020 207c 0a7c 6f34 203d 2069 6465       |.|o4 = ide
│ │ │ │ -0007be60: 616c 2028 7820 202d 2033 3733 3132 3835  al (x  - 3731285
│ │ │ │ -0007be70: 7820 202b 2035 3639 3438 3578 2020 2b20  x  + 569485x  + 
│ │ │ │ -0007be80: 3432 3535 3230 3178 2020 2d20 3230 3938  4255201x  - 2098
│ │ │ │ -0007be90: 3731 3278 2020 2d20 3432 3438 3939 3078  712x  - 4248990x
│ │ │ │ -0007bea0: 2020 2d20 207c 0a7c 2020 2020 2020 2020    -  |.|        
│ │ │ │ -0007beb0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -0007bec0: 2032 2020 2020 2020 2020 2020 3320 2020   2          3   
│ │ │ │ -0007bed0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ -0007bee0: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ -0007bef0: 3620 2020 207c 0a7c 2020 2020 202d 2d2d  6    |.|     ---
│ │ │ │ -0007bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007be50: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +0007be60: 203d 2069 6465 616c 2028 7820 202d 2033   = ideal (x  - 3
│ │ │ │ +0007be70: 3733 3132 3835 7820 202b 2035 3639 3438  731285x  + 56948
│ │ │ │ +0007be80: 3578 2020 2b20 3432 3535 3230 3178 2020  5x  + 4255201x  
│ │ │ │ +0007be90: 2d20 3230 3938 3731 3278 2020 2d20 3432  - 2098712x  - 42
│ │ │ │ +0007bea0: 3438 3939 3078 2020 2d20 207c 0a7c 2020  48990x  -  |.|  
│ │ │ │ +0007beb0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +0007bec0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0007bed0: 2020 3320 2020 2020 2020 2020 2020 3420    3           4 
│ │ │ │ +0007bee0: 2020 2020 2020 2020 2020 3520 2020 2020            5     
│ │ │ │ +0007bef0: 2020 2020 2020 3620 2020 207c 0a7c 2020        6    |.|  
│ │ │ │ +0007bf00: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007bf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007bf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007bf40: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ +0007bf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │  0007bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bf90: 2020 2020 207c 0a7c 2020 2020 2031 3830       |.|     180
│ │ │ │ -0007bfa0: 3133 3432 7820 202b 2034 3035 3032 3239  1342x  + 4050229
│ │ │ │ -0007bfb0: 7820 202d 2032 3331 3932 3633 7820 2c20  x  - 2319263x , 
│ │ │ │ -0007bfc0: 7820 202d 2033 3039 3436 3839 7820 202d  x  - 3094689x  -
│ │ │ │ -0007bfd0: 2034 3431 3031 3836 7820 202b 2020 2020   4410186x  +    
│ │ │ │ -0007bfe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007bff0: 2020 2020 2037 2020 2020 2020 2020 2020       7          
│ │ │ │ -0007c000: 2038 2020 2020 2020 2020 2020 2039 2020   8           9  
│ │ │ │ -0007c010: 2030 2020 2020 2020 2020 2020 2032 2020   0           2  
│ │ │ │ -0007c020: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ -0007c030: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │ -0007c040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007bf90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007bfa0: 2020 2031 3830 3133 3432 7820 202b 2034     1801342x  + 4
│ │ │ │ +0007bfb0: 3035 3032 3239 7820 202d 2032 3331 3932  050229x  - 23192
│ │ │ │ +0007bfc0: 3633 7820 2c20 7820 202d 2033 3039 3436  63x , x  - 30946
│ │ │ │ +0007bfd0: 3839 7820 202d 2034 3431 3031 3836 7820  89x  - 4410186x 
│ │ │ │ +0007bfe0: 202b 2020 2020 2020 2020 207c 0a7c 2020   +         |.|  
│ │ │ │ +0007bff0: 2020 2020 2020 2020 2020 2037 2020 2020             7    
│ │ │ │ +0007c000: 2020 2020 2020 2038 2020 2020 2020 2020         8        
│ │ │ │ +0007c010: 2020 2039 2020 2030 2020 2020 2020 2020     9   0        
│ │ │ │ +0007c020: 2020 2032 2020 2020 2020 2020 2020 2033     2           3
│ │ │ │ +0007c030: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007c040: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007c050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007c080: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ +0007c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │  0007c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c0d0: 2020 2020 327c 0a7c 2020 2020 2033 3139      2|.|     319
│ │ │ │ -0007c0e0: 3631 3436 7820 202b 2032 3731 3337 3731  6146x  + 2713771
│ │ │ │ -0007c0f0: 7820 202b 2032 3236 3134 3132 7820 202d  x  + 2261412x  -
│ │ │ │ -0007c100: 2031 3236 3731 3936 7820 202d 2034 3231   1267196x  - 421
│ │ │ │ -0007c110: 3034 3033 7820 202b 2032 3835 3933 3278  0403x  + 285932x
│ │ │ │ -0007c120: 202c 2078 207c 0a7c 2020 2020 2020 2020   , x |.|        
│ │ │ │ -0007c130: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ -0007c140: 2035 2020 2020 2020 2020 2020 2036 2020   5           6  
│ │ │ │ -0007c150: 2020 2020 2020 2020 2037 2020 2020 2020           7      
│ │ │ │ -0007c160: 2020 2020 2038 2020 2020 2020 2020 2020       8          
│ │ │ │ -0007c170: 3920 2020 327c 0a7c 2020 2020 202d 2d2d  9   2|.|     ---
│ │ │ │ -0007c180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007c0d0: 2020 2020 2020 2020 2020 327c 0a7c 2020            2|.|  
│ │ │ │ +0007c0e0: 2020 2033 3139 3631 3436 7820 202b 2032     3196146x  + 2
│ │ │ │ +0007c0f0: 3731 3337 3731 7820 202b 2032 3236 3134  713771x  + 22614
│ │ │ │ +0007c100: 3132 7820 202d 2031 3236 3731 3936 7820  12x  - 1267196x 
│ │ │ │ +0007c110: 202d 2034 3231 3034 3033 7820 202b 2032   - 4210403x  + 2
│ │ │ │ +0007c120: 3835 3933 3278 202c 2078 207c 0a7c 2020  85932x , x |.|  
│ │ │ │ +0007c130: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ +0007c140: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ +0007c150: 2020 2036 2020 2020 2020 2020 2020 2037     6           7
│ │ │ │ +0007c160: 2020 2020 2020 2020 2020 2038 2020 2020             8    
│ │ │ │ +0007c170: 2020 2020 2020 3920 2020 327c 0a7c 2020        9   2|.|  
│ │ │ │ +0007c180: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007c190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007c1c0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ +0007c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │  0007c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c1e0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0007c1e0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0007c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c200: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0007c210: 2020 2020 207c 0a7c 2020 2020 202b 2031       |.|     + 1
│ │ │ │ -0007c220: 3034 3534 3231 7820 7820 202b 2037 3138  045421x x  + 718
│ │ │ │ -0007c230: 3533 3278 2020 2d20 3337 3031 3632 3878  532x  - 3701628x
│ │ │ │ -0007c240: 2078 2020 2b20 3339 3033 3739 3878 2078   x  + 3903798x x
│ │ │ │ -0007c250: 2020 2b20 3238 3432 3339 3778 2020 2d20    + 2842397x  - 
│ │ │ │ -0007c260: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c270: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ -0007c280: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -0007c290: 3220 3420 2020 2020 2020 2020 2020 3320  2 4           3 
│ │ │ │ -0007c2a0: 3420 2020 2020 2020 2020 2020 3420 2020  4           4   
│ │ │ │ -0007c2b0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │ -0007c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007c210: 2020 3220 2020 2020 2020 207c 0a7c 2020    2        |.|  
│ │ │ │ +0007c220: 2020 202b 2031 3034 3534 3231 7820 7820     + 1045421x x 
│ │ │ │ +0007c230: 202b 2037 3138 3533 3278 2020 2d20 3337   + 718532x  - 37
│ │ │ │ +0007c240: 3031 3632 3878 2078 2020 2b20 3339 3033  01628x x  + 3903
│ │ │ │ +0007c250: 3739 3878 2078 2020 2b20 3238 3432 3339  798x x  + 284239
│ │ │ │ +0007c260: 3778 2020 2d20 2020 2020 207c 0a7c 2020  7x  -      |.|  
│ │ │ │ +0007c270: 2020 2020 2020 2020 2020 2020 2032 2033               2 3
│ │ │ │ +0007c280: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +0007c290: 2020 2020 2020 3220 3420 2020 2020 2020        2 4       
│ │ │ │ +0007c2a0: 2020 2020 3320 3420 2020 2020 2020 2020      3 4         
│ │ │ │ +0007c2b0: 2020 3420 2020 2020 2020 207c 0a7c 2020    4        |.|  
│ │ │ │ +0007c2c0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007c300: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ +0007c300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │  0007c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c330: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0007c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c350: 2020 2020 207c 0a7c 2020 2020 2032 3939       |.|     299
│ │ │ │ -0007c360: 3739 3632 7820 7820 202b 2034 3138 3938  7962x x  + 41898
│ │ │ │ -0007c370: 3335 7820 7820 202b 2031 3438 3932 3235  35x x  + 1489225
│ │ │ │ -0007c380: 7820 7820 202d 2032 3237 3939 3535 7820  x x  - 2279955x 
│ │ │ │ -0007c390: 202b 2032 3532 3037 3832 7820 7820 202b   + 2520782x x  +
│ │ │ │ -0007c3a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c3b0: 2020 2020 2032 2035 2020 2020 2020 2020       2 5        
│ │ │ │ -0007c3c0: 2020 2033 2035 2020 2020 2020 2020 2020     3 5          
│ │ │ │ -0007c3d0: 2034 2035 2020 2020 2020 2020 2020 2035   4 5           5
│ │ │ │ -0007c3e0: 2020 2020 2020 2020 2020 2032 2036 2020             2 6  
│ │ │ │ -0007c3f0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │ -0007c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007c340: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0007c350: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007c360: 2020 2032 3939 3739 3632 7820 7820 202b     2997962x x  +
│ │ │ │ +0007c370: 2034 3138 3938 3335 7820 7820 202b 2031   4189835x x  + 1
│ │ │ │ +0007c380: 3438 3932 3235 7820 7820 202d 2032 3237  489225x x  - 227
│ │ │ │ +0007c390: 3939 3535 7820 202b 2032 3532 3037 3832  9955x  + 2520782
│ │ │ │ +0007c3a0: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ +0007c3b0: 2020 2020 2020 2020 2020 2032 2035 2020             2 5  
│ │ │ │ +0007c3c0: 2020 2020 2020 2020 2033 2035 2020 2020           3 5    
│ │ │ │ +0007c3d0: 2020 2020 2020 2034 2035 2020 2020 2020         4 5      
│ │ │ │ +0007c3e0: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │ +0007c3f0: 2032 2036 2020 2020 2020 207c 0a7c 2020   2 6       |.|  
│ │ │ │ +0007c400: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007c440: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ +0007c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │  0007c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c470: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0007c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c490: 2020 2020 207c 0a7c 2020 2020 2034 3439       |.|     449
│ │ │ │ -0007c4a0: 3432 3830 7820 7820 202b 2033 3130 3132  4280x x  + 31012
│ │ │ │ -0007c4b0: 3535 7820 7820 202d 2036 3831 3935 3078  55x x  - 681950x
│ │ │ │ -0007c4c0: 2078 2020 2b20 3133 3037 3439 3078 2020   x  + 1307490x  
│ │ │ │ -0007c4d0: 2b20 3236 3930 3736 3778 2078 2020 2b20  + 2690767x x  + 
│ │ │ │ -0007c4e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c4f0: 2020 2020 2033 2036 2020 2020 2020 2020       3 6        
│ │ │ │ -0007c500: 2020 2034 2036 2020 2020 2020 2020 2020     4 6          
│ │ │ │ -0007c510: 3520 3620 2020 2020 2020 2020 2020 3620  5 6           6 
│ │ │ │ -0007c520: 2020 2020 2020 2020 2020 3220 3720 2020            2 7   
│ │ │ │ -0007c530: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │ -0007c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007c480: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0007c490: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007c4a0: 2020 2034 3439 3432 3830 7820 7820 202b     4494280x x  +
│ │ │ │ +0007c4b0: 2033 3130 3132 3535 7820 7820 202d 2036   3101255x x  - 6
│ │ │ │ +0007c4c0: 3831 3935 3078 2078 2020 2b20 3133 3037  81950x x  + 1307
│ │ │ │ +0007c4d0: 3439 3078 2020 2b20 3236 3930 3736 3778  490x  + 2690767x
│ │ │ │ +0007c4e0: 2078 2020 2b20 2020 2020 207c 0a7c 2020   x  +      |.|  
│ │ │ │ +0007c4f0: 2020 2020 2020 2020 2020 2033 2036 2020             3 6  
│ │ │ │ +0007c500: 2020 2020 2020 2020 2034 2036 2020 2020           4 6    
│ │ │ │ +0007c510: 2020 2020 2020 3520 3620 2020 2020 2020        5 6       
│ │ │ │ +0007c520: 2020 2020 3620 2020 2020 2020 2020 2020      6           
│ │ │ │ +0007c530: 3220 3720 2020 2020 2020 207c 0a7c 2020  2 7        |.|  
│ │ │ │ +0007c540: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007c580: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ +0007c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │  0007c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c5c0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0007c5d0: 2020 2020 207c 0a7c 2020 2020 2034 3530       |.|     450
│ │ │ │ -0007c5e0: 3336 3531 7820 7820 202b 2031 3736 3235  3651x x  + 17625
│ │ │ │ -0007c5f0: 3238 7820 7820 202b 2031 3337 3638 3278  28x x  + 137682x
│ │ │ │ -0007c600: 2078 2020 2d20 3232 3239 3039 3378 2078   x  - 2229093x x
│ │ │ │ -0007c610: 2020 2d20 3430 3138 3936 3778 2020 2b20    - 4018967x  + 
│ │ │ │ -0007c620: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c630: 2020 2020 2033 2037 2020 2020 2020 2020       3 7        
│ │ │ │ -0007c640: 2020 2034 2037 2020 2020 2020 2020 2020     4 7          
│ │ │ │ -0007c650: 3520 3720 2020 2020 2020 2020 2020 3620  5 7           6 
│ │ │ │ -0007c660: 3720 2020 2020 2020 2020 2020 3720 2020  7           7   
│ │ │ │ -0007c670: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │ -0007c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0007c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007c5d0: 2020 3220 2020 2020 2020 207c 0a7c 2020    2        |.|  
│ │ │ │ +0007c5e0: 2020 2034 3530 3336 3531 7820 7820 202b     4503651x x  +
│ │ │ │ +0007c5f0: 2031 3736 3235 3238 7820 7820 202b 2031   1762528x x  + 1
│ │ │ │ +0007c600: 3337 3638 3278 2078 2020 2d20 3232 3239  37682x x  - 2229
│ │ │ │ +0007c610: 3039 3378 2078 2020 2d20 3430 3138 3936  093x x  - 401896
│ │ │ │ +0007c620: 3778 2020 2b20 2020 2020 207c 0a7c 2020  7x  +      |.|  
│ │ │ │ +0007c630: 2020 2020 2020 2020 2020 2033 2037 2020             3 7  
│ │ │ │ +0007c640: 2020 2020 2020 2020 2034 2037 2020 2020           4 7    
│ │ │ │ +0007c650: 2020 2020 2020 3520 3720 2020 2020 2020        5 7       
│ │ │ │ +0007c660: 2020 2020 3620 3720 2020 2020 2020 2020      6 7         
│ │ │ │ +0007c670: 2020 3720 2020 2020 2020 207c 0a7c 2020    7        |.|  
│ │ │ │ +0007c680: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007c6c0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
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│ │ │ │  0007c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0007c720: 3631 3137 7820 7820 202d 2032 3534 3133  6117x x  - 25413
│ │ │ │ -0007c730: 3039 7820 7820 202b 2033 3831 3039 3638  09x x  + 3810968
│ │ │ │ -0007c740: 7820 7820 202d 2034 3230 3831 3934 7820  x x  - 4208194x 
│ │ │ │ -0007c750: 7820 202d 2031 3634 3335 3630 7820 7820  x  - 1643560x x 
│ │ │ │ -0007c760: 202b 2020 207c 0a7c 2020 2020 2020 2020   +   |.|        
│ │ │ │ -0007c770: 2020 2020 2032 2038 2020 2020 2020 2020       2 8        
│ │ │ │ -0007c780: 2020 2033 2038 2020 2020 2020 2020 2020     3 8          
│ │ │ │ -0007c790: 2034 2038 2020 2020 2020 2020 2020 2035   4 8           5
│ │ │ │ -0007c7a0: 2038 2020 2020 2020 2020 2020 2036 2038   8           6 8
│ │ │ │ -0007c7b0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
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│ │ │ │ +0007c720: 2020 2034 3533 3631 3137 7820 7820 202d     4536117x x  -
│ │ │ │ +0007c730: 2032 3534 3133 3039 7820 7820 202b 2033   2541309x x  + 3
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│ │ │ │ +0007c770: 2020 2020 2020 2020 2020 2032 2038 2020             2 8  
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│ │ │ │ +0007c7c0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  0007c7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0007c860: 3035 3733 7820 7820 202d 2032 3238 3035  0573x x  - 22805
│ │ │ │ -0007c870: 3136 7820 202d 2031 3533 3230 3536 7820  16x  - 1532056x 
│ │ │ │ -0007c880: 7820 202b 2031 3838 3339 3335 7820 7820  x  + 1883935x x 
│ │ │ │ -0007c890: 202b 2031 3838 3736 3637 7820 7820 202b   + 1887667x x  +
│ │ │ │ -0007c8a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c8b0: 2020 2020 2037 2038 2020 2020 2020 2020       7 8        
│ │ │ │ -0007c8c0: 2020 2038 2020 2020 2020 2020 2020 2032     8           2
│ │ │ │ -0007c8d0: 2039 2020 2020 2020 2020 2020 2033 2039   9           3 9
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│ │ │ │ -0007c8f0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
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│ │ │ │ +0007c870: 2032 3238 3035 3136 7820 202d 2031 3533   2280516x  - 153
│ │ │ │ +0007c880: 3230 3536 7820 7820 202b 2031 3838 3339  2056x x  + 18839
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│ │ │ │ +0007c8a0: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
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│ │ │ │ -0007c9b0: 3934 7820 7820 202d 2031 3830 3137 3632  94x x  - 1801762
│ │ │ │ -0007c9c0: 7820 7820 202b 2033 3032 3232 3432 7820  x x  + 3022242x 
│ │ │ │ -0007c9d0: 7820 202b 2033 3631 3837 3839 7820 2920  x  + 3618789x ) 
│ │ │ │ -0007c9e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c9f0: 2020 2020 2035 2039 2020 2020 2020 2020       5 9        
│ │ │ │ -0007ca00: 2020 2036 2039 2020 2020 2020 2020 2020     6 9          
│ │ │ │ -0007ca10: 2037 2039 2020 2020 2020 2020 2020 2038   7 9           8
│ │ │ │ -0007ca20: 2039 2020 2020 2020 2020 2020 2039 2020   9           9  
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│ │ │ │ +0007c9d0: 3232 3432 7820 7820 202b 2033 3631 3837  2242x x  + 36187
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│ │ │ │ +0007ca90: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ +0007caa0: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │  0007cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cad0: 2020 2020 207c 0a7c 6f34 203a 2049 6465       |.|o4 : Ide
│ │ │ │ -0007cae0: 616c 206f 6620 2d2d 2d2d 2d2d 2d2d 5b78  al of --------[x
│ │ │ │ -0007caf0: 202e 2e78 205d 2020 2020 2020 2020 2020   ..x ]          
│ │ │ │ +0007cad0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +0007cae0: 203a 2049 6465 616c 206f 6620 2d2d 2d2d   : Ideal of ----
│ │ │ │ +0007caf0: 2d2d 2d2d 5b78 202e 2e78 205d 2020 2020  ----[x ..x ]    
│ │ │ │  0007cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0007cb30: 2020 2020 2020 3130 3030 3030 3139 2020        10000019  
│ │ │ │ -0007cb40: 3020 2020 3920 2020 2020 2020 2020 2020  0   9           
│ │ │ │ +0007cb20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007cb30: 2020 2020 2020 2020 2020 2020 3130 3030              1000
│ │ │ │ +0007cb40: 3030 3139 2020 3020 2020 3920 2020 2020  0019  0   9     
│ │ │ │  0007cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cb70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0007cb70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0007cb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007cbc0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 696d  -----+.|i5 : tim
│ │ │ │ -0007cbd0: 6520 7061 7261 6d65 7472 697a 6520 5120  e parametrize Q 
│ │ │ │ -0007cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +0007cbd0: 203a 2074 696d 6520 7061 7261 6d65 7472   : time parametr
│ │ │ │ +0007cbe0: 697a 6520 5120 2020 2020 2020 2020 2020  ize Q           
│ │ │ │  0007cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cc10: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0007cc20: 2030 2e35 3637 3233 3773 2028 6370 7529   0.567237s (cpu)
│ │ │ │ -0007cc30: 3b20 302e 3334 3732 3935 7320 2874 6872  ; 0.347295s (thr
│ │ │ │ -0007cc40: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -0007cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cc60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007cc10: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +0007cc20: 2d20 7573 6564 2030 2e36 3137 3438 3573  - used 0.617485s
│ │ │ │ +0007cc30: 2028 6370 7529 3b20 302e 3434 3134 3231   (cpu); 0.441421
│ │ │ │ +0007cc40: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +0007cc50: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +0007cc60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ccb0: 2020 2020 207c 0a7c 6f35 203d 202d 2d20       |.|o5 = -- 
│ │ │ │ -0007ccc0: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ -0007ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ccb0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +0007ccc0: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +0007ccd0: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  0007cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cd00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007cd10: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ -0007cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007cd00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007cd20: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  0007cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cd50: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -0007cd60: 7263 653a 2050 726f 6a28 2d2d 2d2d 2d2d  rce: Proj(------
│ │ │ │ -0007cd70: 2d2d 5b74 202c 2074 202c 2074 202c 2074  --[t , t , t , t
│ │ │ │ -0007cd80: 202c 2074 202c 2074 202c 2074 205d 2920   , t , t , t ]) 
│ │ │ │ -0007cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cda0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007cdb0: 2020 2020 2020 2020 2020 3130 3030 3030            100000
│ │ │ │ -0007cdc0: 3139 2020 3020 2020 3120 2020 3220 2020  19  0   1   2   
│ │ │ │ -0007cdd0: 3320 2020 3420 2020 3520 2020 3620 2020  3   4   5   6   
│ │ │ │ -0007cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cdf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007ce00: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ -0007ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007cd50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007cd60: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ +0007cd70: 2d2d 2d2d 2d2d 2d2d 5b74 202c 2074 202c  --------[t , t ,
│ │ │ │ +0007cd80: 2074 202c 2074 202c 2074 202c 2074 202c   t , t , t , t ,
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│ │ │ │ +0007cda0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007cdc0: 3130 3030 3030 3139 2020 3020 2020 3120  10000019  0   1 
│ │ │ │ +0007cdd0: 2020 3220 2020 3320 2020 3420 2020 3520    2   3   4   5 
│ │ │ │ +0007cde0: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ +0007cdf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ce10: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  0007ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ce40: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -0007ce50: 6765 743a 2050 726f 6a28 2d2d 2d2d 2d2d  get: Proj(------
│ │ │ │ -0007ce60: 2d2d 5b78 202c 2078 202c 2078 202c 2078  --[x , x , x , x
│ │ │ │ -0007ce70: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ -0007ce80: 202c 2078 202c 2078 205d 2920 2020 2020   , x , x ])     
│ │ │ │ -0007ce90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007cea0: 2020 2020 2020 2020 2020 3130 3030 3030            100000
│ │ │ │ -0007ceb0: 3139 2020 3020 2020 3120 2020 3220 2020  19  0   1   2   
│ │ │ │ -0007cec0: 3320 2020 3420 2020 3520 2020 3620 2020  3   4   5   6   
│ │ │ │ -0007ced0: 3720 2020 3820 2020 3920 2020 2020 2020  7   8   9       
│ │ │ │ -0007cee0: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -0007cef0: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ -0007cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ce40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007ce50: 2020 2074 6172 6765 743a 2050 726f 6a28     target: Proj(
│ │ │ │ +0007ce60: 2d2d 2d2d 2d2d 2d2d 5b78 202c 2078 202c  --------[x , x ,
│ │ │ │ +0007ce70: 2078 202c 2078 202c 2078 202c 2078 202c   x , x , x , x ,
│ │ │ │ +0007ce80: 2078 202c 2078 202c 2078 202c 2078 205d   x , x , x , x ]
│ │ │ │ +0007ce90: 2920 2020 2020 2020 2020 207c 0a7c 2020  )          |.|  
│ │ │ │ +0007cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ceb0: 3130 3030 3030 3139 2020 3020 2020 3120  10000019  0   1 
│ │ │ │ +0007cec0: 2020 3220 2020 3320 2020 3420 2020 3520    2   3   4   5 
│ │ │ │ +0007ced0: 2020 3620 2020 3720 2020 3820 2020 3920    6   7   8   9 
│ │ │ │ +0007cee0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007cef0: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
│ │ │ │ +0007cf00: 733a 207b 2020 2020 2020 2020 2020 2020  s: {            
│ │ │ │  0007cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cf30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007cf30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cf50: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0007cf60: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0007cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cf80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007cf90: 2020 2020 2020 2020 2020 2020 2020 3831                81
│ │ │ │ -0007cfa0: 3739 3938 7420 202b 2031 3332 3932 3136  7998t  + 1329216
│ │ │ │ -0007cfb0: 7420 7420 202d 2031 3131 3431 3832 7420  t t  - 1114182t 
│ │ │ │ -0007cfc0: 202b 2031 3334 3438 3231 7420 7420 202d   + 1344821t t  -
│ │ │ │ -0007cfd0: 2031 3738 307c 0a7c 2020 2020 2020 2020   1780|.|        
│ │ │ │ +0007cf50: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0007cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007cf70: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0007cf80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007cfa0: 2020 2020 3831 3739 3938 7420 202b 2031      817998t  + 1
│ │ │ │ +0007cfb0: 3332 3932 3136 7420 7420 202d 2031 3131  329216t t  - 111
│ │ │ │ +0007cfc0: 3431 3832 7420 202b 2031 3334 3438 3231  4182t  + 1344821
│ │ │ │ +0007cfd0: 7420 7420 202d 2031 3738 307c 0a7c 2020  t t  - 1780|.|  
│ │ │ │  0007cfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cff0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0007d000: 2030 2031 2020 2020 2020 2020 2020 2031   0 1           1
│ │ │ │ -0007d010: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ -0007d020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007cff0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +0007d000: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +0007d010: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0007d020: 2030 2032 2020 2020 2020 207c 0a7c 2020   0 2       |.|  
│ │ │ │  0007d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d070: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d090: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0007d0a0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0007d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d0c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007d0d0: 2020 2020 2020 2020 2020 2020 2020 3236                26
│ │ │ │ -0007d0e0: 3939 3239 3274 2020 2d20 3931 3332 3836  99292t  - 913286
│ │ │ │ -0007d0f0: 7420 7420 202b 2031 3330 3835 3637 7420  t t  + 1308567t 
│ │ │ │ -0007d100: 202b 2033 3734 3231 3635 7420 7420 202b   + 3742165t t  +
│ │ │ │ -0007d110: 2032 3233 347c 0a7c 2020 2020 2020 2020   2234|.|        
│ │ │ │ +0007d090: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0007d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d0b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0007d0c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007d0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d0e0: 2020 2020 3236 3939 3239 3274 2020 2d20      2699292t  - 
│ │ │ │ +0007d0f0: 3931 3332 3836 7420 7420 202b 2031 3330  913286t t  + 130
│ │ │ │ +0007d100: 3835 3637 7420 202b 2033 3734 3231 3635  8567t  + 3742165
│ │ │ │ +0007d110: 7420 7420 202b 2032 3233 347c 0a7c 2020  t t  + 2234|.|  
│ │ │ │  0007d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d130: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -0007d140: 2030 2031 2020 2020 2020 2020 2020 2031   0 1           1
│ │ │ │ -0007d150: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ -0007d160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d130: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0007d140: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +0007d150: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0007d160: 2030 2032 2020 2020 2020 207c 0a7c 2020   0 2       |.|  
│ │ │ │  0007d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d1b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d1b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d1d0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0007d1e0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0007d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007d210: 2020 2020 2020 2020 2020 2020 2020 3234                24
│ │ │ │ -0007d220: 3039 3732 3574 2020 2d20 3532 3833 3431  09725t  - 528341
│ │ │ │ -0007d230: 7420 7420 202b 2031 3637 3534 3032 7420  t t  + 1675402t 
│ │ │ │ -0007d240: 202d 2037 3037 3631 3674 2074 2020 2d20   - 707616t t  - 
│ │ │ │ -0007d250: 3132 3831 387c 0a7c 2020 2020 2020 2020  12818|.|        
│ │ │ │ +0007d1d0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0007d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d1f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0007d200: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d220: 2020 2020 3234 3039 3732 3574 2020 2d20      2409725t  - 
│ │ │ │ +0007d230: 3532 3833 3431 7420 7420 202b 2031 3637  528341t t  + 167
│ │ │ │ +0007d240: 3534 3032 7420 202d 2037 3037 3631 3674  5402t  - 707616t
│ │ │ │ +0007d250: 2074 2020 2d20 3132 3831 387c 0a7c 2020   t  - 12818|.|  
│ │ │ │  0007d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d270: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -0007d280: 2030 2031 2020 2020 2020 2020 2020 2031   0 1           1
│ │ │ │ -0007d290: 2020 2020 2020 2020 2020 3020 3220 2020            0 2   
│ │ │ │ -0007d2a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d270: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0007d280: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +0007d290: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0007d2a0: 3020 3220 2020 2020 2020 207c 0a7c 2020  0 2        |.|  
│ │ │ │  0007d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d2f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d2f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d310: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0007d310: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0007d320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d330: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0007d340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007d350: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0007d360: 3332 3537 3337 3774 2020 2d20 3138 3935  3257377t  - 1895
│ │ │ │ -0007d370: 3235 3274 2074 2020 2b20 3235 3932 3733  252t t  + 259273
│ │ │ │ -0007d380: 3074 2020 2b20 3236 3031 3833 3274 2074  0t  + 2601832t t
│ │ │ │ -0007d390: 2020 2d20 327c 0a7c 2020 2020 2020 2020    - 2|.|        
│ │ │ │ +0007d330: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0007d340: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007d350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d360: 2020 2020 2d20 3332 3537 3337 3774 2020      - 3257377t  
│ │ │ │ +0007d370: 2d20 3138 3935 3235 3274 2074 2020 2b20  - 1895252t t  + 
│ │ │ │ +0007d380: 3235 3932 3733 3074 2020 2b20 3236 3031  2592730t  + 2601
│ │ │ │ +0007d390: 3833 3274 2074 2020 2d20 327c 0a7c 2020  832t t  - 2|.|  
│ │ │ │  0007d3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d3b0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -0007d3c0: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ -0007d3d0: 2020 3120 2020 2020 2020 2020 2020 3020    1           0 
│ │ │ │ -0007d3e0: 3220 2020 207c 0a7c 2020 2020 2020 2020  2    |.|        
│ │ │ │ +0007d3b0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0007d3c0: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ +0007d3d0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +0007d3e0: 2020 2020 3020 3220 2020 207c 0a7c 2020      0 2    |.|  
│ │ │ │  0007d3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d430: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007d440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d450: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0007d460: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0007d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007d490: 2020 2020 2020 2020 2020 2020 2020 3336                36
│ │ │ │ -0007d4a0: 3634 3436 7420 202b 2034 3239 3732 3133  6446t  + 4297213
│ │ │ │ -0007d4b0: 7420 7420 202b 2034 3730 3836 3433 7420  t t  + 4708643t 
│ │ │ │ -0007d4c0: 202b 2032 3734 3634 3933 7420 7420 202d   + 2746493t t  -
│ │ │ │ -0007d4d0: 2033 3031 347c 0a7c 2020 2020 2020 2020   3014|.|        
│ │ │ │ +0007d450: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0007d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d470: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0007d480: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007d490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d4a0: 2020 2020 3336 3634 3436 7420 202b 2034      366446t  + 4
│ │ │ │ +0007d4b0: 3239 3732 3133 7420 7420 202b 2034 3730  297213t t  + 470
│ │ │ │ +0007d4c0: 3836 3433 7420 202b 2032 3734 3634 3933  8643t  + 2746493
│ │ │ │ +0007d4d0: 7420 7420 202d 2033 3031 347c 0a7c 2020  t t  - 3014|.|  
│ │ │ │  0007d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d4f0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0007d500: 2030 2031 2020 2020 2020 2020 2020 2031   0 1           1
│ │ │ │ -0007d510: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ -0007d520: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d4f0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +0007d500: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +0007d510: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0007d520: 2030 2032 2020 2020 2020 207c 0a7c 2020   0 2       |.|  
│ │ │ │  0007d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d570: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d590: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0007d5a0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0007d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d5c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007d5d0: 2020 2020 2020 2020 2020 2020 2020 3136                16
│ │ │ │ -0007d5e0: 3332 3733 7420 202d 2031 3138 3030 3735  3273t  - 1180075
│ │ │ │ -0007d5f0: 7420 7420 202d 2032 3135 3035 3332 7420  t t  - 2150532t 
│ │ │ │ -0007d600: 202b 2032 3938 3835 3938 7420 7420 202b   + 2988598t t  +
│ │ │ │ -0007d610: 2034 3338 357c 0a7c 2020 2020 2020 2020   4385|.|        
│ │ │ │ +0007d590: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0007d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d5b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0007d5c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d5e0: 2020 2020 3136 3332 3733 7420 202d 2031      163273t  - 1
│ │ │ │ +0007d5f0: 3138 3030 3735 7420 7420 202d 2032 3135  180075t t  - 215
│ │ │ │ +0007d600: 3035 3332 7420 202b 2032 3938 3835 3938  0532t  + 2988598
│ │ │ │ +0007d610: 7420 7420 202b 2034 3338 357c 0a7c 2020  t t  + 4385|.|  
│ │ │ │  0007d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d630: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -0007d640: 2030 2031 2020 2020 2020 2020 2020 2031   0 1           1
│ │ │ │ -0007d650: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ -0007d660: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d630: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +0007d640: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +0007d650: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0007d660: 2030 2032 2020 2020 2020 207c 0a7c 2020   0 2       |.|  
│ │ │ │  0007d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d6b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d6b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d6d0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0007d6d0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  0007d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d6f0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0007d700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007d710: 2020 2020 2020 2020 2020 2020 2020 3231                21
│ │ │ │ -0007d720: 3239 3339 3274 2020 2d20 3237 3632 3634  29392t  - 276264
│ │ │ │ -0007d730: 3074 2074 2020 2b20 3230 3638 3933 3574  0t t  + 2068935t
│ │ │ │ -0007d740: 2020 2d20 3636 3533 3335 7420 7420 202b    - 665335t t  +
│ │ │ │ -0007d750: 2031 3635 347c 0a7c 2020 2020 2020 2020   1654|.|        
│ │ │ │ +0007d6f0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0007d700: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d720: 2020 2020 3231 3239 3339 3274 2020 2d20      2129392t  - 
│ │ │ │ +0007d730: 3237 3632 3634 3074 2074 2020 2b20 3230  2762640t t  + 20
│ │ │ │ +0007d740: 3638 3933 3574 2020 2d20 3636 3533 3335  68935t  - 665335
│ │ │ │ +0007d750: 7420 7420 202b 2031 3635 347c 0a7c 2020  t t  + 1654|.|  
│ │ │ │  0007d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d770: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -0007d780: 2020 3020 3120 2020 2020 2020 2020 2020    0 1           
│ │ │ │ -0007d790: 3120 2020 2020 2020 2020 2030 2032 2020  1          0 2  
│ │ │ │ -0007d7a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d770: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0007d780: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ +0007d790: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +0007d7a0: 2030 2032 2020 2020 2020 207c 0a7c 2020   0 2       |.|  
│ │ │ │  0007d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d7f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d7f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d810: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0007d810: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0007d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d830: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0007d840: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007d850: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0007d860: 3231 3430 3534 3674 2020 2d20 3433 3535  2140546t  - 4355
│ │ │ │ -0007d870: 3231 3174 2074 2020 2d20 3334 3635 3936  211t t  - 346596
│ │ │ │ -0007d880: 7420 202d 2035 3838 3635 3274 2074 2020  t  - 588652t t  
│ │ │ │ -0007d890: 2b20 3434 387c 0a7c 2020 2020 2020 2020  + 448|.|        
│ │ │ │ +0007d830: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0007d840: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d860: 2020 2020 2d20 3231 3430 3534 3674 2020      - 2140546t  
│ │ │ │ +0007d870: 2d20 3433 3535 3231 3174 2074 2020 2d20  - 4355211t t  - 
│ │ │ │ +0007d880: 3334 3635 3936 7420 202d 2035 3838 3635  346596t  - 58865
│ │ │ │ +0007d890: 3274 2074 2020 2b20 3434 387c 0a7c 2020  2t t  + 448|.|  
│ │ │ │  0007d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d8b0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -0007d8c0: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ -0007d8d0: 2031 2020 2020 2020 2020 2020 3020 3220   1          0 2 
│ │ │ │ -0007d8e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d8b0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0007d8c0: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ +0007d8d0: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +0007d8e0: 2020 3020 3220 2020 2020 207c 0a7c 2020    0 2      |.|  
│ │ │ │  0007d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d930: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d930: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d950: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0007d950: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0007d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d970: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0007d980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007d990: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -0007d9a0: 3237 3539 3932 3374 2020 2b20 3337 3139  2759923t  + 3719
│ │ │ │ -0007d9b0: 3736 3774 2074 2020 2d20 3139 3639 3031  767t t  - 196901
│ │ │ │ -0007d9c0: 3574 2020 2d20 3936 3932 3632 7420 7420  5t  - 969262t t 
│ │ │ │ -0007d9d0: 202d 2038 317c 0a7c 2020 2020 2020 2020   - 81|.|        
│ │ │ │ +0007d970: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0007d980: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007d9a0: 2020 2020 2d20 3237 3539 3932 3374 2020      - 2759923t  
│ │ │ │ +0007d9b0: 2b20 3337 3139 3736 3774 2074 2020 2d20  + 3719767t t  - 
│ │ │ │ +0007d9c0: 3139 3639 3031 3574 2020 2d20 3936 3932  1969015t  - 9692
│ │ │ │ +0007d9d0: 3632 7420 7420 202d 2038 317c 0a7c 2020  62t t  - 81|.|  
│ │ │ │  0007d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007d9f0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -0007da00: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ -0007da10: 2020 3120 2020 2020 2020 2020 2030 2032    1          0 2
│ │ │ │ -0007da20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007d9f0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0007da00: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ +0007da10: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +0007da20: 2020 2030 2032 2020 2020 207c 0a7c 2020     0 2     |.|  
│ │ │ │  0007da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007da70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007da80: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0007da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007daa0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0007da70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007da90: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0007daa0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0007dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007dac0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007dad0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -0007dae0: 202b 2032 3535 3732 3031 7420 7420 202b   + 2557201t t  +
│ │ │ │ -0007daf0: 2038 3937 3538 3274 2020 2d20 3139 3535   897582t  - 1955
│ │ │ │ -0007db00: 3636 3774 2074 2020 2d20 3431 3632 3833  667t t  - 416283
│ │ │ │ -0007db10: 3274 2074 207c 0a7c 2020 2020 2020 2020  2t t |.|        
│ │ │ │ -0007db20: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -0007db30: 2020 2020 2020 2020 2020 2030 2031 2020             0 1  
│ │ │ │ -0007db40: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -0007db50: 2020 2020 3020 3220 2020 2020 2020 2020      0 2         
│ │ │ │ -0007db60: 2020 3120 327c 0a7c 2020 2020 2020 2020    1 2|.|        
│ │ │ │ -0007db70: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ -0007db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007dac0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007dae0: 2020 2020 7420 202b 2032 3535 3732 3031      t  + 2557201
│ │ │ │ +0007daf0: 7420 7420 202b 2038 3937 3538 3274 2020  t t  + 897582t  
│ │ │ │ +0007db00: 2d20 3139 3535 3636 3774 2074 2020 2d20  - 1955667t t  - 
│ │ │ │ +0007db10: 3431 3632 3833 3274 2074 207c 0a7c 2020  4162832t t |.|  
│ │ │ │ +0007db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007db30: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +0007db40: 2030 2031 2020 2020 2020 2020 2020 3120   0 1          1 
│ │ │ │ +0007db50: 2020 2020 2020 2020 2020 3020 3220 2020            0 2   
│ │ │ │ +0007db60: 2020 2020 2020 2020 3120 327c 0a7c 2020          1 2|.|  
│ │ │ │ +0007db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007db80: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  0007db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007dbb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007dbb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007dc00: 2020 2020 207c 0a7c 6f35 203a 2052 6174       |.|o5 : Rat
│ │ │ │ -0007dc10: 696f 6e61 6c4d 6170 2028 7175 6164 7261  ionalMap (quadra
│ │ │ │ -0007dc20: 7469 6320 7261 7469 6f6e 616c 206d 6170  tic rational map
│ │ │ │ -0007dc30: 2066 726f 6d20 5050 5e36 2074 6f20 5050   from PP^6 to PP
│ │ │ │ -0007dc40: 5e39 2920 2020 2020 2020 2020 2020 2020  ^9)             
│ │ │ │ -0007dc50: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0007dc00: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +0007dc10: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +0007dc20: 7175 6164 7261 7469 6320 7261 7469 6f6e  quadratic ration
│ │ │ │ +0007dc30: 616c 206d 6170 2066 726f 6d20 5050 5e36  al map from PP^6
│ │ │ │ +0007dc40: 2074 6f20 5050 5e39 2920 2020 2020 2020   to PP^9)       
│ │ │ │ +0007dc50: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  0007dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007dc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007dc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007dca0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0007dcb0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0007dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0007dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007dcc0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0007dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007dce0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0007dcf0: 2020 2020 207c 0a7c 3439 3474 2074 2020       |.|494t t  
│ │ │ │ -0007dd00: 2b20 3233 3630 3936 3174 2020 2d20 3233  + 2360961t  - 23
│ │ │ │ -0007dd10: 3030 3138 7420 7420 202b 2034 3132 3839  0018t t  + 41289
│ │ │ │ -0007dd20: 3136 7420 7420 202d 2034 3231 3030 3274  16t t  - 421002t
│ │ │ │ -0007dd30: 2074 2020 2b20 3438 3233 3531 3074 2020   t  + 4823510t  
│ │ │ │ -0007dd40: 2b20 3139 307c 0a7c 2020 2020 3120 3220  + 190|.|    1 2 
│ │ │ │ -0007dd50: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0007dd60: 2020 2020 2030 2033 2020 2020 2020 2020       0 3        
│ │ │ │ -0007dd70: 2020 2031 2033 2020 2020 2020 2020 2020     1 3          
│ │ │ │ -0007dd80: 3220 3320 2020 2020 2020 2020 2020 3320  2 3           3 
│ │ │ │ -0007dd90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007dcf0: 2020 2020 3220 2020 2020 207c 0a7c 3439      2      |.|49
│ │ │ │ +0007dd00: 3474 2074 2020 2b20 3233 3630 3936 3174  4t t  + 2360961t
│ │ │ │ +0007dd10: 2020 2d20 3233 3030 3138 7420 7420 202b    - 230018t t  +
│ │ │ │ +0007dd20: 2034 3132 3839 3136 7420 7420 202d 2034   4128916t t  - 4
│ │ │ │ +0007dd30: 3231 3030 3274 2074 2020 2b20 3438 3233  21002t t  + 4823
│ │ │ │ +0007dd40: 3531 3074 2020 2b20 3139 307c 0a7c 2020  510t  + 190|.|  
│ │ │ │ +0007dd50: 2020 3120 3220 2020 2020 2020 2020 2020    1 2           
│ │ │ │ +0007dd60: 3220 2020 2020 2020 2020 2030 2033 2020  2          0 3  
│ │ │ │ +0007dd70: 2020 2020 2020 2020 2031 2033 2020 2020           1 3    
│ │ │ │ +0007dd80: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ +0007dd90: 2020 2020 3320 2020 2020 207c 0a7c 2020      3      |.|  
│ │ │ │  0007dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007dde0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007ddf0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0007de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007dde0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007de00: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0007de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007de20: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0007de30: 2020 2020 207c 0a7c 3734 3874 2074 2020       |.|748t t  
│ │ │ │ -0007de40: 2d20 3130 3930 3834 3774 2020 2b20 3234  - 1090847t  + 24
│ │ │ │ -0007de50: 3634 3074 2074 2020 2b20 3333 3537 3233  640t t  + 335723
│ │ │ │ -0007de60: 3874 2074 2020 2b20 3334 3031 3435 3174  8t t  + 3401451t
│ │ │ │ -0007de70: 2074 2020 2d20 3639 3036 3931 7420 202d   t  - 690691t  -
│ │ │ │ -0007de80: 2031 3039 327c 0a7c 2020 2020 3120 3220   1092|.|    1 2 
│ │ │ │ -0007de90: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0007dea0: 2020 2020 3020 3320 2020 2020 2020 2020      0 3         
│ │ │ │ -0007deb0: 2020 3120 3320 2020 2020 2020 2020 2020    1 3           
│ │ │ │ -0007dec0: 3220 3320 2020 2020 2020 2020 2033 2020  2 3          3  
│ │ │ │ -0007ded0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007de30: 2020 2032 2020 2020 2020 207c 0a7c 3734     2       |.|74
│ │ │ │ +0007de40: 3874 2074 2020 2d20 3130 3930 3834 3774  8t t  - 1090847t
│ │ │ │ +0007de50: 2020 2b20 3234 3634 3074 2074 2020 2b20    + 24640t t  + 
│ │ │ │ +0007de60: 3333 3537 3233 3874 2074 2020 2b20 3334  3357238t t  + 34
│ │ │ │ +0007de70: 3031 3435 3174 2074 2020 2d20 3639 3036  01451t t  - 6906
│ │ │ │ +0007de80: 3931 7420 202d 2031 3039 327c 0a7c 2020  91t  - 1092|.|  
│ │ │ │ +0007de90: 2020 3120 3220 2020 2020 2020 2020 2020    1 2           
│ │ │ │ +0007dea0: 3220 2020 2020 2020 2020 3020 3320 2020  2         0 3   
│ │ │ │ +0007deb0: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ +0007dec0: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ +0007ded0: 2020 2033 2020 2020 2020 207c 0a7c 2020     3       |.|  
│ │ │ │  0007dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007df20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007df30: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0007df20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007df30: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  0007df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007df60: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0007df70: 2020 2020 207c 0a7c 3036 7420 7420 202b       |.|06t t  +
│ │ │ │ -0007df80: 2031 3533 3733 3030 7420 202d 2033 3533   1537300t  - 353
│ │ │ │ -0007df90: 3530 3032 7420 7420 202d 2032 3931 3331  5002t t  - 29131
│ │ │ │ -0007dfa0: 3735 7420 7420 202b 2031 3836 3935 3936  75t t  + 1869596
│ │ │ │ -0007dfb0: 7420 7420 202d 2032 3735 3132 3436 7420  t t  - 2751246t 
│ │ │ │ -0007dfc0: 202d 2036 357c 0a7c 2020 2031 2032 2020   - 65|.|   1 2  
│ │ │ │ -0007dfd0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -0007dfe0: 2020 2020 2030 2033 2020 2020 2020 2020       0 3        
│ │ │ │ -0007dff0: 2020 2031 2033 2020 2020 2020 2020 2020     1 3          
│ │ │ │ -0007e000: 2032 2033 2020 2020 2020 2020 2020 2033   2 3           3
│ │ │ │ -0007e010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007df70: 2020 2020 2032 2020 2020 207c 0a7c 3036       2     |.|06
│ │ │ │ +0007df80: 7420 7420 202b 2031 3533 3733 3030 7420  t t  + 1537300t 
│ │ │ │ +0007df90: 202d 2033 3533 3530 3032 7420 7420 202d   - 3535002t t  -
│ │ │ │ +0007dfa0: 2032 3931 3331 3735 7420 7420 202b 2031   2913175t t  + 1
│ │ │ │ +0007dfb0: 3836 3935 3936 7420 7420 202d 2032 3735  869596t t  - 275
│ │ │ │ +0007dfc0: 3132 3436 7420 202d 2036 357c 0a7c 2020  1246t  - 65|.|  
│ │ │ │ +0007dfd0: 2031 2032 2020 2020 2020 2020 2020 2032   1 2           2
│ │ │ │ +0007dfe0: 2020 2020 2020 2020 2020 2030 2033 2020             0 3  
│ │ │ │ +0007dff0: 2020 2020 2020 2020 2031 2033 2020 2020           1 3    
│ │ │ │ +0007e000: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ +0007e010: 2020 2020 2033 2020 2020 207c 0a7c 2020       3     |.|  
│ │ │ │  0007e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e060: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007e070: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0007e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e060: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e080: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0007e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e0b0: 2020 2032 207c 0a7c 3535 3231 3237 7420     2 |.|552127t 
│ │ │ │ -0007e0c0: 7420 202d 2033 3333 3438 3530 7420 202d  t  - 3334850t  -
│ │ │ │ -0007e0d0: 2032 3534 3632 3331 7420 7420 202d 2034   2546231t t  - 4
│ │ │ │ -0007e0e0: 3636 3630 3338 7420 7420 202d 2033 3237  666038t t  - 327
│ │ │ │ -0007e0f0: 3437 3933 7420 7420 202d 2032 3537 3133  4793t t  - 25713
│ │ │ │ -0007e100: 3837 7420 207c 0a7c 2020 2020 2020 2031  87t  |.|       1
│ │ │ │ -0007e110: 2032 2020 2020 2020 2020 2020 2032 2020   2           2  
│ │ │ │ -0007e120: 2020 2020 2020 2020 2030 2033 2020 2020           0 3    
│ │ │ │ -0007e130: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ -0007e140: 2020 2020 2032 2033 2020 2020 2020 2020       2 3        
│ │ │ │ -0007e150: 2020 2033 207c 0a7c 2020 2020 2020 2020     3 |.|        
│ │ │ │ +0007e0b0: 2020 2020 2020 2020 2032 207c 0a7c 3535           2 |.|55
│ │ │ │ +0007e0c0: 3231 3237 7420 7420 202d 2033 3333 3438  2127t t  - 33348
│ │ │ │ +0007e0d0: 3530 7420 202d 2032 3534 3632 3331 7420  50t  - 2546231t 
│ │ │ │ +0007e0e0: 7420 202d 2034 3636 3630 3338 7420 7420  t  - 4666038t t 
│ │ │ │ +0007e0f0: 202d 2033 3237 3437 3933 7420 7420 202d   - 3274793t t  -
│ │ │ │ +0007e100: 2032 3537 3133 3837 7420 207c 0a7c 2020   2571387t  |.|  
│ │ │ │ +0007e110: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ +0007e120: 2020 2032 2020 2020 2020 2020 2020 2030     2           0
│ │ │ │ +0007e130: 2033 2020 2020 2020 2020 2020 2031 2033   3           1 3
│ │ │ │ +0007e140: 2020 2020 2020 2020 2020 2032 2033 2020             2 3  
│ │ │ │ +0007e150: 2020 2020 2020 2020 2033 207c 0a7c 2020           3 |.|  
│ │ │ │  0007e160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e1a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007e1b0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0007e1a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007e1b0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  0007e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e1e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0007e1f0: 2020 2020 207c 0a7c 3639 3074 2074 2020       |.|690t t  
│ │ │ │ -0007e200: 2d20 3936 3738 3335 7420 202b 2034 3534  - 967835t  + 454
│ │ │ │ -0007e210: 3637 3038 7420 7420 202d 2032 3335 3732  6708t t  - 23572
│ │ │ │ -0007e220: 3074 2074 2020 2b20 3339 3430 3730 3374  0t t  + 3940703t
│ │ │ │ -0007e230: 2074 2020 2b20 3437 3134 3131 3974 2020   t  + 4714119t  
│ │ │ │ -0007e240: 2d20 3232 367c 0a7c 2020 2020 3120 3220  - 226|.|    1 2 
│ │ │ │ -0007e250: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -0007e260: 2020 2020 2030 2033 2020 2020 2020 2020       0 3        
│ │ │ │ -0007e270: 2020 3120 3320 2020 2020 2020 2020 2020    1 3           
│ │ │ │ -0007e280: 3220 3320 2020 2020 2020 2020 2020 3320  2 3           3 
│ │ │ │ -0007e290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e1f0: 2020 2020 3220 2020 2020 207c 0a7c 3639      2      |.|69
│ │ │ │ +0007e200: 3074 2074 2020 2d20 3936 3738 3335 7420  0t t  - 967835t 
│ │ │ │ +0007e210: 202b 2034 3534 3637 3038 7420 7420 202d   + 4546708t t  -
│ │ │ │ +0007e220: 2032 3335 3732 3074 2074 2020 2b20 3339   235720t t  + 39
│ │ │ │ +0007e230: 3430 3730 3374 2074 2020 2b20 3437 3134  40703t t  + 4714
│ │ │ │ +0007e240: 3131 3974 2020 2d20 3232 367c 0a7c 2020  119t  - 226|.|  
│ │ │ │ +0007e250: 2020 3120 3220 2020 2020 2020 2020 2032    1 2          2
│ │ │ │ +0007e260: 2020 2020 2020 2020 2020 2030 2033 2020             0 3  
│ │ │ │ +0007e270: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ +0007e280: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ +0007e290: 2020 2020 3320 2020 2020 207c 0a7c 2020      3      |.|  
│ │ │ │  0007e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e2e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007e2f0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0007e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e2e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e300: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0007e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e320: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0007e330: 2020 2020 207c 0a7c 3634 3974 2074 2020       |.|649t t  
│ │ │ │ -0007e340: 2d20 3435 3336 3331 3574 2020 2b20 3233  - 4536315t  + 23
│ │ │ │ -0007e350: 3739 3630 3074 2074 2020 2d20 3331 3230  79600t t  - 3120
│ │ │ │ -0007e360: 3133 7420 7420 202d 2036 3239 3937 3674  13t t  - 629976t
│ │ │ │ -0007e370: 2074 2020 2b20 3137 3730 3839 3774 2020   t  + 1770897t  
│ │ │ │ -0007e380: 2d20 3435 337c 0a7c 2020 2020 3120 3220  - 453|.|    1 2 
│ │ │ │ -0007e390: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0007e3a0: 2020 2020 2020 3020 3320 2020 2020 2020        0 3       
│ │ │ │ -0007e3b0: 2020 2031 2033 2020 2020 2020 2020 2020     1 3          
│ │ │ │ -0007e3c0: 3220 3320 2020 2020 2020 2020 2020 3320  2 3           3 
│ │ │ │ -0007e3d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e330: 2020 2020 3220 2020 2020 207c 0a7c 3634      2      |.|64
│ │ │ │ +0007e340: 3974 2074 2020 2d20 3435 3336 3331 3574  9t t  - 4536315t
│ │ │ │ +0007e350: 2020 2b20 3233 3739 3630 3074 2074 2020    + 2379600t t  
│ │ │ │ +0007e360: 2d20 3331 3230 3133 7420 7420 202d 2036  - 312013t t  - 6
│ │ │ │ +0007e370: 3239 3937 3674 2074 2020 2b20 3137 3730  29976t t  + 1770
│ │ │ │ +0007e380: 3839 3774 2020 2d20 3435 337c 0a7c 2020  897t  - 453|.|  
│ │ │ │ +0007e390: 2020 3120 3220 2020 2020 2020 2020 2020    1 2           
│ │ │ │ +0007e3a0: 3220 2020 2020 2020 2020 2020 3020 3320  2           0 3 
│ │ │ │ +0007e3b0: 2020 2020 2020 2020 2031 2033 2020 2020           1 3    
│ │ │ │ +0007e3c0: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ +0007e3d0: 2020 2020 3320 2020 2020 207c 0a7c 2020      3      |.|  
│ │ │ │  0007e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e420: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007e430: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0007e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e440: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0007e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e470: 3220 2020 207c 0a7c 3831 3374 2074 2020  2    |.|813t t  
│ │ │ │ -0007e480: 2b20 3330 3937 3837 3074 2020 2b20 3433  + 3097870t  + 43
│ │ │ │ -0007e490: 3330 3737 3274 2074 2020 2b20 3431 3036  30772t t  + 4106
│ │ │ │ -0007e4a0: 3438 3074 2074 2020 2d20 3230 3538 3833  480t t  - 205883
│ │ │ │ -0007e4b0: 3874 2074 2020 2b20 3439 3232 3830 3574  8t t  + 4922805t
│ │ │ │ -0007e4c0: 2020 2b20 337c 0a7c 2020 2020 3120 3220    + 3|.|    1 2 
│ │ │ │ -0007e4d0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0007e4e0: 2020 2020 2020 3020 3320 2020 2020 2020        0 3       
│ │ │ │ -0007e4f0: 2020 2020 3120 3320 2020 2020 2020 2020      1 3         
│ │ │ │ -0007e500: 2020 3220 3320 2020 2020 2020 2020 2020    2 3           
│ │ │ │ -0007e510: 3320 2020 207c 0a7c 2020 2020 2020 2020  3    |.|        
│ │ │ │ +0007e470: 2020 2020 2020 3220 2020 207c 0a7c 3831        2    |.|81
│ │ │ │ +0007e480: 3374 2074 2020 2b20 3330 3937 3837 3074  3t t  + 3097870t
│ │ │ │ +0007e490: 2020 2b20 3433 3330 3737 3274 2074 2020    + 4330772t t  
│ │ │ │ +0007e4a0: 2b20 3431 3036 3438 3074 2074 2020 2d20  + 4106480t t  - 
│ │ │ │ +0007e4b0: 3230 3538 3833 3874 2074 2020 2b20 3439  2058838t t  + 49
│ │ │ │ +0007e4c0: 3232 3830 3574 2020 2b20 337c 0a7c 2020  22805t  + 3|.|  
│ │ │ │ +0007e4d0: 2020 3120 3220 2020 2020 2020 2020 2020    1 2           
│ │ │ │ +0007e4e0: 3220 2020 2020 2020 2020 2020 3020 3320  2           0 3 
│ │ │ │ +0007e4f0: 2020 2020 2020 2020 2020 3120 3320 2020            1 3   
│ │ │ │ +0007e500: 2020 2020 2020 2020 3220 3320 2020 2020          2 3     
│ │ │ │ +0007e510: 2020 2020 2020 3320 2020 207c 0a7c 2020        3    |.|  
│ │ │ │  0007e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e560: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007e570: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0007e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e560: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e580: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0007e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e5b0: 2032 2020 207c 0a7c 3135 3031 7420 7420   2   |.|1501t t 
│ │ │ │ -0007e5c0: 202b 2032 3039 3131 3330 7420 202d 2031   + 2091130t  - 1
│ │ │ │ -0007e5d0: 3937 3536 3934 7420 7420 202b 2032 3431  975694t t  + 241
│ │ │ │ -0007e5e0: 3230 3032 7420 7420 202b 2034 3735 3939  2002t t  + 47599
│ │ │ │ -0007e5f0: 3939 7420 7420 202b 2032 3933 3236 3130  99t t  + 2932610
│ │ │ │ -0007e600: 7420 202b 207c 0a7c 2020 2020 2031 2032  t  + |.|     1 2
│ │ │ │ -0007e610: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0007e620: 2020 2020 2020 2030 2033 2020 2020 2020         0 3      
│ │ │ │ -0007e630: 2020 2020 2031 2033 2020 2020 2020 2020       1 3        
│ │ │ │ -0007e640: 2020 2032 2033 2020 2020 2020 2020 2020     2 3          
│ │ │ │ -0007e650: 2033 2020 207c 0a7c 2020 2020 2020 2020   3   |.|        
│ │ │ │ +0007e5b0: 2020 2020 2020 2032 2020 207c 0a7c 3135         2   |.|15
│ │ │ │ +0007e5c0: 3031 7420 7420 202b 2032 3039 3131 3330  01t t  + 2091130
│ │ │ │ +0007e5d0: 7420 202d 2031 3937 3536 3934 7420 7420  t  - 1975694t t 
│ │ │ │ +0007e5e0: 202b 2032 3431 3230 3032 7420 7420 202b   + 2412002t t  +
│ │ │ │ +0007e5f0: 2034 3735 3939 3939 7420 7420 202b 2032   4759999t t  + 2
│ │ │ │ +0007e600: 3933 3236 3130 7420 202b 207c 0a7c 2020  932610t  + |.|  
│ │ │ │ +0007e610: 2020 2031 2032 2020 2020 2020 2020 2020     1 2          
│ │ │ │ +0007e620: 2032 2020 2020 2020 2020 2020 2030 2033   2           0 3
│ │ │ │ +0007e630: 2020 2020 2020 2020 2020 2031 2033 2020             1 3  
│ │ │ │ +0007e640: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ +0007e650: 2020 2020 2020 2033 2020 207c 0a7c 2020         3   |.|  
│ │ │ │  0007e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e6a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007e6b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0007e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e6a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007e6c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0007e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e6f0: 2032 2020 207c 0a7c 3932 3931 7420 7420   2   |.|9291t t 
│ │ │ │ -0007e700: 202b 2033 3632 3831 3234 7420 202d 2034   + 3628124t  - 4
│ │ │ │ -0007e710: 3738 3131 3935 7420 7420 202d 2034 3136  781195t t  - 416
│ │ │ │ -0007e720: 3935 3537 7420 7420 202d 2032 3730 3739  9557t t  - 27079
│ │ │ │ -0007e730: 3834 7420 7420 202d 2032 3637 3333 3935  84t t  - 2673395
│ │ │ │ -0007e740: 7420 202d 207c 0a7c 2020 2020 2031 2032  t  - |.|     1 2
│ │ │ │ -0007e750: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0007e760: 2020 2020 2020 2030 2033 2020 2020 2020         0 3      
│ │ │ │ -0007e770: 2020 2020 2031 2033 2020 2020 2020 2020       1 3        
│ │ │ │ -0007e780: 2020 2032 2033 2020 2020 2020 2020 2020     2 3          
│ │ │ │ -0007e790: 2033 2020 207c 0a7c 2020 2020 2020 2020   3   |.|        
│ │ │ │ +0007e6f0: 2020 2020 2020 2032 2020 207c 0a7c 3932         2   |.|92
│ │ │ │ +0007e700: 3931 7420 7420 202b 2033 3632 3831 3234  91t t  + 3628124
│ │ │ │ +0007e710: 7420 202d 2034 3738 3131 3935 7420 7420  t  - 4781195t t 
│ │ │ │ +0007e720: 202d 2034 3136 3935 3537 7420 7420 202d   - 4169557t t  -
│ │ │ │ +0007e730: 2032 3730 3739 3834 7420 7420 202d 2032   2707984t t  - 2
│ │ │ │ +0007e740: 3637 3333 3935 7420 202d 207c 0a7c 2020  673395t  - |.|  
│ │ │ │ +0007e750: 2020 2031 2032 2020 2020 2020 2020 2020     1 2          
│ │ │ │ +0007e760: 2032 2020 2020 2020 2020 2020 2030 2033   2           0 3
│ │ │ │ +0007e770: 2020 2020 2020 2020 2020 2031 2033 2020             1 3  
│ │ │ │ +0007e780: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ +0007e790: 2020 2020 2020 2033 2020 207c 0a7c 2020         3   |.|  
│ │ │ │  0007e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e7e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007e7f0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0007e7e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007e7f0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  0007e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e820: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0007e830: 2020 2020 207c 0a7c 202b 2032 3931 3739       |.| + 29179
│ │ │ │ -0007e840: 3734 7420 202d 2032 3431 3430 3435 7420  74t  - 2414045t 
│ │ │ │ -0007e850: 7420 202d 2033 3133 3334 3536 7420 7420  t  - 3133456t t 
│ │ │ │ -0007e860: 202b 2031 3330 3236 3974 2074 2020 2d20   + 130269t t  - 
│ │ │ │ -0007e870: 3932 3130 3235 7420 202b 2034 3932 3437  921025t  + 49247
│ │ │ │ -0007e880: 3637 7420 747c 0a7c 2020 2020 2020 2020  67t t|.|        
│ │ │ │ -0007e890: 2020 2032 2020 2020 2020 2020 2020 2030     2           0
│ │ │ │ -0007e8a0: 2033 2020 2020 2020 2020 2020 2031 2033   3           1 3
│ │ │ │ -0007e8b0: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -0007e8c0: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -0007e8d0: 2020 2030 207c 0a7c 2d2d 2d2d 2d2d 2d2d     0 |.|--------
│ │ │ │ +0007e820: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0007e830: 2020 2020 2020 2020 2020 207c 0a7c 202b             |.| +
│ │ │ │ +0007e840: 2032 3931 3739 3734 7420 202d 2032 3431   2917974t  - 241
│ │ │ │ +0007e850: 3430 3435 7420 7420 202d 2033 3133 3334  4045t t  - 31334
│ │ │ │ +0007e860: 3536 7420 7420 202b 2031 3330 3236 3974  56t t  + 130269t
│ │ │ │ +0007e870: 2074 2020 2d20 3932 3130 3235 7420 202b   t  - 921025t  +
│ │ │ │ +0007e880: 2034 3932 3437 3637 7420 747c 0a7c 2020   4924767t t|.|  
│ │ │ │ +0007e890: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0007e8a0: 2020 2020 2030 2033 2020 2020 2020 2020       0 3        
│ │ │ │ +0007e8b0: 2020 2031 2033 2020 2020 2020 2020 2020     1 3          
│ │ │ │ +0007e8c0: 3220 3320 2020 2020 2020 2020 2033 2020  2 3          3  
│ │ │ │ +0007e8d0: 2020 2020 2020 2020 2030 207c 0a7c 2d2d           0 |.|--
│ │ │ │  0007e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007e920: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ +0007e920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │  0007e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e970: 2020 2020 207c 0a7c 3234 3330 7420 7420       |.|2430t t 
│ │ │ │ -0007e980: 202b 2033 3433 3634 3833 7420 7420 202b   + 3436483t t  +
│ │ │ │ -0007e990: 2031 3533 3535 3238 7420 7420 202d 2031   1535528t t  - 1
│ │ │ │ -0007e9a0: 3038 3335 3837 7420 7420 2020 2020 2020  083587t t       
│ │ │ │ +0007e970: 2020 2020 2020 2020 2020 207c 0a7c 3234             |.|24
│ │ │ │ +0007e980: 3330 7420 7420 202b 2033 3433 3634 3833  30t t  + 3436483
│ │ │ │ +0007e990: 7420 7420 202b 2031 3533 3535 3238 7420  t t  + 1535528t 
│ │ │ │ +0007e9a0: 7420 202d 2031 3038 3335 3837 7420 7420  t  - 1083587t t 
│ │ │ │  0007e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e9c0: 2020 2020 207c 0a7c 2020 2020 2030 2034       |.|     0 4
│ │ │ │ -0007e9d0: 2020 2020 2020 2020 2020 2031 2034 2020             1 4  
│ │ │ │ -0007e9e0: 2020 2020 2020 2020 2032 2034 2020 2020           2 4    
│ │ │ │ -0007e9f0: 2020 2020 2020 2033 2034 2020 2020 2020         3 4      
│ │ │ │ +0007e9c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007e9d0: 2020 2030 2034 2020 2020 2020 2020 2020     0 4          
│ │ │ │ +0007e9e0: 2031 2034 2020 2020 2020 2020 2020 2032   1 4           2
│ │ │ │ +0007e9f0: 2034 2020 2020 2020 2020 2020 2033 2034   4           3 4
│ │ │ │  0007ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ea10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007ea10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ea60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007ea60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007eab0: 2020 2020 207c 0a7c 3432 3074 2074 2020       |.|420t t  
│ │ │ │ -0007eac0: 2d20 3239 3230 3336 3374 2074 2020 2b20  - 2920363t t  + 
│ │ │ │ -0007ead0: 3237 3036 3934 3474 2074 2020 2b20 3436  2706944t t  + 46
│ │ │ │ -0007eae0: 3236 3934 3974 2074 2020 2d20 2020 2020  26949t t  -     
│ │ │ │ -0007eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007eb00: 2020 2020 207c 0a7c 2020 2020 3020 3420       |.|    0 4 
│ │ │ │ -0007eb10: 2020 2020 2020 2020 2020 3120 3420 2020            1 4   
│ │ │ │ -0007eb20: 2020 2020 2020 2020 3220 3420 2020 2020          2 4     
│ │ │ │ -0007eb30: 2020 2020 2020 3320 3420 2020 2020 2020        3 4       
│ │ │ │ +0007eab0: 2020 2020 2020 2020 2020 207c 0a7c 3432             |.|42
│ │ │ │ +0007eac0: 3074 2074 2020 2d20 3239 3230 3336 3374  0t t  - 2920363t
│ │ │ │ +0007ead0: 2074 2020 2b20 3237 3036 3934 3474 2074   t  + 2706944t t
│ │ │ │ +0007eae0: 2020 2b20 3436 3236 3934 3974 2074 2020    + 4626949t t  
│ │ │ │ +0007eaf0: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ +0007eb00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007eb10: 2020 3020 3420 2020 2020 2020 2020 2020    0 4           
│ │ │ │ +0007eb20: 3120 3420 2020 2020 2020 2020 2020 3220  1 4           2 
│ │ │ │ +0007eb30: 3420 2020 2020 2020 2020 2020 3320 3420  4           3 4 
│ │ │ │  0007eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007eb50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007eb50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007eba0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007eba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ebd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ebf0: 2020 2020 207c 0a7c 3434 3734 7420 7420       |.|4474t t 
│ │ │ │ -0007ec00: 202d 2033 3930 3039 3639 7420 7420 202d   - 3900969t t  -
│ │ │ │ -0007ec10: 2032 3137 3732 3930 7420 7420 202b 2034   2177290t t  + 4
│ │ │ │ -0007ec20: 3931 3135 3636 7420 7420 2020 2020 2020  911566t t       
│ │ │ │ +0007ebf0: 2020 2020 2020 2020 2020 207c 0a7c 3434             |.|44
│ │ │ │ +0007ec00: 3734 7420 7420 202d 2033 3930 3039 3639  74t t  - 3900969
│ │ │ │ +0007ec10: 7420 7420 202d 2032 3137 3732 3930 7420  t t  - 2177290t 
│ │ │ │ +0007ec20: 7420 202b 2034 3931 3135 3636 7420 7420  t  + 4911566t t 
│ │ │ │  0007ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ec40: 2020 2020 207c 0a7c 2020 2020 2030 2034       |.|     0 4
│ │ │ │ -0007ec50: 2020 2020 2020 2020 2020 2031 2034 2020             1 4  
│ │ │ │ -0007ec60: 2020 2020 2020 2020 2032 2034 2020 2020           2 4    
│ │ │ │ -0007ec70: 2020 2020 2020 2033 2034 2020 2020 2020         3 4      
│ │ │ │ +0007ec40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007ec50: 2020 2030 2034 2020 2020 2020 2020 2020     0 4          
│ │ │ │ +0007ec60: 2031 2034 2020 2020 2020 2020 2020 2032   1 4           2
│ │ │ │ +0007ec70: 2034 2020 2020 2020 2020 2020 2033 2034   4           3 4
│ │ │ │  0007ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ec90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007ec90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ece0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007ece0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ed30: 2020 2020 207c 0a7c 2d20 3232 3736 3735       |.|- 227675
│ │ │ │ -0007ed40: 3874 2074 2020 2d20 3435 3938 3332 3074  8t t  - 4598320t
│ │ │ │ -0007ed50: 2074 2020 2d20 3730 3135 3274 2074 2020   t  - 70152t t  
│ │ │ │ -0007ed60: 2b20 3231 3530 3239 3374 2020 2020 2020  + 2150293t      
│ │ │ │ +0007ed30: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ +0007ed40: 3232 3736 3735 3874 2074 2020 2d20 3435  2276758t t  - 45
│ │ │ │ +0007ed50: 3938 3332 3074 2074 2020 2d20 3730 3135  98320t t  - 7015
│ │ │ │ +0007ed60: 3274 2074 2020 2b20 3231 3530 3239 3374  2t t  + 2150293t
│ │ │ │  0007ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ed80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007ed90: 2020 3020 3420 2020 2020 2020 2020 2020    0 4           
│ │ │ │ -0007eda0: 3120 3420 2020 2020 2020 2020 3220 3420  1 4         2 4 
│ │ │ │ -0007edb0: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ -0007edc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007edd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007ed80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007ed90: 2020 2020 2020 2020 3020 3420 2020 2020          0 4     
│ │ │ │ +0007eda0: 2020 2020 2020 3120 3420 2020 2020 2020        1 4       
│ │ │ │ +0007edb0: 2020 3220 3420 2020 2020 2020 2020 2020    2 4           
│ │ │ │ +0007edc0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0007edd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ee20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007ee20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ee70: 2020 2020 207c 0a7c 3037 3732 7420 7420       |.|0772t t 
│ │ │ │ -0007ee80: 202d 2033 3337 3837 3634 7420 7420 202d   - 3378764t t  -
│ │ │ │ -0007ee90: 2034 3234 3736 3137 7420 7420 202b 2031   4247617t t  + 1
│ │ │ │ -0007eea0: 3733 3133 3674 2074 2020 2b20 2020 2020  73136t t  +     
│ │ │ │ -0007eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007eec0: 2020 2020 207c 0a7c 2020 2020 2030 2034       |.|     0 4
│ │ │ │ -0007eed0: 2020 2020 2020 2020 2020 2031 2034 2020             1 4  
│ │ │ │ -0007eee0: 2020 2020 2020 2020 2032 2034 2020 2020           2 4    
│ │ │ │ -0007eef0: 2020 2020 2020 3320 3420 2020 2020 2020        3 4       
│ │ │ │ +0007ee70: 2020 2020 2020 2020 2020 207c 0a7c 3037             |.|07
│ │ │ │ +0007ee80: 3732 7420 7420 202d 2033 3337 3837 3634  72t t  - 3378764
│ │ │ │ +0007ee90: 7420 7420 202d 2034 3234 3736 3137 7420  t t  - 4247617t 
│ │ │ │ +0007eea0: 7420 202b 2031 3733 3133 3674 2074 2020  t  + 173136t t  
│ │ │ │ +0007eeb0: 2b20 2020 2020 2020 2020 2020 2020 2020  +               
│ │ │ │ +0007eec0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007eed0: 2020 2030 2034 2020 2020 2020 2020 2020     0 4          
│ │ │ │ +0007eee0: 2031 2034 2020 2020 2020 2020 2020 2032   1 4           2
│ │ │ │ +0007eef0: 2034 2020 2020 2020 2020 2020 3320 3420   4          3 4 
│ │ │ │  0007ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ef10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007ef10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ef60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007ef60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007efb0: 2020 2020 207c 0a7c 3631 7420 7420 202b       |.|61t t  +
│ │ │ │ -0007efc0: 2032 3438 3638 3436 7420 7420 202d 2032   2486846t t  - 2
│ │ │ │ -0007efd0: 3334 3434 3937 7420 7420 202d 2034 3136  344497t t  - 416
│ │ │ │ -0007efe0: 3132 3533 7420 7420 202b 2020 2020 2020  1253t t  +      
│ │ │ │ +0007efb0: 2020 2020 2020 2020 2020 207c 0a7c 3631             |.|61
│ │ │ │ +0007efc0: 7420 7420 202b 2032 3438 3638 3436 7420  t t  + 2486846t 
│ │ │ │ +0007efd0: 7420 202d 2032 3334 3434 3937 7420 7420  t  - 2344497t t 
│ │ │ │ +0007efe0: 202d 2034 3136 3132 3533 7420 7420 202b   - 4161253t t  +
│ │ │ │  0007eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f000: 2020 2020 207c 0a7c 2020 2030 2034 2020       |.|   0 4  
│ │ │ │ -0007f010: 2020 2020 2020 2020 2031 2034 2020 2020           1 4    
│ │ │ │ -0007f020: 2020 2020 2020 2032 2034 2020 2020 2020         2 4      
│ │ │ │ -0007f030: 2020 2020 2033 2034 2020 2020 2020 2020       3 4        
│ │ │ │ +0007f000: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007f010: 2030 2034 2020 2020 2020 2020 2020 2031   0 4           1
│ │ │ │ +0007f020: 2034 2020 2020 2020 2020 2020 2032 2034   4           2 4
│ │ │ │ +0007f030: 2020 2020 2020 2020 2020 2033 2034 2020             3 4  
│ │ │ │  0007f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f050: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f0a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f0a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f0f0: 2020 2020 207c 0a7c 3533 3230 3435 7420       |.|532045t 
│ │ │ │ -0007f100: 7420 202b 2031 3131 3438 3836 7420 7420  t  + 1114886t t 
│ │ │ │ -0007f110: 202b 2031 3530 3338 3835 7420 7420 202d   + 1503885t t  -
│ │ │ │ -0007f120: 2034 3135 3134 3531 7420 7420 2020 2020   4151451t t     
│ │ │ │ -0007f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f140: 2020 2020 207c 0a7c 2020 2020 2020 2030       |.|       0
│ │ │ │ -0007f150: 2034 2020 2020 2020 2020 2020 2031 2034   4           1 4
│ │ │ │ -0007f160: 2020 2020 2020 2020 2020 2032 2034 2020             2 4  
│ │ │ │ -0007f170: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +0007f0f0: 2020 2020 2020 2020 2020 207c 0a7c 3533             |.|53
│ │ │ │ +0007f100: 3230 3435 7420 7420 202b 2031 3131 3438  2045t t  + 11148
│ │ │ │ +0007f110: 3836 7420 7420 202b 2031 3530 3338 3835  86t t  + 1503885
│ │ │ │ +0007f120: 7420 7420 202d 2034 3135 3134 3531 7420  t t  - 4151451t 
│ │ │ │ +0007f130: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +0007f140: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007f150: 2020 2020 2030 2034 2020 2020 2020 2020       0 4        
│ │ │ │ +0007f160: 2020 2031 2034 2020 2020 2020 2020 2020     1 4          
│ │ │ │ +0007f170: 2032 2034 2020 2020 2020 2020 2020 2033   2 4           3
│ │ │ │  0007f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f190: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f190: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f1e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f1e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f230: 2020 2020 207c 0a7c 3432 3138 3931 3374       |.|4218913t
│ │ │ │ -0007f240: 2074 2020 2d20 3232 3934 3936 7420 7420   t  - 229496t t 
│ │ │ │ -0007f250: 202d 2031 3531 3638 3139 7420 7420 202d   - 1516819t t  -
│ │ │ │ -0007f260: 2032 3938 3637 3031 7420 7420 2020 2020   2986701t t     
│ │ │ │ -0007f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f280: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007f290: 3020 3420 2020 2020 2020 2020 2031 2034  0 4          1 4
│ │ │ │ -0007f2a0: 2020 2020 2020 2020 2020 2032 2034 2020             2 4  
│ │ │ │ -0007f2b0: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +0007f230: 2020 2020 2020 2020 2020 207c 0a7c 3432             |.|42
│ │ │ │ +0007f240: 3138 3931 3374 2074 2020 2d20 3232 3934  18913t t  - 2294
│ │ │ │ +0007f250: 3936 7420 7420 202d 2031 3531 3638 3139  96t t  - 1516819
│ │ │ │ +0007f260: 7420 7420 202d 2032 3938 3637 3031 7420  t t  - 2986701t 
│ │ │ │ +0007f270: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +0007f280: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007f290: 2020 2020 2020 3020 3420 2020 2020 2020        0 4       
│ │ │ │ +0007f2a0: 2020 2031 2034 2020 2020 2020 2020 2020     1 4          
│ │ │ │ +0007f2b0: 2032 2034 2020 2020 2020 2020 2020 2033   2 4           3
│ │ │ │  0007f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f2d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f2d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f320: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f320: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f370: 2020 2020 207c 0a7c 3237 3739 3036 3874       |.|2779068t
│ │ │ │ -0007f380: 2074 2020 2d20 3338 3239 3430 3674 2074   t  - 3829406t t
│ │ │ │ -0007f390: 2020 2b20 3237 3931 3938 3774 2074 2020    + 2791987t t  
│ │ │ │ -0007f3a0: 2b20 3132 3239 3630 3274 2020 2020 2020  + 1229602t      
│ │ │ │ +0007f370: 2020 2020 2020 2020 2020 207c 0a7c 3237             |.|27
│ │ │ │ +0007f380: 3739 3036 3874 2074 2020 2d20 3338 3239  79068t t  - 3829
│ │ │ │ +0007f390: 3430 3674 2074 2020 2b20 3237 3931 3938  406t t  + 279198
│ │ │ │ +0007f3a0: 3774 2074 2020 2b20 3132 3239 3630 3274  7t t  + 1229602t
│ │ │ │  0007f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f3c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007f3d0: 3020 3420 2020 2020 2020 2020 2020 3120  0 4           1 
│ │ │ │ -0007f3e0: 3420 2020 2020 2020 2020 2020 3220 3420  4           2 4 
│ │ │ │ -0007f3f0: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ -0007f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f410: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f3c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007f3d0: 2020 2020 2020 3020 3420 2020 2020 2020        0 4       
│ │ │ │ +0007f3e0: 2020 2020 3120 3420 2020 2020 2020 2020      1 4         
│ │ │ │ +0007f3f0: 2020 3220 3420 2020 2020 2020 2020 2020    2 4           
│ │ │ │ +0007f400: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0007f410: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f460: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f4b0: 2020 2020 207c 0a7c 2020 2b20 3236 3838       |.|  + 2688
│ │ │ │ -0007f4c0: 3434 7420 7420 202d 2031 3137 3938 3831  44t t  - 1179881
│ │ │ │ -0007f4d0: 7420 7420 202d 2031 3938 3432 3532 7420  t t  - 1984252t 
│ │ │ │ -0007f4e0: 7420 202b 2038 3532 3539 7420 2020 2020  t  + 85259t     
│ │ │ │ -0007f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f500: 2020 2020 207c 0a7c 3420 2020 2020 2020       |.|4       
│ │ │ │ -0007f510: 2020 2031 2034 2020 2020 2020 2020 2020     1 4          
│ │ │ │ -0007f520: 2032 2034 2020 2020 2020 2020 2020 2033   2 4           3
│ │ │ │ -0007f530: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0007f4b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007f4c0: 2b20 3236 3838 3434 7420 7420 202d 2031  + 268844t t  - 1
│ │ │ │ +0007f4d0: 3137 3938 3831 7420 7420 202d 2031 3938  179881t t  - 198
│ │ │ │ +0007f4e0: 3432 3532 7420 7420 202b 2038 3532 3539  4252t t  + 85259
│ │ │ │ +0007f4f0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +0007f500: 2020 2020 2020 2020 2020 207c 0a7c 3420             |.|4 
│ │ │ │ +0007f510: 2020 2020 2020 2020 2031 2034 2020 2020           1 4    
│ │ │ │ +0007f520: 2020 2020 2020 2032 2034 2020 2020 2020         2 4      
│ │ │ │ +0007f530: 2020 2020 2033 2034 2020 2020 2020 2020       3 4        
│ │ │ │  0007f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f550: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0007f550: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  0007f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007f570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007f580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007f590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007f5a0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0007f5b0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0007f5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0007f5b0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0007f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f5f0: 2020 2020 207c 0a7c 2b20 3135 3538 3937       |.|+ 155897
│ │ │ │ -0007f600: 3674 2020 2d20 3331 3036 3839 3074 2074  6t  - 3106890t t
│ │ │ │ -0007f610: 2020 2d20 3138 3237 3435 3574 2074 2020    - 1827455t t  
│ │ │ │ -0007f620: 2b20 3439 3533 3931 3174 2074 2020 2b20  + 4953911t t  + 
│ │ │ │ -0007f630: 3336 3638 3039 3374 2074 2020 2d20 3338  3668093t t  - 38
│ │ │ │ -0007f640: 3831 3237 377c 0a7c 2020 2020 2020 2020  81277|.|        
│ │ │ │ -0007f650: 2020 3420 2020 2020 2020 2020 2020 3020    4           0 
│ │ │ │ -0007f660: 3520 2020 2020 2020 2020 2020 3120 3520  5           1 5 
│ │ │ │ -0007f670: 2020 2020 2020 2020 2020 3220 3520 2020            2 5   
│ │ │ │ -0007f680: 2020 2020 2020 2020 3320 3520 2020 2020          3 5     
│ │ │ │ -0007f690: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f5f0: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ +0007f600: 3135 3538 3937 3674 2020 2d20 3331 3036  1558976t  - 3106
│ │ │ │ +0007f610: 3839 3074 2074 2020 2d20 3138 3237 3435  890t t  - 182745
│ │ │ │ +0007f620: 3574 2074 2020 2b20 3439 3533 3931 3174  5t t  + 4953911t
│ │ │ │ +0007f630: 2074 2020 2b20 3336 3638 3039 3374 2074   t  + 3668093t t
│ │ │ │ +0007f640: 2020 2d20 3338 3831 3237 377c 0a7c 2020    - 3881277|.|  
│ │ │ │ +0007f650: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +0007f660: 2020 2020 3020 3520 2020 2020 2020 2020      0 5         
│ │ │ │ +0007f670: 2020 3120 3520 2020 2020 2020 2020 2020    1 5           
│ │ │ │ +0007f680: 3220 3520 2020 2020 2020 2020 2020 3320  2 5           3 
│ │ │ │ +0007f690: 3520 2020 2020 2020 2020 207c 0a7c 2020  5          |.|  
│ │ │ │  0007f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f6e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007f6f0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0007f6e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007f6f0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  0007f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f730: 2020 2020 207c 0a7c 2031 3530 3036 3338       |.| 1500638
│ │ │ │ -0007f740: 7420 202d 2031 3736 3336 3974 2074 2020  t  - 176369t t  
│ │ │ │ -0007f750: 2d20 3231 3932 3332 3974 2074 2020 2d20  - 2192329t t  - 
│ │ │ │ -0007f760: 3233 3235 3438 3974 2074 2020 2b20 3131  2325489t t  + 11
│ │ │ │ -0007f770: 3832 3935 3274 2074 2020 2b20 3731 3632  82952t t  + 7162
│ │ │ │ -0007f780: 3539 7420 747c 0a7c 2020 2020 2020 2020  59t t|.|        
│ │ │ │ -0007f790: 2034 2020 2020 2020 2020 2020 3020 3520   4          0 5 
│ │ │ │ -0007f7a0: 2020 2020 2020 2020 2020 3120 3520 2020            1 5   
│ │ │ │ -0007f7b0: 2020 2020 2020 2020 3220 3520 2020 2020          2 5     
│ │ │ │ -0007f7c0: 2020 2020 2020 3320 3520 2020 2020 2020        3 5       
│ │ │ │ -0007f7d0: 2020 2034 207c 0a7c 2020 2020 2020 2020     4 |.|        
│ │ │ │ +0007f730: 2020 2020 2020 2020 2020 207c 0a7c 2031             |.| 1
│ │ │ │ +0007f740: 3530 3036 3338 7420 202d 2031 3736 3336  500638t  - 17636
│ │ │ │ +0007f750: 3974 2074 2020 2d20 3231 3932 3332 3974  9t t  - 2192329t
│ │ │ │ +0007f760: 2074 2020 2d20 3233 3235 3438 3974 2074   t  - 2325489t t
│ │ │ │ +0007f770: 2020 2b20 3131 3832 3935 3274 2074 2020    + 1182952t t  
│ │ │ │ +0007f780: 2b20 3731 3632 3539 7420 747c 0a7c 2020  + 716259t t|.|  
│ │ │ │ +0007f790: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +0007f7a0: 2020 3020 3520 2020 2020 2020 2020 2020    0 5           
│ │ │ │ +0007f7b0: 3120 3520 2020 2020 2020 2020 2020 3220  1 5           2 
│ │ │ │ +0007f7c0: 3520 2020 2020 2020 2020 2020 3320 3520  5           3 5 
│ │ │ │ +0007f7d0: 2020 2020 2020 2020 2034 207c 0a7c 2020           4 |.|  
│ │ │ │  0007f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f820: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007f830: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0007f820: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007f830: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0007f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f870: 2020 2020 207c 0a7c 2d20 3230 3432 3735       |.|- 204275
│ │ │ │ -0007f880: 3574 2020 2d20 3232 3138 3039 3874 2074  5t  - 2218098t t
│ │ │ │ -0007f890: 2020 2d20 3337 3935 3831 3874 2074 2020    - 3795818t t  
│ │ │ │ -0007f8a0: 2b20 3238 3039 3331 3674 2074 2020 2b20  + 2809316t t  + 
│ │ │ │ -0007f8b0: 3130 3331 3937 3174 2074 2020 2d20 3536  1031971t t  - 56
│ │ │ │ -0007f8c0: 3938 3837 747c 0a7c 2020 2020 2020 2020  9887t|.|        
│ │ │ │ -0007f8d0: 2020 3420 2020 2020 2020 2020 2020 3020    4           0 
│ │ │ │ -0007f8e0: 3520 2020 2020 2020 2020 2020 3120 3520  5           1 5 
│ │ │ │ -0007f8f0: 2020 2020 2020 2020 2020 3220 3520 2020            2 5   
│ │ │ │ -0007f900: 2020 2020 2020 2020 3320 3520 2020 2020          3 5     
│ │ │ │ -0007f910: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007f870: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ +0007f880: 3230 3432 3735 3574 2020 2d20 3232 3138  2042755t  - 2218
│ │ │ │ +0007f890: 3039 3874 2074 2020 2d20 3337 3935 3831  098t t  - 379581
│ │ │ │ +0007f8a0: 3874 2074 2020 2b20 3238 3039 3331 3674  8t t  + 2809316t
│ │ │ │ +0007f8b0: 2074 2020 2b20 3130 3331 3937 3174 2074   t  + 1031971t t
│ │ │ │ +0007f8c0: 2020 2d20 3536 3938 3837 747c 0a7c 2020    - 569887t|.|  
│ │ │ │ +0007f8d0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +0007f8e0: 2020 2020 3020 3520 2020 2020 2020 2020      0 5         
│ │ │ │ +0007f8f0: 2020 3120 3520 2020 2020 2020 2020 2020    1 5           
│ │ │ │ +0007f900: 3220 3520 2020 2020 2020 2020 2020 3320  2 5           3 
│ │ │ │ +0007f910: 3520 2020 2020 2020 2020 207c 0a7c 2020  5          |.|  
│ │ │ │  0007f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f960: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007f970: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0007f960: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007f970: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0007f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f9b0: 2020 2020 207c 0a7c 7420 202d 2031 3737       |.|t  - 177
│ │ │ │ -0007f9c0: 3937 3574 2020 2d20 3831 3330 3574 2074  975t  - 81305t t
│ │ │ │ -0007f9d0: 2020 2d20 3433 3030 3130 3674 2074 2020    - 4300106t t  
│ │ │ │ -0007f9e0: 2b20 3736 3939 3974 2074 2020 2b20 3138  + 76999t t  + 18
│ │ │ │ -0007f9f0: 3034 3132 3574 2074 2020 2b20 3231 3833  04125t t  + 2183
│ │ │ │ -0007fa00: 3536 3574 207c 0a7c 2034 2020 2020 2020  565t |.| 4      
│ │ │ │ -0007fa10: 2020 2020 3420 2020 2020 2020 2020 3020      4         0 
│ │ │ │ -0007fa20: 3520 2020 2020 2020 2020 2020 3120 3520  5           1 5 
│ │ │ │ -0007fa30: 2020 2020 2020 2020 3220 3520 2020 2020          2 5     
│ │ │ │ -0007fa40: 2020 2020 2020 3320 3520 2020 2020 2020        3 5       
│ │ │ │ -0007fa50: 2020 2020 347c 0a7c 2020 2020 2020 2020      4|.|        
│ │ │ │ +0007f9b0: 2020 2020 2020 2020 2020 207c 0a7c 7420             |.|t 
│ │ │ │ +0007f9c0: 202d 2031 3737 3937 3574 2020 2d20 3831   - 177975t  - 81
│ │ │ │ +0007f9d0: 3330 3574 2074 2020 2d20 3433 3030 3130  305t t  - 430010
│ │ │ │ +0007f9e0: 3674 2074 2020 2b20 3736 3939 3974 2074  6t t  + 76999t t
│ │ │ │ +0007f9f0: 2020 2b20 3138 3034 3132 3574 2074 2020    + 1804125t t  
│ │ │ │ +0007fa00: 2b20 3231 3833 3536 3574 207c 0a7c 2034  + 2183565t |.| 4
│ │ │ │ +0007fa10: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0007fa20: 2020 2020 3020 3520 2020 2020 2020 2020      0 5         
│ │ │ │ +0007fa30: 2020 3120 3520 2020 2020 2020 2020 3220    1 5         2 
│ │ │ │ +0007fa40: 3520 2020 2020 2020 2020 2020 3320 3520  5           3 5 
│ │ │ │ +0007fa50: 2020 2020 2020 2020 2020 347c 0a7c 2020            4|.|  
│ │ │ │  0007fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007faa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007fab0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0007faa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007fab0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  0007fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007faf0: 2020 2020 207c 0a7c 2032 3736 3031 3538       |.| 2760158
│ │ │ │ -0007fb00: 7420 202d 2032 3534 3633 3530 7420 7420  t  - 2546350t t 
│ │ │ │ -0007fb10: 202b 2032 3236 3733 3432 7420 7420 202b   + 2267342t t  +
│ │ │ │ -0007fb20: 2033 3736 3730 3531 7420 7420 202d 2031   3767051t t  - 1
│ │ │ │ -0007fb30: 3736 3739 3335 7420 7420 202d 2034 3039  767935t t  - 409
│ │ │ │ -0007fb40: 3030 3337 747c 0a7c 2020 2020 2020 2020  0037t|.|        
│ │ │ │ -0007fb50: 2034 2020 2020 2020 2020 2020 2030 2035   4           0 5
│ │ │ │ -0007fb60: 2020 2020 2020 2020 2020 2031 2035 2020             1 5  
│ │ │ │ -0007fb70: 2020 2020 2020 2020 2032 2035 2020 2020           2 5    
│ │ │ │ -0007fb80: 2020 2020 2020 2033 2035 2020 2020 2020         3 5      
│ │ │ │ -0007fb90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007faf0: 2020 2020 2020 2020 2020 207c 0a7c 2032             |.| 2
│ │ │ │ +0007fb00: 3736 3031 3538 7420 202d 2032 3534 3633  760158t  - 25463
│ │ │ │ +0007fb10: 3530 7420 7420 202b 2032 3236 3733 3432  50t t  + 2267342
│ │ │ │ +0007fb20: 7420 7420 202b 2033 3736 3730 3531 7420  t t  + 3767051t 
│ │ │ │ +0007fb30: 7420 202d 2031 3736 3739 3335 7420 7420  t  - 1767935t t 
│ │ │ │ +0007fb40: 202d 2034 3039 3030 3337 747c 0a7c 2020   - 4090037t|.|  
│ │ │ │ +0007fb50: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +0007fb60: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ +0007fb70: 2031 2035 2020 2020 2020 2020 2020 2032   1 5           2
│ │ │ │ +0007fb80: 2035 2020 2020 2020 2020 2020 2033 2035   5           3 5
│ │ │ │ +0007fb90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0007fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fbe0: 2020 2020 207c 0a7c 2020 2020 2020 2032       |.|       2
│ │ │ │ -0007fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007fbe0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007fbf0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  0007fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fc30: 2020 2020 207c 0a7c 3436 3632 3539 7420       |.|466259t 
│ │ │ │ -0007fc40: 202b 2035 3730 3931 3974 2074 2020 2b20   + 570919t t  + 
│ │ │ │ -0007fc50: 3434 3236 3638 3674 2074 2020 2d20 3234  4426686t t  - 24
│ │ │ │ -0007fc60: 3836 3935 7420 7420 202d 2034 3531 3433  8695t t  - 45143
│ │ │ │ -0007fc70: 3374 2074 2020 2b20 3430 3233 3239 3874  3t t  + 4023298t
│ │ │ │ -0007fc80: 2074 2020 2b7c 0a7c 2020 2020 2020 2034   t  +|.|       4
│ │ │ │ -0007fc90: 2020 2020 2020 2020 2020 3020 3520 2020            0 5   
│ │ │ │ -0007fca0: 2020 2020 2020 2020 3120 3520 2020 2020          1 5     
│ │ │ │ -0007fcb0: 2020 2020 2032 2035 2020 2020 2020 2020       2 5        
│ │ │ │ -0007fcc0: 2020 3320 3520 2020 2020 2020 2020 2020    3 5           
│ │ │ │ -0007fcd0: 3420 3520 207c 0a7c 2020 2020 2020 2020  4 5  |.|        
│ │ │ │ +0007fc30: 2020 2020 2020 2020 2020 207c 0a7c 3436             |.|46
│ │ │ │ +0007fc40: 3632 3539 7420 202b 2035 3730 3931 3974  6259t  + 570919t
│ │ │ │ +0007fc50: 2074 2020 2b20 3434 3236 3638 3674 2074   t  + 4426686t t
│ │ │ │ +0007fc60: 2020 2d20 3234 3836 3935 7420 7420 202d    - 248695t t  -
│ │ │ │ +0007fc70: 2034 3531 3433 3374 2074 2020 2b20 3430   451433t t  + 40
│ │ │ │ +0007fc80: 3233 3239 3874 2074 2020 2b7c 0a7c 2020  23298t t  +|.|  
│ │ │ │ +0007fc90: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ +0007fca0: 3020 3520 2020 2020 2020 2020 2020 3120  0 5           1 
│ │ │ │ +0007fcb0: 3520 2020 2020 2020 2020 2032 2035 2020  5          2 5  
│ │ │ │ +0007fcc0: 2020 2020 2020 2020 3320 3520 2020 2020          3 5     
│ │ │ │ +0007fcd0: 2020 2020 2020 3420 3520 207c 0a7c 2020        4 5  |.|  
│ │ │ │  0007fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fd20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007fd30: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0007fd20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007fd30: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0007fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fd70: 2020 2020 207c 0a7c 2020 2d20 3239 3733       |.|  - 2973
│ │ │ │ -0007fd80: 3530 3074 2020 2d20 3432 3138 3133 3674  500t  - 4218136t
│ │ │ │ -0007fd90: 2074 2020 2d20 3338 3034 3432 3374 2074   t  - 3804423t t
│ │ │ │ -0007fda0: 2020 2b20 3237 3236 3236 3574 2074 2020    + 2726265t t  
│ │ │ │ -0007fdb0: 2b20 3435 3233 3931 3874 2074 2020 2d20  + 4523918t t  - 
│ │ │ │ -0007fdc0: 3231 3033 327c 0a7c 3420 2020 2020 2020  21032|.|4       
│ │ │ │ -0007fdd0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ -0007fde0: 3020 3520 2020 2020 2020 2020 2020 3120  0 5           1 
│ │ │ │ -0007fdf0: 3520 2020 2020 2020 2020 2020 3220 3520  5           2 5 
│ │ │ │ -0007fe00: 2020 2020 2020 2020 2020 3320 3520 2020            3 5   
│ │ │ │ -0007fe10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007fd70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007fd80: 2d20 3239 3733 3530 3074 2020 2d20 3432  - 2973500t  - 42
│ │ │ │ +0007fd90: 3138 3133 3674 2074 2020 2d20 3338 3034  18136t t  - 3804
│ │ │ │ +0007fda0: 3432 3374 2074 2020 2b20 3237 3236 3236  423t t  + 272626
│ │ │ │ +0007fdb0: 3574 2074 2020 2b20 3435 3233 3931 3874  5t t  + 4523918t
│ │ │ │ +0007fdc0: 2074 2020 2d20 3231 3033 327c 0a7c 3420   t  - 21032|.|4 
│ │ │ │ +0007fdd0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0007fde0: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ +0007fdf0: 2020 2020 3120 3520 2020 2020 2020 2020      1 5         
│ │ │ │ +0007fe00: 2020 3220 3520 2020 2020 2020 2020 2020    2 5           
│ │ │ │ +0007fe10: 3320 3520 2020 2020 2020 207c 0a7c 2020  3 5        |.|  
│ │ │ │  0007fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fe60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007fe70: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0007fe60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007fe70: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0007fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007feb0: 2020 2020 207c 0a7c 2020 2d20 3333 3532       |.|  - 3352
│ │ │ │ -0007fec0: 3539 3474 2020 2d20 3434 3731 3839 3974  594t  - 4471899t
│ │ │ │ -0007fed0: 2074 2020 2d20 3236 3239 3533 3074 2074   t  - 2629530t t
│ │ │ │ -0007fee0: 2020 2b20 3637 3637 3239 7420 7420 202d    + 676729t t  -
│ │ │ │ -0007fef0: 2032 3333 3832 3635 7420 7420 202d 2034   2338265t t  - 4
│ │ │ │ -0007ff00: 3037 3833 307c 0a7c 3420 2020 2020 2020  07830|.|4       
│ │ │ │ -0007ff10: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ -0007ff20: 3020 3520 2020 2020 2020 2020 2020 3120  0 5           1 
│ │ │ │ -0007ff30: 3520 2020 2020 2020 2020 2032 2035 2020  5          2 5  
│ │ │ │ -0007ff40: 2020 2020 2020 2020 2033 2035 2020 2020           3 5    
│ │ │ │ -0007ff50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007feb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007fec0: 2d20 3333 3532 3539 3474 2020 2d20 3434  - 3352594t  - 44
│ │ │ │ +0007fed0: 3731 3839 3974 2074 2020 2d20 3236 3239  71899t t  - 2629
│ │ │ │ +0007fee0: 3533 3074 2074 2020 2b20 3637 3637 3239  530t t  + 676729
│ │ │ │ +0007fef0: 7420 7420 202d 2032 3333 3832 3635 7420  t t  - 2338265t 
│ │ │ │ +0007ff00: 7420 202d 2034 3037 3833 307c 0a7c 3420  t  - 407830|.|4 
│ │ │ │ +0007ff10: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0007ff20: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ +0007ff30: 2020 2020 3120 3520 2020 2020 2020 2020      1 5         
│ │ │ │ +0007ff40: 2032 2035 2020 2020 2020 2020 2020 2033   2 5           3
│ │ │ │ +0007ff50: 2035 2020 2020 2020 2020 207c 0a7c 2020   5         |.|  
│ │ │ │  0007ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ffa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007ffb0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0007ffa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007ffb0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  0007ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fff0: 2020 2020 207c 0a7c 7420 202b 2034 3930       |.|t  + 490
│ │ │ │ -00080000: 3238 3730 7420 202b 2038 3030 3935 3274  2870t  + 800952t
│ │ │ │ -00080010: 2074 2020 2d20 3338 3231 3632 3174 2074   t  - 3821621t t
│ │ │ │ -00080020: 2020 2d20 3836 3835 3637 7420 7420 202b    - 868567t t  +
│ │ │ │ -00080030: 2039 3834 3933 3174 2074 2020 2b20 3937   984931t t  + 97
│ │ │ │ -00080040: 3435 3431 747c 0a7c 2034 2020 2020 2020  4541t|.| 4      
│ │ │ │ -00080050: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ -00080060: 3020 3520 2020 2020 2020 2020 2020 3120  0 5           1 
│ │ │ │ -00080070: 3520 2020 2020 2020 2020 2032 2035 2020  5          2 5  
│ │ │ │ -00080080: 2020 2020 2020 2020 3320 3520 2020 2020          3 5     
│ │ │ │ -00080090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007fff0: 2020 2020 2020 2020 2020 207c 0a7c 7420             |.|t 
│ │ │ │ +00080000: 202b 2034 3930 3238 3730 7420 202b 2038   + 4902870t  + 8
│ │ │ │ +00080010: 3030 3935 3274 2074 2020 2d20 3338 3231  00952t t  - 3821
│ │ │ │ +00080020: 3632 3174 2074 2020 2d20 3836 3835 3637  621t t  - 868567
│ │ │ │ +00080030: 7420 7420 202b 2039 3834 3933 3174 2074  t t  + 984931t t
│ │ │ │ +00080040: 2020 2b20 3937 3435 3431 747c 0a7c 2034    + 974541t|.| 4
│ │ │ │ +00080050: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ +00080060: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ +00080070: 2020 2020 3120 3520 2020 2020 2020 2020      1 5         
│ │ │ │ +00080080: 2032 2035 2020 2020 2020 2020 2020 3320   2 5          3 
│ │ │ │ +00080090: 3520 2020 2020 2020 2020 207c 0a7c 2020  5          |.|  
│ │ │ │  000800a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000800b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000800c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000800d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000800e0: 2020 2020 207c 0a7c 3220 2020 2020 2020       |.|2       
│ │ │ │ +000800e0: 2020 2020 2020 2020 2020 207c 0a7c 3220             |.|2 
│ │ │ │  000800f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080130: 2020 2020 207c 0a7c 2020 2d20 3938 3935       |.|  - 9895
│ │ │ │ -00080140: 3638 7420 7420 202d 2032 3330 3339 3132  68t t  - 2303912
│ │ │ │ -00080150: 7420 7420 202d 2034 3037 3435 3936 7420  t t  - 4074596t 
│ │ │ │ -00080160: 7420 202d 2031 3734 3136 3239 7420 7420  t  - 1741629t t 
│ │ │ │ -00080170: 202d 2031 3235 3136 3835 7420 7420 202d   - 1251685t t  -
│ │ │ │ -00080180: 2032 3139 357c 0a7c 3420 2020 2020 2020   2195|.|4       
│ │ │ │ -00080190: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ -000801a0: 2031 2035 2020 2020 2020 2020 2020 2032   1 5           2
│ │ │ │ -000801b0: 2035 2020 2020 2020 2020 2020 2033 2035   5           3 5
│ │ │ │ -000801c0: 2020 2020 2020 2020 2020 2034 2035 2020             4 5  
│ │ │ │ -000801d0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00080130: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00080140: 2d20 3938 3935 3638 7420 7420 202d 2032  - 989568t t  - 2
│ │ │ │ +00080150: 3330 3339 3132 7420 7420 202d 2034 3037  303912t t  - 407
│ │ │ │ +00080160: 3435 3936 7420 7420 202d 2031 3734 3136  4596t t  - 17416
│ │ │ │ +00080170: 3239 7420 7420 202d 2031 3235 3136 3835  29t t  - 1251685
│ │ │ │ +00080180: 7420 7420 202d 2032 3139 357c 0a7c 3420  t t  - 2195|.|4 
│ │ │ │ +00080190: 2020 2020 2020 2020 2030 2035 2020 2020           0 5    
│ │ │ │ +000801a0: 2020 2020 2020 2031 2035 2020 2020 2020         1 5      
│ │ │ │ +000801b0: 2020 2020 2032 2035 2020 2020 2020 2020       2 5        
│ │ │ │ +000801c0: 2020 2033 2035 2020 2020 2020 2020 2020     3 5          
│ │ │ │ +000801d0: 2034 2035 2020 2020 2020 207c 0a7c 2d2d   4 5       |.|--
│ │ │ │  000801e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000801f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00080200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00080210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00080220: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00080230: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00080220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +00080230: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  00080240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080270: 2020 2020 207c 0a7c 7420 7420 202b 2032       |.|t t  + 2
│ │ │ │ -00080280: 3434 3333 3037 7420 202b 2034 3531 3438  443307t  + 45148
│ │ │ │ -00080290: 3839 7420 7420 202d 2032 3037 3434 3738  89t t  - 2074478
│ │ │ │ -000802a0: 7420 7420 202b 2034 3332 3838 3933 7420  t t  + 4328893t 
│ │ │ │ -000802b0: 7420 202d 2036 3130 3434 3874 2074 2020  t  - 610448t t  
│ │ │ │ -000802c0: 2d20 3237 367c 0a7c 2034 2035 2020 2020  - 276|.| 4 5    
│ │ │ │ -000802d0: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ -000802e0: 2020 2030 2036 2020 2020 2020 2020 2020     0 6          
│ │ │ │ -000802f0: 2031 2036 2020 2020 2020 2020 2020 2032   1 6           2
│ │ │ │ -00080300: 2036 2020 2020 2020 2020 2020 3320 3620   6          3 6 
│ │ │ │ -00080310: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00080270: 2020 2020 2020 2020 2020 207c 0a7c 7420             |.|t 
│ │ │ │ +00080280: 7420 202b 2032 3434 3333 3037 7420 202b  t  + 2443307t  +
│ │ │ │ +00080290: 2034 3531 3438 3839 7420 7420 202d 2032   4514889t t  - 2
│ │ │ │ +000802a0: 3037 3434 3738 7420 7420 202b 2034 3332  074478t t  + 432
│ │ │ │ +000802b0: 3838 3933 7420 7420 202d 2036 3130 3434  8893t t  - 61044
│ │ │ │ +000802c0: 3874 2074 2020 2d20 3237 367c 0a7c 2034  8t t  - 276|.| 4
│ │ │ │ +000802d0: 2035 2020 2020 2020 2020 2020 2035 2020   5           5  
│ │ │ │ +000802e0: 2020 2020 2020 2020 2030 2036 2020 2020           0 6    
│ │ │ │ +000802f0: 2020 2020 2020 2031 2036 2020 2020 2020         1 6      
│ │ │ │ +00080300: 2020 2020 2032 2036 2020 2020 2020 2020       2 6        
│ │ │ │ +00080310: 2020 3320 3620 2020 2020 207c 0a7c 2020    3 6      |.|  
│ │ │ │  00080320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00080370: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00080360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00080370: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  00080380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000803a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000803b0: 2020 2020 207c 0a7c 2020 2b20 3230 3234       |.|  + 2024
│ │ │ │ -000803c0: 3134 3574 2020 2b20 3831 3032 3332 7420  145t  + 810232t 
│ │ │ │ -000803d0: 7420 202d 2034 3633 3434 3033 7420 7420  t  - 4634403t t 
│ │ │ │ -000803e0: 202d 2031 3037 3936 3437 7420 7420 202d   - 1079647t t  -
│ │ │ │ -000803f0: 2034 3732 3933 3536 7420 7420 202d 2032   4729356t t  - 2
│ │ │ │ -00080400: 3632 3039 377c 0a7c 3520 2020 2020 2020  62097|.|5       
│ │ │ │ -00080410: 2020 2020 3520 2020 2020 2020 2020 2030      5          0
│ │ │ │ -00080420: 2036 2020 2020 2020 2020 2020 2031 2036   6           1 6
│ │ │ │ -00080430: 2020 2020 2020 2020 2020 2032 2036 2020             2 6  
│ │ │ │ -00080440: 2020 2020 2020 2020 2033 2036 2020 2020           3 6    
│ │ │ │ -00080450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000803b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000803c0: 2b20 3230 3234 3134 3574 2020 2b20 3831  + 2024145t  + 81
│ │ │ │ +000803d0: 3032 3332 7420 7420 202d 2034 3633 3434  0232t t  - 46344
│ │ │ │ +000803e0: 3033 7420 7420 202d 2031 3037 3936 3437  03t t  - 1079647
│ │ │ │ +000803f0: 7420 7420 202d 2034 3732 3933 3536 7420  t t  - 4729356t 
│ │ │ │ +00080400: 7420 202d 2032 3632 3039 377c 0a7c 3520  t  - 262097|.|5 
│ │ │ │ +00080410: 2020 2020 2020 2020 2020 3520 2020 2020            5     
│ │ │ │ +00080420: 2020 2020 2030 2036 2020 2020 2020 2020       0 6        
│ │ │ │ +00080430: 2020 2031 2036 2020 2020 2020 2020 2020     1 6          
│ │ │ │ +00080440: 2032 2036 2020 2020 2020 2020 2020 2033   2 6           3
│ │ │ │ +00080450: 2036 2020 2020 2020 2020 207c 0a7c 2020   6         |.|  
│ │ │ │  00080460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000804a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000804b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000804a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000804b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  000804c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000804d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000804e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000804f0: 2020 2020 207c 0a7c 2074 2020 2b20 3632       |.| t  + 62
│ │ │ │ -00080500: 3031 3231 7420 202d 2031 3330 3132 3674  0121t  - 130126t
│ │ │ │ -00080510: 2074 2020 2d20 3333 3031 3433 3774 2074   t  - 3301437t t
│ │ │ │ -00080520: 2020 2d20 3237 3936 3739 3774 2074 2020    - 2796797t t  
│ │ │ │ -00080530: 2d20 3232 3137 3038 3474 2074 2020 2b20  - 2217084t t  + 
│ │ │ │ -00080540: 3336 3531 337c 0a7c 3420 3520 2020 2020  36513|.|4 5     
│ │ │ │ -00080550: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │ -00080560: 3020 3620 2020 2020 2020 2020 2020 3120  0 6           1 
│ │ │ │ -00080570: 3620 2020 2020 2020 2020 2020 3220 3620  6           2 6 
│ │ │ │ -00080580: 2020 2020 2020 2020 2020 3320 3620 2020            3 6   
│ │ │ │ -00080590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000804f0: 2020 2020 2020 2020 2020 207c 0a7c 2074             |.| t
│ │ │ │ +00080500: 2020 2b20 3632 3031 3231 7420 202d 2031    + 620121t  - 1
│ │ │ │ +00080510: 3330 3132 3674 2074 2020 2d20 3333 3031  30126t t  - 3301
│ │ │ │ +00080520: 3433 3774 2074 2020 2d20 3237 3936 3739  437t t  - 279679
│ │ │ │ +00080530: 3774 2074 2020 2d20 3232 3137 3038 3474  7t t  - 2217084t
│ │ │ │ +00080540: 2074 2020 2b20 3336 3531 337c 0a7c 3420   t  + 36513|.|4 
│ │ │ │ +00080550: 3520 2020 2020 2020 2020 2035 2020 2020  5          5    
│ │ │ │ +00080560: 2020 2020 2020 3020 3620 2020 2020 2020        0 6       
│ │ │ │ +00080570: 2020 2020 3120 3620 2020 2020 2020 2020      1 6         
│ │ │ │ +00080580: 2020 3220 3620 2020 2020 2020 2020 2020    2 6           
│ │ │ │ +00080590: 3320 3620 2020 2020 2020 207c 0a7c 2020  3 6        |.|  
│ │ │ │  000805a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000805b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000805c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000805d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000805e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000805f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000805e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000805f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  00080600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080630: 2020 2020 207c 0a7c 7420 202b 2032 3736       |.|t  + 276
│ │ │ │ -00080640: 3239 3632 7420 202d 2031 3037 3333 3435  2962t  - 1073345
│ │ │ │ -00080650: 7420 7420 202b 2032 3237 3330 3030 7420  t t  + 2273000t 
│ │ │ │ -00080660: 7420 202d 2031 3634 3431 3937 7420 7420  t  - 1644197t t 
│ │ │ │ -00080670: 202d 2038 3830 3430 3874 2074 2020 2d20   - 880408t t  - 
│ │ │ │ -00080680: 3238 3930 357c 0a7c 2035 2020 2020 2020  28905|.| 5      
│ │ │ │ -00080690: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │ -000806a0: 2030 2036 2020 2020 2020 2020 2020 2031   0 6           1
│ │ │ │ -000806b0: 2036 2020 2020 2020 2020 2020 2032 2036   6           2 6
│ │ │ │ -000806c0: 2020 2020 2020 2020 2020 3320 3620 2020            3 6   
│ │ │ │ -000806d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00080630: 2020 2020 2020 2020 2020 207c 0a7c 7420             |.|t 
│ │ │ │ +00080640: 202b 2032 3736 3239 3632 7420 202d 2031   + 2762962t  - 1
│ │ │ │ +00080650: 3037 3333 3435 7420 7420 202b 2032 3237  073345t t  + 227
│ │ │ │ +00080660: 3330 3030 7420 7420 202d 2031 3634 3431  3000t t  - 16441
│ │ │ │ +00080670: 3937 7420 7420 202d 2038 3830 3430 3874  97t t  - 880408t
│ │ │ │ +00080680: 2074 2020 2d20 3238 3930 357c 0a7c 2035   t  - 28905|.| 5
│ │ │ │ +00080690: 2020 2020 2020 2020 2020 2035 2020 2020             5    
│ │ │ │ +000806a0: 2020 2020 2020 2030 2036 2020 2020 2020         0 6      
│ │ │ │ +000806b0: 2020 2020 2031 2036 2020 2020 2020 2020       1 6        
│ │ │ │ +000806c0: 2020 2032 2036 2020 2020 2020 2020 2020     2 6          
│ │ │ │ +000806d0: 3320 3620 2020 2020 2020 207c 0a7c 2020  3 6        |.|  
│ │ │ │  000806e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000806f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080720: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00080730: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00080720: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00080730: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00080740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080770: 2020 2020 207c 0a7c 2074 2020 2b20 3230       |.| t  + 20
│ │ │ │ -00080780: 3036 3730 3674 2020 2d20 3937 3233 3531  06706t  - 972351
│ │ │ │ -00080790: 7420 7420 202d 2033 3838 3330 3039 7420  t t  - 3883009t 
│ │ │ │ -000807a0: 7420 202b 2034 3833 3131 3331 7420 7420  t  + 4831131t t 
│ │ │ │ -000807b0: 202b 2034 3433 3833 3234 7420 7420 202d   + 4438324t t  -
│ │ │ │ -000807c0: 2037 3536 377c 0a7c 3420 3520 2020 2020   7567|.|4 5     
│ │ │ │ -000807d0: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ -000807e0: 2030 2036 2020 2020 2020 2020 2020 2031   0 6           1
│ │ │ │ -000807f0: 2036 2020 2020 2020 2020 2020 2032 2036   6           2 6
│ │ │ │ -00080800: 2020 2020 2020 2020 2020 2033 2036 2020             3 6  
│ │ │ │ -00080810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00080770: 2020 2020 2020 2020 2020 207c 0a7c 2074             |.| t
│ │ │ │ +00080780: 2020 2b20 3230 3036 3730 3674 2020 2d20    + 2006706t  - 
│ │ │ │ +00080790: 3937 3233 3531 7420 7420 202d 2033 3838  972351t t  - 388
│ │ │ │ +000807a0: 3330 3039 7420 7420 202b 2034 3833 3131  3009t t  + 48311
│ │ │ │ +000807b0: 3331 7420 7420 202b 2034 3433 3833 3234  31t t  + 4438324
│ │ │ │ +000807c0: 7420 7420 202d 2037 3536 377c 0a7c 3420  t t  - 7567|.|4 
│ │ │ │ +000807d0: 3520 2020 2020 2020 2020 2020 3520 2020  5           5   
│ │ │ │ +000807e0: 2020 2020 2020 2030 2036 2020 2020 2020         0 6      
│ │ │ │ +000807f0: 2020 2020 2031 2036 2020 2020 2020 2020       1 6        
│ │ │ │ +00080800: 2020 2032 2036 2020 2020 2020 2020 2020     2 6          
│ │ │ │ +00080810: 2033 2036 2020 2020 2020 207c 0a7c 2020   3 6       |.|  
│ │ │ │  00080820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080860: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00080870: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00080860: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00080870: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00080880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000808a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000808b0: 2020 2020 207c 0a7c 2031 3534 3431 3333       |.| 1544133
│ │ │ │ -000808c0: 7420 202b 2031 3233 3437 3539 7420 7420  t  + 1234759t t 
│ │ │ │ -000808d0: 202b 2039 3032 3930 7420 7420 202d 2032   + 90290t t  - 2
│ │ │ │ -000808e0: 3534 3535 3836 7420 7420 202d 2031 3135  545586t t  - 115
│ │ │ │ -000808f0: 3037 3339 7420 7420 202d 2031 3738 3737  0739t t  - 17877
│ │ │ │ -00080900: 3737 7420 747c 0a7c 2020 2020 2020 2020  77t t|.|        
│ │ │ │ -00080910: 2035 2020 2020 2020 2020 2020 2030 2036   5           0 6
│ │ │ │ -00080920: 2020 2020 2020 2020 2031 2036 2020 2020           1 6    
│ │ │ │ -00080930: 2020 2020 2020 2032 2036 2020 2020 2020         2 6      
│ │ │ │ -00080940: 2020 2020 2033 2036 2020 2020 2020 2020       3 6        
│ │ │ │ -00080950: 2020 2034 207c 0a7c 2020 2020 2020 2020     4 |.|        
│ │ │ │ +000808b0: 2020 2020 2020 2020 2020 207c 0a7c 2031             |.| 1
│ │ │ │ +000808c0: 3534 3431 3333 7420 202b 2031 3233 3437  544133t  + 12347
│ │ │ │ +000808d0: 3539 7420 7420 202b 2039 3032 3930 7420  59t t  + 90290t 
│ │ │ │ +000808e0: 7420 202d 2032 3534 3535 3836 7420 7420  t  - 2545586t t 
│ │ │ │ +000808f0: 202d 2031 3135 3037 3339 7420 7420 202d   - 1150739t t  -
│ │ │ │ +00080900: 2031 3738 3737 3737 7420 747c 0a7c 2020   1787777t t|.|  
│ │ │ │ +00080910: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ +00080920: 2020 2030 2036 2020 2020 2020 2020 2031     0 6         1
│ │ │ │ +00080930: 2036 2020 2020 2020 2020 2020 2032 2036   6           2 6
│ │ │ │ +00080940: 2020 2020 2020 2020 2020 2033 2036 2020             3 6  
│ │ │ │ +00080950: 2020 2020 2020 2020 2034 207c 0a7c 2020           4 |.|  
│ │ │ │  00080960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000809a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000809b0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000809a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000809b0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  000809c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000809d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000809e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000809f0: 2020 2020 207c 0a7c 3174 2074 2020 2b20       |.|1t t  + 
│ │ │ │ -00080a00: 3337 3636 3833 3974 2020 2b20 3136 3934  3766839t  + 1694
│ │ │ │ -00080a10: 3237 3974 2074 2020 2b20 3234 3930 3533  279t t  + 249053
│ │ │ │ -00080a20: 3874 2074 2020 2b20 3735 3834 3736 7420  8t t  + 758476t 
│ │ │ │ -00080a30: 7420 202b 2034 3537 3535 3437 7420 7420  t  + 4575547t t 
│ │ │ │ -00080a40: 202d 2034 317c 0a7c 2020 3420 3520 2020   - 41|.|  4 5   
│ │ │ │ -00080a50: 2020 2020 2020 2020 3520 2020 2020 2020          5       
│ │ │ │ -00080a60: 2020 2020 3020 3620 2020 2020 2020 2020      0 6         
│ │ │ │ -00080a70: 2020 3120 3620 2020 2020 2020 2020 2032    1 6          2
│ │ │ │ -00080a80: 2036 2020 2020 2020 2020 2020 2033 2036   6           3 6
│ │ │ │ -00080a90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000809f0: 2020 2020 2020 2020 2020 207c 0a7c 3174             |.|1t
│ │ │ │ +00080a00: 2074 2020 2b20 3337 3636 3833 3974 2020   t  + 3766839t  
│ │ │ │ +00080a10: 2b20 3136 3934 3237 3974 2074 2020 2b20  + 1694279t t  + 
│ │ │ │ +00080a20: 3234 3930 3533 3874 2074 2020 2b20 3735  2490538t t  + 75
│ │ │ │ +00080a30: 3834 3736 7420 7420 202b 2034 3537 3535  8476t t  + 45755
│ │ │ │ +00080a40: 3437 7420 7420 202d 2034 317c 0a7c 2020  47t t  - 41|.|  
│ │ │ │ +00080a50: 3420 3520 2020 2020 2020 2020 2020 3520  4 5           5 
│ │ │ │ +00080a60: 2020 2020 2020 2020 2020 3020 3620 2020            0 6   
│ │ │ │ +00080a70: 2020 2020 2020 2020 3120 3620 2020 2020          1 6     
│ │ │ │ +00080a80: 2020 2020 2032 2036 2020 2020 2020 2020       2 6        
│ │ │ │ +00080a90: 2020 2033 2036 2020 2020 207c 0a7c 2020     3 6     |.|  
│ │ │ │  00080aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080ae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00080af0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00080ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00080af0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00080b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080b30: 2020 2020 207c 0a7c 3874 2074 2020 2b20       |.|8t t  + 
│ │ │ │ -00080b40: 3336 3434 3530 3574 2020 2d20 3137 3338  3644505t  - 1738
│ │ │ │ -00080b50: 3031 3774 2074 2020 2d20 3338 3835 3033  017t t  - 388503
│ │ │ │ -00080b60: 3174 2074 2020 2d20 3437 3430 3935 3974  1t t  - 4740959t
│ │ │ │ -00080b70: 2074 2020 2d20 3334 3539 3134 3374 2074   t  - 3459143t t
│ │ │ │ -00080b80: 2020 2d20 317c 0a7c 2020 3420 3520 2020    - 1|.|  4 5   
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│ │ │ │ -00080ba0: 2020 2020 3020 3620 2020 2020 2020 2020      0 6         
│ │ │ │ -00080bb0: 2020 3120 3620 2020 2020 2020 2020 2020    1 6           
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│ │ │ │ +00080b30: 2020 2020 2020 2020 2020 207c 0a7c 3874             |.|8t
│ │ │ │ +00080b40: 2074 2020 2b20 3336 3434 3530 3574 2020   t  + 3644505t  
│ │ │ │ +00080b50: 2d20 3137 3338 3031 3774 2074 2020 2d20  - 1738017t t  - 
│ │ │ │ +00080b60: 3338 3835 3033 3174 2074 2020 2d20 3437  3885031t t  - 47
│ │ │ │ +00080b70: 3430 3935 3974 2074 2020 2d20 3334 3539  40959t t  - 3459
│ │ │ │ +00080b80: 3134 3374 2074 2020 2d20 317c 0a7c 2020  143t t  - 1|.|  
│ │ │ │ +00080b90: 3420 3520 2020 2020 2020 2020 2020 3520  4 5           5 
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│ │ │ │ +00080bb0: 2020 2020 2020 2020 3120 3620 2020 2020          1 6     
│ │ │ │ +00080bc0: 2020 2020 2020 3220 3620 2020 2020 2020        2 6       
│ │ │ │ +00080bd0: 2020 2020 3320 3620 2020 207c 0a7c 2020      3 6    |.|  
│ │ │ │  00080be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080c20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00080c30: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00080c20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00080c30: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00080c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080c70: 2020 2020 207c 0a7c 2074 2020 2d20 3432       |.| t  - 42
│ │ │ │ -00080c80: 3933 3433 3474 2020 2d20 3136 3333 3630  93434t  - 163360
│ │ │ │ -00080c90: 3774 2074 2020 2b20 3230 3330 3635 3174  7t t  + 2030651t
│ │ │ │ -00080ca0: 2074 2020 2d20 3335 3634 3431 3474 2074   t  - 3564414t t
│ │ │ │ -00080cb0: 2020 2d20 3438 3539 3439 3174 2074 2020    - 4859491t t  
│ │ │ │ -00080cc0: 2b20 3331 357c 0a7c 3420 3520 2020 2020  + 315|.|4 5     
│ │ │ │ -00080cd0: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ -00080ce0: 2020 3020 3620 2020 2020 2020 2020 2020    0 6           
│ │ │ │ -00080cf0: 3120 3620 2020 2020 2020 2020 2020 3220  1 6           2 
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│ │ │ │ -00080d10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00080c70: 2020 2020 2020 2020 2020 207c 0a7c 2074             |.| t
│ │ │ │ +00080c80: 2020 2d20 3432 3933 3433 3474 2020 2d20    - 4293434t  - 
│ │ │ │ +00080c90: 3136 3333 3630 3774 2074 2020 2b20 3230  1633607t t  + 20
│ │ │ │ +00080ca0: 3330 3635 3174 2074 2020 2d20 3335 3634  30651t t  - 3564
│ │ │ │ +00080cb0: 3431 3474 2074 2020 2d20 3438 3539 3439  414t t  - 485949
│ │ │ │ +00080cc0: 3174 2074 2020 2b20 3331 357c 0a7c 3420  1t t  + 315|.|4 
│ │ │ │ +00080cd0: 3520 2020 2020 2020 2020 2020 3520 2020  5           5   
│ │ │ │ +00080ce0: 2020 2020 2020 2020 3020 3620 2020 2020          0 6     
│ │ │ │ +00080cf0: 2020 2020 2020 3120 3620 2020 2020 2020        1 6       
│ │ │ │ +00080d00: 2020 2020 3220 3620 2020 2020 2020 2020      2 6         
│ │ │ │ +00080d10: 2020 3320 3620 2020 2020 207c 0a7c 2020    3 6      |.|  
│ │ │ │  00080d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080d60: 2020 2020 207c 0a7c 2020 2020 3220 2020       |.|    2   
│ │ │ │ -00080d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080d60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00080d70: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00080d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080db0: 2020 2020 207c 0a7c 3532 3074 2020 2b20       |.|520t  + 
│ │ │ │ -00080dc0: 3130 3930 3339 3274 2074 2020 2d20 3432  1090392t t  - 42
│ │ │ │ -00080dd0: 3231 3335 3074 2074 2020 2d20 3336 3333  21350t t  - 3633
│ │ │ │ -00080de0: 3837 3574 2074 2020 2b20 3238 3633 3835  875t t  + 286385
│ │ │ │ -00080df0: 7420 7420 202d 2034 3132 3832 3538 7420  t t  - 4128258t 
│ │ │ │ -00080e00: 7420 202d 207c 0a7c 2020 2020 3520 2020  t  - |.|    5   
│ │ │ │ -00080e10: 2020 2020 2020 2020 3020 3620 2020 2020          0 6     
│ │ │ │ -00080e20: 2020 2020 2020 3120 3620 2020 2020 2020        1 6       
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│ │ │ │ +00080db0: 2020 2020 2020 2020 2020 207c 0a7c 3532             |.|52
│ │ │ │ +00080dc0: 3074 2020 2b20 3130 3930 3339 3274 2074  0t  + 1090392t t
│ │ │ │ +00080dd0: 2020 2d20 3432 3231 3335 3074 2074 2020    - 4221350t t  
│ │ │ │ +00080de0: 2d20 3336 3333 3837 3574 2074 2020 2b20  - 3633875t t  + 
│ │ │ │ +00080df0: 3238 3633 3835 7420 7420 202d 2034 3132  286385t t  - 412
│ │ │ │ +00080e00: 3832 3538 7420 7420 202d 207c 0a7c 2020  8258t t  - |.|  
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│ │ │ │ +00080ef0: 2020 2020 2020 2020 2020 207c 0a7c 3735             |.|75
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│ │ │ │ -00081640: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00081650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081650: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00081660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081670: 2020 2020 207c 0a7c 3036 3036 3374 2074       |.|06063t t
│ │ │ │ -00081680: 2020 2b20 3138 3634 3332 3074 2074 2020    + 1864320t t  
│ │ │ │ -00081690: 2b20 3231 3238 3039 3274 202c 2020 2020  + 2128092t ,    
│ │ │ │ -000816a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081670: 2020 2020 2020 2020 2020 207c 0a7c 3036             |.|06
│ │ │ │ +00081680: 3036 3374 2074 2020 2b20 3138 3634 3332  063t t  + 186432
│ │ │ │ +00081690: 3074 2074 2020 2b20 3231 3238 3039 3274  0t t  + 2128092t
│ │ │ │ +000816a0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  000816b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000816c0: 2020 2020 207c 0a7c 2020 2020 2020 3420       |.|      4 
│ │ │ │ -000816d0: 3620 2020 2020 2020 2020 2020 3520 3620  6           5 6 
│ │ │ │ -000816e0: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ -000816f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000816c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000816d0: 2020 2020 3420 3620 2020 2020 2020 2020      4 6         
│ │ │ │ +000816e0: 2020 3520 3620 2020 2020 2020 2020 2020    5 6           
│ │ │ │ +000816f0: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │  00081700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081710: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00081710: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00081720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081760: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00081760: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00081770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081780: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00081790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081790: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  000817a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000817b0: 2020 2020 207c 0a7c 3134 3233 3438 7420       |.|142348t 
│ │ │ │ -000817c0: 7420 202d 2032 3637 3832 3437 7420 7420  t  - 2678247t t 
│ │ │ │ -000817d0: 202b 2035 3137 3531 3474 202c 2020 2020   + 517514t ,    
│ │ │ │ -000817e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000817b0: 2020 2020 2020 2020 2020 207c 0a7c 3134             |.|14
│ │ │ │ +000817c0: 3233 3438 7420 7420 202d 2032 3637 3832  2348t t  - 26782
│ │ │ │ +000817d0: 3437 7420 7420 202b 2035 3137 3531 3474  47t t  + 517514t
│ │ │ │ +000817e0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  000817f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081800: 2020 2020 207c 0a7c 2020 2020 2020 2034       |.|       4
│ │ │ │ -00081810: 2036 2020 2020 2020 2020 2020 2035 2036   6           5 6
│ │ │ │ -00081820: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ -00081830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081800: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00081810: 2020 2020 2034 2036 2020 2020 2020 2020       4 6        
│ │ │ │ +00081820: 2020 2035 2036 2020 2020 2020 2020 2020     5 6          
│ │ │ │ +00081830: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │  00081840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081850: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00081850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00081860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000818a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000818a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000818b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000818c0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000818c0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  000818d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000818e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000818f0: 2020 2020 207c 0a7c 3334 3338 7420 7420       |.|3438t t 
│ │ │ │ -00081900: 202d 2036 3536 3231 3974 2074 2020 2b20   - 656219t t  + 
│ │ │ │ -00081910: 3339 3939 3438 7420 2c20 2020 2020 2020  399948t ,       
│ │ │ │ +000818f0: 2020 2020 2020 2020 2020 207c 0a7c 3334             |.|34
│ │ │ │ +00081900: 3338 7420 7420 202d 2036 3536 3231 3974  38t t  - 656219t
│ │ │ │ +00081910: 2074 2020 2b20 3339 3939 3438 7420 2c20   t  + 399948t , 
│ │ │ │  00081920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081940: 2020 2020 207c 0a7c 2020 2020 2034 2036       |.|     4 6
│ │ │ │ -00081950: 2020 2020 2020 2020 2020 3520 3620 2020            5 6   
│ │ │ │ -00081960: 2020 2020 2020 2036 2020 2020 2020 2020         6        
│ │ │ │ +00081940: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00081950: 2020 2034 2036 2020 2020 2020 2020 2020     4 6          
│ │ │ │ +00081960: 3520 3620 2020 2020 2020 2020 2036 2020  5 6          6  
│ │ │ │  00081970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00081990: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000819a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000819b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000819c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000819d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000819e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000819f0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00081a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000819e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000819f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081a00: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00081a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081a30: 2020 2020 207c 0a7c 3238 3132 3736 3874       |.|2812768t
│ │ │ │ -00081a40: 2074 2020 2d20 3236 3532 3534 3274 2020   t  - 2652542t  
│ │ │ │ -00081a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081a30: 2020 2020 2020 2020 2020 207c 0a7c 3238             |.|28
│ │ │ │ +00081a40: 3132 3736 3874 2074 2020 2d20 3236 3532  12768t t  - 2652
│ │ │ │ +00081a50: 3534 3274 2020 2020 2020 2020 2020 2020  542t            
│ │ │ │  00081a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081a80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00081a90: 3520 3620 2020 2020 2020 2020 2020 3620  5 6           6 
│ │ │ │ -00081aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081a80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00081a90: 2020 2020 2020 3520 3620 2020 2020 2020        5 6       
│ │ │ │ +00081aa0: 2020 2020 3620 2020 2020 2020 2020 2020      6           
│ │ │ │  00081ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081ad0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00081ad0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00081ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00081af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00081b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00081b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00081b20: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20  -----+..Ways to 
│ │ │ │ -00081b30: 7573 6520 7468 6973 206d 6574 686f 643a  use this method:
│ │ │ │ -00081b40: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00081b50: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  =========..  * *
│ │ │ │ -00081b60: 6e6f 7465 2070 6172 616d 6574 7269 7a65  note parametrize
│ │ │ │ -00081b70: 2849 6465 616c 293a 2070 6172 616d 6574  (Ideal): paramet
│ │ │ │ -00081b80: 7269 7a65 5f6c 7049 6465 616c 5f72 702c  rize_lpIdeal_rp,
│ │ │ │ -00081b90: 202d 2d20 7061 7261 6d65 7472 697a 6174   -- parametrizat
│ │ │ │ -00081ba0: 696f 6e20 6f66 0a20 2020 206c 696e 6561  ion of.    linea
│ │ │ │ -00081bb0: 7220 7661 7269 6574 6965 7320 616e 6420  r varieties and 
│ │ │ │ -00081bc0: 6879 7065 7271 7561 6472 6963 730a 2020  hyperquadrics.  
│ │ │ │ -00081bd0: 2a20 2270 6172 616d 6574 7269 7a65 2850  * "parametrize(P
│ │ │ │ -00081be0: 6f6c 796e 6f6d 6961 6c52 696e 6729 220a  olynomialRing)".
│ │ │ │ -00081bf0: 2020 2a20 2270 6172 616d 6574 7269 7a65    * "parametrize
│ │ │ │ -00081c00: 2851 756f 7469 656e 7452 696e 6729 220a  (QuotientRing)".
│ │ │ │ -00081c10: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -00081c20: 696e 666f 2c20 4e6f 6465 3a20 706f 696e  info, Node: poin
│ │ │ │ -00081c30: 742c 204e 6578 743a 2070 6f69 6e74 5f6c  t, Next: point_l
│ │ │ │ -00081c40: 7051 756f 7469 656e 7452 696e 675f 7270  pQuotientRing_rp
│ │ │ │ -00081c50: 2c20 5072 6576 3a20 7061 7261 6d65 7472  , Prev: parametr
│ │ │ │ -00081c60: 697a 655f 6c70 4964 6561 6c5f 7270 2c20  ize_lpIdeal_rp, 
│ │ │ │ -00081c70: 5570 3a20 546f 700a 0a70 6f69 6e74 202d  Up: Top..point -
│ │ │ │ -00081c80: 2d20 7069 636b 2061 2072 616e 646f 6d20  - pick a random 
│ │ │ │ -00081c90: 7261 7469 6f6e 616c 2070 6f69 6e74 206f  rational point o
│ │ │ │ -00081ca0: 6e20 6120 7072 6f6a 6563 7469 7665 2076  n a projective v
│ │ │ │ -00081cb0: 6172 6965 7479 0a2a 2a2a 2a2a 2a2a 2a2a  ariety.*********
│ │ │ │ +00081b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761  -----------+..Wa
│ │ │ │ +00081b30: 7973 2074 6f20 7573 6520 7468 6973 206d  ys to use this m
│ │ │ │ +00081b40: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d  ethod:.=========
│ │ │ │ +00081b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00081b60: 0a20 202a 202a 6e6f 7465 2070 6172 616d  .  * *note param
│ │ │ │ +00081b70: 6574 7269 7a65 2849 6465 616c 293a 2070  etrize(Ideal): p
│ │ │ │ +00081b80: 6172 616d 6574 7269 7a65 5f6c 7049 6465  arametrize_lpIde
│ │ │ │ +00081b90: 616c 5f72 702c 202d 2d20 7061 7261 6d65  al_rp, -- parame
│ │ │ │ +00081ba0: 7472 697a 6174 696f 6e20 6f66 0a20 2020  trization of.   
│ │ │ │ +00081bb0: 206c 696e 6561 7220 7661 7269 6574 6965   linear varietie
│ │ │ │ +00081bc0: 7320 616e 6420 6879 7065 7271 7561 6472  s and hyperquadr
│ │ │ │ +00081bd0: 6963 730a 2020 2a20 2270 6172 616d 6574  ics.  * "paramet
│ │ │ │ +00081be0: 7269 7a65 2850 6f6c 796e 6f6d 6961 6c52  rize(PolynomialR
│ │ │ │ +00081bf0: 696e 6729 220a 2020 2a20 2270 6172 616d  ing)".  * "param
│ │ │ │ +00081c00: 6574 7269 7a65 2851 756f 7469 656e 7452  etrize(QuotientR
│ │ │ │ +00081c10: 696e 6729 220a 1f0a 4669 6c65 3a20 4372  ing)"...File: Cr
│ │ │ │ +00081c20: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ +00081c30: 3a20 706f 696e 742c 204e 6578 743a 2070  : point, Next: p
│ │ │ │ +00081c40: 6f69 6e74 5f6c 7051 756f 7469 656e 7452  oint_lpQuotientR
│ │ │ │ +00081c50: 696e 675f 7270 2c20 5072 6576 3a20 7061  ing_rp, Prev: pa
│ │ │ │ +00081c60: 7261 6d65 7472 697a 655f 6c70 4964 6561  rametrize_lpIdea
│ │ │ │ +00081c70: 6c5f 7270 2c20 5570 3a20 546f 700a 0a70  l_rp, Up: Top..p
│ │ │ │ +00081c80: 6f69 6e74 202d 2d20 7069 636b 2061 2072  oint -- pick a r
│ │ │ │ +00081c90: 616e 646f 6d20 7261 7469 6f6e 616c 2070  andom rational p
│ │ │ │ +00081ca0: 6f69 6e74 206f 6e20 6120 7072 6f6a 6563  oint on a projec
│ │ │ │ +00081cb0: 7469 7665 2076 6172 6965 7479 0a2a 2a2a  tive variety.***
│ │ │ │  00081cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00081cd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00081ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00081cf0: 2a2a 2a2a 0a0a 4465 7363 7269 7074 696f  ****..Descriptio
│ │ │ │ -00081d00: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a53  n.===========..S
│ │ │ │ -00081d10: 6565 2070 6f69 6e74 284d 756c 7469 7072  ee point(Multipr
│ │ │ │ -00081d20: 6f6a 6563 7469 7665 5661 7269 6574 6965  ojectiveVarietie
│ │ │ │ -00081d30: 7329 2028 6d69 7373 696e 6720 646f 6375  s) (missing docu
│ │ │ │ -00081d40: 6d65 6e74 6174 696f 6e29 2061 6e64 202a  mentation) and *
│ │ │ │ -00081d50: 6e6f 7465 0a70 6f69 6e74 2851 756f 7469  note.point(Quoti
│ │ │ │ -00081d60: 656e 7452 696e 6729 3a20 706f 696e 745f  entRing): point_
│ │ │ │ -00081d70: 6c70 5175 6f74 6965 6e74 5269 6e67 5f72  lpQuotientRing_r
│ │ │ │ -00081d80: 702c 2e0a 0a57 6179 7320 746f 2075 7365  p,...Ways to use
│ │ │ │ -00081d90: 2070 6f69 6e74 3a0a 3d3d 3d3d 3d3d 3d3d   point:.========
│ │ │ │ -00081da0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00081db0: 2270 6f69 6e74 2850 6f6c 796e 6f6d 6961  "point(Polynomia
│ │ │ │ -00081dc0: 6c52 696e 6729 2220 2d2d 2073 6565 202a  lRing)" -- see *
│ │ │ │ -00081dd0: 6e6f 7465 2070 6f69 6e74 2851 756f 7469  note point(Quoti
│ │ │ │ -00081de0: 656e 7452 696e 6729 3a0a 2020 2020 706f  entRing):.    po
│ │ │ │ -00081df0: 696e 745f 6c70 5175 6f74 6965 6e74 5269  int_lpQuotientRi
│ │ │ │ -00081e00: 6e67 5f72 702c 202d 2d20 7069 636b 2061  ng_rp, -- pick a
│ │ │ │ -00081e10: 2072 616e 646f 6d20 7261 7469 6f6e 616c   random rational
│ │ │ │ -00081e20: 2070 6f69 6e74 206f 6e20 6120 7072 6f6a   point on a proj
│ │ │ │ -00081e30: 6563 7469 7665 0a20 2020 2076 6172 6965  ective.    varie
│ │ │ │ -00081e40: 7479 0a20 202a 202a 6e6f 7465 2070 6f69  ty.  * *note poi
│ │ │ │ -00081e50: 6e74 2851 756f 7469 656e 7452 696e 6729  nt(QuotientRing)
│ │ │ │ -00081e60: 3a20 706f 696e 745f 6c70 5175 6f74 6965  : point_lpQuotie
│ │ │ │ -00081e70: 6e74 5269 6e67 5f72 702c 202d 2d20 7069  ntRing_rp, -- pi
│ │ │ │ -00081e80: 636b 2061 2072 616e 646f 6d0a 2020 2020  ck a random.    
│ │ │ │ -00081e90: 7261 7469 6f6e 616c 2070 6f69 6e74 206f  rational point o
│ │ │ │ -00081ea0: 6e20 6120 7072 6f6a 6563 7469 7665 2076  n a projective v
│ │ │ │ -00081eb0: 6172 6965 7479 0a0a 466f 7220 7468 6520  ariety..For the 
│ │ │ │ -00081ec0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00081ed0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -00081ee0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -00081ef0: 706f 696e 743a 2070 6f69 6e74 2c20 6973  point: point, is
│ │ │ │ -00081f00: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ -00081f10: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ -00081f20: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ -00081f30: 6e63 7469 6f6e 2c2e 0a1f 0a46 696c 653a  nction,....File:
│ │ │ │ -00081f40: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ -00081f50: 6f64 653a 2070 6f69 6e74 5f6c 7051 756f  ode: point_lpQuo
│ │ │ │ -00081f60: 7469 656e 7452 696e 675f 7270 2c20 4e65  tientRing_rp, Ne
│ │ │ │ -00081f70: 7874 3a20 7072 6f6a 6563 7469 7665 4465  xt: projectiveDe
│ │ │ │ -00081f80: 6772 6565 732c 2050 7265 763a 2070 6f69  grees, Prev: poi
│ │ │ │ -00081f90: 6e74 2c20 5570 3a20 546f 700a 0a70 6f69  nt, Up: Top..poi
│ │ │ │ -00081fa0: 6e74 2851 756f 7469 656e 7452 696e 6729  nt(QuotientRing)
│ │ │ │ -00081fb0: 202d 2d20 7069 636b 2061 2072 616e 646f   -- pick a rando
│ │ │ │ -00081fc0: 6d20 7261 7469 6f6e 616c 2070 6f69 6e74  m rational point
│ │ │ │ -00081fd0: 206f 6e20 6120 7072 6f6a 6563 7469 7665   on a projective
│ │ │ │ -00081fe0: 2076 6172 6965 7479 0a2a 2a2a 2a2a 2a2a   variety.*******
│ │ │ │ +00081cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363  **********..Desc
│ │ │ │ +00081d00: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00081d10: 3d3d 3d0a 0a53 6565 2070 6f69 6e74 284d  ===..See point(M
│ │ │ │ +00081d20: 756c 7469 7072 6f6a 6563 7469 7665 5661  ultiprojectiveVa
│ │ │ │ +00081d30: 7269 6574 6965 7329 2028 6d69 7373 696e  rieties) (missin
│ │ │ │ +00081d40: 6720 646f 6375 6d65 6e74 6174 696f 6e29  g documentation)
│ │ │ │ +00081d50: 2061 6e64 202a 6e6f 7465 0a70 6f69 6e74   and *note.point
│ │ │ │ +00081d60: 2851 756f 7469 656e 7452 696e 6729 3a20  (QuotientRing): 
│ │ │ │ +00081d70: 706f 696e 745f 6c70 5175 6f74 6965 6e74  point_lpQuotient
│ │ │ │ +00081d80: 5269 6e67 5f72 702c 2e0a 0a57 6179 7320  Ring_rp,...Ways 
│ │ │ │ +00081d90: 746f 2075 7365 2070 6f69 6e74 3a0a 3d3d  to use point:.==
│ │ │ │ +00081da0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00081db0: 0a0a 2020 2a20 2270 6f69 6e74 2850 6f6c  ..  * "point(Pol
│ │ │ │ +00081dc0: 796e 6f6d 6961 6c52 696e 6729 2220 2d2d  ynomialRing)" --
│ │ │ │ +00081dd0: 2073 6565 202a 6e6f 7465 2070 6f69 6e74   see *note point
│ │ │ │ +00081de0: 2851 756f 7469 656e 7452 696e 6729 3a0a  (QuotientRing):.
│ │ │ │ +00081df0: 2020 2020 706f 696e 745f 6c70 5175 6f74      point_lpQuot
│ │ │ │ +00081e00: 6965 6e74 5269 6e67 5f72 702c 202d 2d20  ientRing_rp, -- 
│ │ │ │ +00081e10: 7069 636b 2061 2072 616e 646f 6d20 7261  pick a random ra
│ │ │ │ +00081e20: 7469 6f6e 616c 2070 6f69 6e74 206f 6e20  tional point on 
│ │ │ │ +00081e30: 6120 7072 6f6a 6563 7469 7665 0a20 2020  a projective.   
│ │ │ │ +00081e40: 2076 6172 6965 7479 0a20 202a 202a 6e6f   variety.  * *no
│ │ │ │ +00081e50: 7465 2070 6f69 6e74 2851 756f 7469 656e  te point(Quotien
│ │ │ │ +00081e60: 7452 696e 6729 3a20 706f 696e 745f 6c70  tRing): point_lp
│ │ │ │ +00081e70: 5175 6f74 6965 6e74 5269 6e67 5f72 702c  QuotientRing_rp,
│ │ │ │ +00081e80: 202d 2d20 7069 636b 2061 2072 616e 646f   -- pick a rando
│ │ │ │ +00081e90: 6d0a 2020 2020 7261 7469 6f6e 616c 2070  m.    rational p
│ │ │ │ +00081ea0: 6f69 6e74 206f 6e20 6120 7072 6f6a 6563  oint on a projec
│ │ │ │ +00081eb0: 7469 7665 2076 6172 6965 7479 0a0a 466f  tive variety..Fo
│ │ │ │ +00081ec0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00081ed0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00081ee0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00081ef0: 2a6e 6f74 6520 706f 696e 743a 2070 6f69  *note point: poi
│ │ │ │ +00081f00: 6e74 2c20 6973 2061 202a 6e6f 7465 206d  nt, is a *note m
│ │ │ │ +00081f10: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a  ethod function:.
│ │ │ │ +00081f20: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ +00081f30: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a1f  thodFunction,...
│ │ │ │ +00081f40: 0a46 696c 653a 2043 7265 6d6f 6e61 2e69  .File: Cremona.i
│ │ │ │ +00081f50: 6e66 6f2c 204e 6f64 653a 2070 6f69 6e74  nfo, Node: point
│ │ │ │ +00081f60: 5f6c 7051 756f 7469 656e 7452 696e 675f  _lpQuotientRing_
│ │ │ │ +00081f70: 7270 2c20 4e65 7874 3a20 7072 6f6a 6563  rp, Next: projec
│ │ │ │ +00081f80: 7469 7665 4465 6772 6565 732c 2050 7265  tiveDegrees, Pre
│ │ │ │ +00081f90: 763a 2070 6f69 6e74 2c20 5570 3a20 546f  v: point, Up: To
│ │ │ │ +00081fa0: 700a 0a70 6f69 6e74 2851 756f 7469 656e  p..point(Quotien
│ │ │ │ +00081fb0: 7452 696e 6729 202d 2d20 7069 636b 2061  tRing) -- pick a
│ │ │ │ +00081fc0: 2072 616e 646f 6d20 7261 7469 6f6e 616c   random rational
│ │ │ │ +00081fd0: 2070 6f69 6e74 206f 6e20 6120 7072 6f6a   point on a proj
│ │ │ │ +00081fe0: 6563 7469 7665 2076 6172 6965 7479 0a2a  ective variety.*
│ │ │ │  00081ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00082000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00082010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00082020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00082030: 2a2a 2a2a 0a0a 5379 6e6f 7073 6973 0a3d  ****..Synopsis.=
│ │ │ │ -00082040: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2046 756e  =======..  * Fun
│ │ │ │ -00082050: 6374 696f 6e3a 202a 6e6f 7465 2070 6f69  ction: *note poi
│ │ │ │ -00082060: 6e74 3a20 706f 696e 742c 0a20 202a 2055  nt: point,.  * U
│ │ │ │ -00082070: 7361 6765 3a20 0a20 2020 2020 2020 2070  sage: .        p
│ │ │ │ -00082080: 6f69 6e74 2052 0a20 202a 2049 6e70 7574  oint R.  * Input
│ │ │ │ -00082090: 733a 0a20 2020 2020 202a 2052 2c20 6120  s:.      * R, a 
│ │ │ │ -000820a0: 2a6e 6f74 6520 7175 6f74 6965 6e74 2072  *note quotient r
│ │ │ │ -000820b0: 696e 673a 2028 4d61 6361 756c 6179 3244  ing: (Macaulay2D
│ │ │ │ -000820c0: 6f63 2951 756f 7469 656e 7452 696e 672c  oc)QuotientRing,
│ │ │ │ -000820d0: 2c20 7468 6520 686f 6d6f 6765 6e65 6f75  , the homogeneou
│ │ │ │ -000820e0: 730a 2020 2020 2020 2020 636f 6f72 6469  s.        coordi
│ │ │ │ -000820f0: 6e61 7465 2072 696e 6720 6f66 2061 2063  nate ring of a c
│ │ │ │ -00082100: 6c6f 7365 6420 7375 6273 6368 656d 6520  losed subscheme 
│ │ │ │ -00082110: 2458 5c73 7562 7365 7465 715c 6d61 7468  $X\subseteq\math
│ │ │ │ -00082120: 6262 7b50 7d5e 6e24 206f 7665 7220 610a  bb{P}^n$ over a.
│ │ │ │ -00082130: 2020 2020 2020 2020 6669 6e69 7465 2067          finite g
│ │ │ │ -00082140: 726f 756e 6420 6669 656c 640a 2020 2a20  round field.  * 
│ │ │ │ -00082150: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00082160: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a   an *note ideal:
│ │ │ │ -00082170: 2028 4d61 6361 756c 6179 3244 6f63 2949   (Macaulay2Doc)I
│ │ │ │ -00082180: 6465 616c 2c2c 2061 6e20 6964 6561 6c20  deal,, an ideal 
│ │ │ │ -00082190: 696e 2052 2064 6566 696e 696e 6720 6120  in R defining a 
│ │ │ │ -000821a0: 706f 696e 7420 6f6e 0a20 2020 2020 2020  point on.       
│ │ │ │ -000821b0: 2024 5824 0a0a 4465 7363 7269 7074 696f   $X$..Descriptio
│ │ │ │ -000821c0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
│ │ │ │ -000821d0: 6869 7320 6675 6e63 7469 6f6e 2069 7320  his function is 
│ │ │ │ -000821e0: 6120 7661 7269 616e 7420 6f66 2074 6865  a variant of the
│ │ │ │ -000821f0: 202a 6e6f 7465 2072 616e 646f 6d4b 5261   *note randomKRa
│ │ │ │ -00082200: 7469 6f6e 616c 506f 696e 743a 0a28 4d61  tionalPoint:.(Ma
│ │ │ │ -00082210: 6361 756c 6179 3244 6f63 2972 616e 646f  caulay2Doc)rando
│ │ │ │ -00082220: 6d4b 5261 7469 6f6e 616c 506f 696e 742c  mKRationalPoint,
│ │ │ │ -00082230: 2066 756e 6374 696f 6e2c 2077 6869 6368   function, which
│ │ │ │ -00082240: 2068 6173 2062 6565 6e20 6675 7274 6865   has been furthe
│ │ │ │ -00082250: 7220 696d 7072 6f76 6564 0a61 6e64 2065  r improved.and e
│ │ │ │ -00082260: 7874 656e 6465 6420 696e 2074 6865 2070  xtended in the p
│ │ │ │ -00082270: 6163 6b61 6765 204d 756c 7469 7072 6f6a  ackage Multiproj
│ │ │ │ -00082280: 6563 7469 7665 5661 7269 6574 6965 7320  ectiveVarieties 
│ │ │ │ -00082290: 286d 6973 7369 6e67 2064 6f63 756d 656e  (missing documen
│ │ │ │ -000822a0: 7461 7469 6f6e 292c 0a73 6565 202a 6e6f  tation),.see *no
│ │ │ │ -000822b0: 7465 2070 6f69 6e74 284d 756c 7469 7072  te point(Multipr
│ │ │ │ -000822c0: 6f6a 6563 7469 7665 5661 7269 6574 7929  ojectiveVariety)
│ │ │ │ -000822d0: 3a0a 284d 756c 7469 7072 6f6a 6563 7469  :.(Multiprojecti
│ │ │ │ -000822e0: 7665 5661 7269 6574 6965 7329 706f 696e  veVarieties)poin
│ │ │ │ -000822f0: 745f 6c70 4d75 6c74 6970 726f 6a65 6374  t_lpMultiproject
│ │ │ │ -00082300: 6976 6556 6172 6965 7479 5f72 702c 2e0a  iveVariety_rp,..
│ │ │ │ -00082310: 0a42 656c 6f77 2077 6520 7665 7269 6679  .Below we verify
│ │ │ │ -00082320: 2074 6865 2062 6972 6174 696f 6e61 6c69   the birationali
│ │ │ │ -00082330: 7479 206f 6620 6120 7261 7469 6f6e 616c  ty of a rational
│ │ │ │ -00082340: 206d 6170 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d   map...+--------
│ │ │ │ +00082030: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ +00082040: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ +00082050: 202a 2046 756e 6374 696f 6e3a 202a 6e6f   * Function: *no
│ │ │ │ +00082060: 7465 2070 6f69 6e74 3a20 706f 696e 742c  te point: point,
│ │ │ │ +00082070: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00082080: 2020 2020 2070 6f69 6e74 2052 0a20 202a       point R.  *
│ │ │ │ +00082090: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +000820a0: 2052 2c20 6120 2a6e 6f74 6520 7175 6f74   R, a *note quot
│ │ │ │ +000820b0: 6965 6e74 2072 696e 673a 2028 4d61 6361  ient ring: (Maca
│ │ │ │ +000820c0: 756c 6179 3244 6f63 2951 756f 7469 656e  ulay2Doc)Quotien
│ │ │ │ +000820d0: 7452 696e 672c 2c20 7468 6520 686f 6d6f  tRing,, the homo
│ │ │ │ +000820e0: 6765 6e65 6f75 730a 2020 2020 2020 2020  geneous.        
│ │ │ │ +000820f0: 636f 6f72 6469 6e61 7465 2072 696e 6720  coordinate ring 
│ │ │ │ +00082100: 6f66 2061 2063 6c6f 7365 6420 7375 6273  of a closed subs
│ │ │ │ +00082110: 6368 656d 6520 2458 5c73 7562 7365 7465  cheme $X\subsete
│ │ │ │ +00082120: 715c 6d61 7468 6262 7b50 7d5e 6e24 206f  q\mathbb{P}^n$ o
│ │ │ │ +00082130: 7665 7220 610a 2020 2020 2020 2020 6669  ver a.        fi
│ │ │ │ +00082140: 6e69 7465 2067 726f 756e 6420 6669 656c  nite ground fiel
│ │ │ │ +00082150: 640a 2020 2a20 4f75 7470 7574 733a 0a20  d.  * Outputs:. 
│ │ │ │ +00082160: 2020 2020 202a 2061 6e20 2a6e 6f74 6520       * an *note 
│ │ │ │ +00082170: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ +00082180: 3244 6f63 2949 6465 616c 2c2c 2061 6e20  2Doc)Ideal,, an 
│ │ │ │ +00082190: 6964 6561 6c20 696e 2052 2064 6566 696e  ideal in R defin
│ │ │ │ +000821a0: 696e 6720 6120 706f 696e 7420 6f6e 0a20  ing a point on. 
│ │ │ │ +000821b0: 2020 2020 2020 2024 5824 0a0a 4465 7363         $X$..Desc
│ │ │ │ +000821c0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +000821d0: 3d3d 3d0a 0a54 6869 7320 6675 6e63 7469  ===..This functi
│ │ │ │ +000821e0: 6f6e 2069 7320 6120 7661 7269 616e 7420  on is a variant 
│ │ │ │ +000821f0: 6f66 2074 6865 202a 6e6f 7465 2072 616e  of the *note ran
│ │ │ │ +00082200: 646f 6d4b 5261 7469 6f6e 616c 506f 696e  domKRationalPoin
│ │ │ │ +00082210: 743a 0a28 4d61 6361 756c 6179 3244 6f63  t:.(Macaulay2Doc
│ │ │ │ +00082220: 2972 616e 646f 6d4b 5261 7469 6f6e 616c  )randomKRational
│ │ │ │ +00082230: 506f 696e 742c 2066 756e 6374 696f 6e2c  Point, function,
│ │ │ │ +00082240: 2077 6869 6368 2068 6173 2062 6565 6e20   which has been 
│ │ │ │ +00082250: 6675 7274 6865 7220 696d 7072 6f76 6564  further improved
│ │ │ │ +00082260: 0a61 6e64 2065 7874 656e 6465 6420 696e  .and extended in
│ │ │ │ +00082270: 2074 6865 2070 6163 6b61 6765 204d 756c   the package Mul
│ │ │ │ +00082280: 7469 7072 6f6a 6563 7469 7665 5661 7269  tiprojectiveVari
│ │ │ │ +00082290: 6574 6965 7320 286d 6973 7369 6e67 2064  eties (missing d
│ │ │ │ +000822a0: 6f63 756d 656e 7461 7469 6f6e 292c 0a73  ocumentation),.s
│ │ │ │ +000822b0: 6565 202a 6e6f 7465 2070 6f69 6e74 284d  ee *note point(M
│ │ │ │ +000822c0: 756c 7469 7072 6f6a 6563 7469 7665 5661  ultiprojectiveVa
│ │ │ │ +000822d0: 7269 6574 7929 3a0a 284d 756c 7469 7072  riety):.(Multipr
│ │ │ │ +000822e0: 6f6a 6563 7469 7665 5661 7269 6574 6965  ojectiveVarietie
│ │ │ │ +000822f0: 7329 706f 696e 745f 6c70 4d75 6c74 6970  s)point_lpMultip
│ │ │ │ +00082300: 726f 6a65 6374 6976 6556 6172 6965 7479  rojectiveVariety
│ │ │ │ +00082310: 5f72 702c 2e0a 0a42 656c 6f77 2077 6520  _rp,...Below we 
│ │ │ │ +00082320: 7665 7269 6679 2074 6865 2062 6972 6174  verify the birat
│ │ │ │ +00082330: 696f 6e61 6c69 7479 206f 6620 6120 7261  ionality of a ra
│ │ │ │ +00082340: 7469 6f6e 616c 206d 6170 2e0a 0a2b 2d2d  tional map...+--
│ │ │ │  00082350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00082390: 2d2d 2d2d 2d2b 0a7c 6931 203a 2066 203d  -----+.|i1 : f =
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│ │ │ │ -000823b0: 6961 6c51 7561 6472 6174 6963 5472 616e  ialQuadraticTran
│ │ │ │ -000823c0: 7366 6f72 6d61 7469 6f6e 2839 2c5a 5a2f  sformation(9,ZZ/
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│ │ │ │ -000823e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00082390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +000823a0: 203a 2066 203d 2069 6e76 6572 7365 4d61   : f = inverseMa
│ │ │ │ +000823b0: 7020 7370 6563 6961 6c51 7561 6472 6174  p specialQuadrat
│ │ │ │ +000823c0: 6963 5472 616e 7366 6f72 6d61 7469 6f6e  icTransformation
│ │ │ │ +000823d0: 2839 2c5a 5a2f 3333 3333 3129 3b20 2020  (9,ZZ/33331);   
│ │ │ │ +000823e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000823f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082430: 2020 2020 207c 0a7c 6f31 203a 2052 6174       |.|o1 : Rat
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│ │ │ │ -00082470: 7375 6276 6172 6965 7479 206f 6620 5050  subvariety of PP
│ │ │ │ -00082480: 5e31 3120 207c 0a7c 2d2d 2d2d 2d2d 2d2d  ^11  |.|--------
│ │ │ │ +00082430: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00082440: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
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│ │ │ │  00082490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000824a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000824b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000824c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000824d0: 2d2d 2d2d 2d7c 0a7c 746f 2050 505e 3829  -----|.|to PP^8)
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│ │ │ │ +000824d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 746f  -----------|.|to
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│ │ │ │  000824f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082520: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00082520: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00082530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000825a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000825b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000825c0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
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│ │ │ │ -00082610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000825c0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +000825d0: 2d20 7573 6564 2030 2e32 3335 3031 3973  - used 0.235019s
│ │ │ │ +000825e0: 2028 6370 7529 3b20 302e 3136 3630 3132   (cpu); 0.166012
│ │ │ │ +000825f0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00082600: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00082610: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00082620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082660: 2020 2020 207c 0a7c 6f32 203d 2069 6465       |.|o2 = ide
│ │ │ │ -00082670: 616c 2028 7920 2020 2d20 3932 3335 7920  al (y   - 9235y 
│ │ │ │ -00082680: 202c 2079 2020 2b20 3131 3037 3579 2020   , y  + 11075y  
│ │ │ │ -00082690: 2c20 7920 202d 2035 3834 3779 2020 2c20  , y  - 5847y  , 
│ │ │ │ -000826a0: 7920 202b 2037 3339 3679 2020 2c20 7920  y  + 7396y  , y 
│ │ │ │ -000826b0: 202b 2020 207c 0a7c 2020 2020 2020 2020   +   |.|        
│ │ │ │ -000826c0: 2020 2020 2031 3020 2020 2020 2020 2031       10        1
│ │ │ │ -000826d0: 3120 2020 3920 2020 2020 2020 2020 3131  1   9         11
│ │ │ │ -000826e0: 2020 2038 2020 2020 2020 2020 3131 2020     8        11  
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│ │ │ │  00082c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00082cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00082d40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00082cf0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
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│ │ │ │ +00082d40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00082d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00082da0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
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│ │ │ │ +00082da0: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │  00082db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082de0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00082de0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │  00082e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00082ea0: 646f 6d20 4b20 7261 7469 6f6e 616c 2070  dom K rational p
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│ │ │ │ -00082ef0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00082f10: 2850 6f6c 796e 6f6d 6961 6c52 696e 6729  (PolynomialRing)
│ │ │ │ -00082f20: 220a 2020 2a20 2a6e 6f74 6520 706f 696e  ".  * *note poin
│ │ │ │ -00082f30: 7428 5175 6f74 6965 6e74 5269 6e67 293a  t(QuotientRing):
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│ │ │ │ -00083070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00082e50: 0a20 202a 202a 6e6f 7465 2072 616e 646f  .  * *note rando
│ │ │ │ +00082e60: 6d4b 5261 7469 6f6e 616c 506f 696e 743a  mKRationalPoint:
│ │ │ │ +00082e70: 2028 4d61 6361 756c 6179 3244 6f63 2972   (Macaulay2Doc)r
│ │ │ │ +00082e80: 616e 646f 6d4b 5261 7469 6f6e 616c 506f  andomKRationalPo
│ │ │ │ +00082e90: 696e 742c 202d 2d20 5069 636b 2061 0a20  int, -- Pick a. 
│ │ │ │ +00082ea0: 2020 2072 616e 646f 6d20 4b20 7261 7469     random K rati
│ │ │ │ +00082eb0: 6f6e 616c 2070 6f69 6e74 206f 6e20 7468  onal point on th
│ │ │ │ +00082ec0: 6520 7363 6865 6d65 2058 2064 6566 696e  e scheme X defin
│ │ │ │ +00082ed0: 6564 2062 7920 490a 0a57 6179 7320 746f  ed by I..Ways to
│ │ │ │ +00082ee0: 2075 7365 2074 6869 7320 6d65 7468 6f64   use this method
│ │ │ │ +00082ef0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +00082f00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00082f10: 2270 6f69 6e74 2850 6f6c 796e 6f6d 6961  "point(Polynomia
│ │ │ │ +00082f20: 6c52 696e 6729 220a 2020 2a20 2a6e 6f74  lRing)".  * *not
│ │ │ │ +00082f30: 6520 706f 696e 7428 5175 6f74 6965 6e74  e point(Quotient
│ │ │ │ +00082f40: 5269 6e67 293a 2070 6f69 6e74 5f6c 7051  Ring): point_lpQ
│ │ │ │ +00082f50: 756f 7469 656e 7452 696e 675f 7270 2c20  uotientRing_rp, 
│ │ │ │ +00082f60: 2d2d 2070 6963 6b20 6120 7261 6e64 6f6d  -- pick a random
│ │ │ │ +00082f70: 0a20 2020 2072 6174 696f 6e61 6c20 706f  .    rational po
│ │ │ │ +00082f80: 696e 7420 6f6e 2061 2070 726f 6a65 6374  int on a project
│ │ │ │ +00082f90: 6976 6520 7661 7269 6574 790a 1f0a 4669  ive variety...Fi
│ │ │ │ +00082fa0: 6c65 3a20 4372 656d 6f6e 612e 696e 666f  le: Cremona.info
│ │ │ │ +00082fb0: 2c20 4e6f 6465 3a20 7072 6f6a 6563 7469  , Node: projecti
│ │ │ │ +00082fc0: 7665 4465 6772 6565 732c 204e 6578 743a  veDegrees, Next:
│ │ │ │ +00082fd0: 2070 726f 6a65 6374 6976 6544 6567 7265   projectiveDegre
│ │ │ │ +00082fe0: 6573 5f6c 7052 6174 696f 6e61 6c4d 6170  es_lpRationalMap
│ │ │ │ +00082ff0: 5f72 702c 2050 7265 763a 2070 6f69 6e74  _rp, Prev: point
│ │ │ │ +00083000: 5f6c 7051 756f 7469 656e 7452 696e 675f  _lpQuotientRing_
│ │ │ │ +00083010: 7270 2c20 5570 3a20 546f 700a 0a70 726f  rp, Up: Top..pro
│ │ │ │ +00083020: 6a65 6374 6976 6544 6567 7265 6573 202d  jectiveDegrees -
│ │ │ │ +00083030: 2d20 7072 6f6a 6563 7469 7665 2064 6567  - projective deg
│ │ │ │ +00083040: 7265 6573 206f 6620 6120 7261 7469 6f6e  rees of a ration
│ │ │ │ +00083050: 616c 206d 6170 2062 6574 7765 656e 2070  al map between p
│ │ │ │ +00083060: 726f 6a65 6374 6976 6520 7661 7269 6574  rojective variet
│ │ │ │ +00083070: 6965 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ies.************
│ │ │ │  00083080: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00083090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000830a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000830b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000830c0: 2a2a 2a2a 0a0a 5379 6e6f 7073 6973 0a3d  ****..Synopsis.=
│ │ │ │ -000830d0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055 7361  =======..  * Usa
│ │ │ │ -000830e0: 6765 3a20 0a20 2020 2020 2020 2070 726f  ge: .        pro
│ │ │ │ -000830f0: 6a65 6374 6976 6544 6567 7265 6573 2070  jectiveDegrees p
│ │ │ │ -00083100: 6869 0a20 202a 2049 6e70 7574 733a 0a20  hi.  * Inputs:. 
│ │ │ │ -00083110: 2020 2020 202a 2070 6869 2c20 6120 2a6e       * phi, a *n
│ │ │ │ -00083120: 6f74 6520 7269 6e67 206d 6170 3a20 284d  ote ring map: (M
│ │ │ │ -00083130: 6163 6175 6c61 7932 446f 6329 5269 6e67  acaulay2Doc)Ring
│ │ │ │ -00083140: 4d61 702c 2c20 7768 6963 6820 7265 7072  Map,, which repr
│ │ │ │ -00083150: 6573 656e 7473 2061 0a20 2020 2020 2020  esents a.       
│ │ │ │ -00083160: 2072 6174 696f 6e61 6c20 6d61 7020 245c   rational map $\
│ │ │ │ -00083170: 5068 6924 2062 6574 7765 656e 2070 726f  Phi$ between pro
│ │ │ │ -00083180: 6a65 6374 6976 6520 7661 7269 6574 6965  jective varietie
│ │ │ │ -00083190: 730a 2020 2a20 2a6e 6f74 6520 4f70 7469  s.  * *note Opti
│ │ │ │ -000831a0: 6f6e 616c 2069 6e70 7574 733a 2028 4d61  onal inputs: (Ma
│ │ │ │ -000831b0: 6361 756c 6179 3244 6f63 2975 7369 6e67  caulay2Doc)using
│ │ │ │ -000831c0: 2066 756e 6374 696f 6e73 2077 6974 6820   functions with 
│ │ │ │ -000831d0: 6f70 7469 6f6e 616c 2069 6e70 7574 732c  optional inputs,
│ │ │ │ -000831e0: 3a0a 2020 2020 2020 2a20 2a6e 6f74 6520  :.      * *note 
│ │ │ │ -000831f0: 426c 6f77 5570 5374 7261 7465 6779 3a20  BlowUpStrategy: 
│ │ │ │ -00083200: 426c 6f77 5570 5374 7261 7465 6779 2c20  BlowUpStrategy, 
│ │ │ │ -00083210: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -00083220: 7661 6c75 650a 2020 2020 2020 2020 2245  value.        "E
│ │ │ │ -00083230: 6c69 6d69 6e61 7465 222c 0a20 2020 2020  liminate",.     
│ │ │ │ -00083240: 202a 202a 6e6f 7465 2043 6572 7469 6679   * *note Certify
│ │ │ │ -00083250: 3a20 4365 7274 6966 792c 203d 3e20 2e2e  : Certify, => ..
│ │ │ │ -00083260: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -00083270: 2066 616c 7365 2c20 7768 6574 6865 7220   false, whether 
│ │ │ │ -00083280: 746f 2065 6e73 7572 650a 2020 2020 2020  to ensure.      
│ │ │ │ -00083290: 2020 636f 7272 6563 746e 6573 7320 6f66    correctness of
│ │ │ │ -000832a0: 206f 7574 7075 740a 2020 2020 2020 2a20   output.      * 
│ │ │ │ -000832b0: 2a6e 6f74 6520 4e75 6d44 6567 7265 6573  *note NumDegrees
│ │ │ │ -000832c0: 3a20 4e75 6d44 6567 7265 6573 2c20 3d3e  : NumDegrees, =>
│ │ │ │ -000832d0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -000832e0: 6c75 6520 696e 6669 6e69 7479 2c20 0a20  lue infinity, . 
│ │ │ │ -000832f0: 2020 2020 202a 202a 6e6f 7465 2056 6572       * *note Ver
│ │ │ │ -00083300: 626f 7365 3a20 696e 7665 7273 654d 6170  bose: inverseMap
│ │ │ │ -00083310: 5f6c 705f 7064 5f70 645f 7064 5f63 6d56  _lp_pd_pd_pd_cmV
│ │ │ │ -00083320: 6572 626f 7365 3d3e 5f70 645f 7064 5f70  erbose=>_pd_pd_p
│ │ │ │ -00083330: 645f 7270 2c20 3d3e 202e 2e2e 2c0a 2020  d_rp, => ...,.  
│ │ │ │ -00083340: 2020 2020 2020 6465 6661 756c 7420 7661        default va
│ │ │ │ -00083350: 6c75 6520 7472 7565 2c0a 2020 2a20 4f75  lue true,.  * Ou
│ │ │ │ -00083360: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ -00083370: 202a 6e6f 7465 206c 6973 743a 2028 4d61   *note list: (Ma
│ │ │ │ -00083380: 6361 756c 6179 3244 6f63 294c 6973 742c  caulay2Doc)List,
│ │ │ │ -00083390: 2c20 7468 6520 6c69 7374 206f 6620 7468  , the list of th
│ │ │ │ -000833a0: 6520 7072 6f6a 6563 7469 7665 2064 6567  e projective deg
│ │ │ │ -000833b0: 7265 6573 0a20 2020 2020 2020 206f 6620  rees.        of 
│ │ │ │ -000833c0: 245c 5068 6924 0a0a 4465 7363 7269 7074  $\Phi$..Descript
│ │ │ │ -000833d0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -000833e0: 0a4c 6574 2024 5c70 6869 3a4b 5b79 5f30  .Let $\phi:K[y_0
│ │ │ │ -000833f0: 2c5c 6c64 6f74 732c 795f 6d5d 2f4a 205c  ,\ldots,y_m]/J \
│ │ │ │ -00083400: 746f 204b 5b78 5f30 2c5c 6c64 6f74 732c  to K[x_0,\ldots,
│ │ │ │ -00083410: 785f 6e5d 2f49 2420 6265 2061 2072 696e  x_n]/I$ be a rin
│ │ │ │ -00083420: 6720 6d61 700a 7265 7072 6573 656e 7469  g map.representi
│ │ │ │ -00083430: 6e67 2061 2072 6174 696f 6e61 6c20 6d61  ng a rational ma
│ │ │ │ -00083440: 7020 245c 5068 693a 2056 2849 2920 5c73  p $\Phi: V(I) \s
│ │ │ │ -00083450: 7562 7365 7465 710a 5c6d 6174 6862 627b  ubseteq.\mathbb{
│ │ │ │ -00083460: 507d 5e6e 3d50 726f 6a28 4b5b 785f 302c  P}^n=Proj(K[x_0,
│ │ │ │ -00083470: 5c6c 646f 7473 2c78 5f6e 5d29 205c 6461  \ldots,x_n]) \da
│ │ │ │ -00083480: 7368 7269 6768 7461 7272 6f77 2056 284a  shrightarrow V(J
│ │ │ │ -00083490: 2920 5c73 7562 7365 7465 710a 5c6d 6174  ) \subseteq.\mat
│ │ │ │ -000834a0: 6862 627b 507d 5e6d 3d50 726f 6a28 4b5b  hbb{P}^m=Proj(K[
│ │ │ │ -000834b0: 795f 302c 5c6c 646f 7473 2c79 5f6d 5d29  y_0,\ldots,y_m])
│ │ │ │ -000834c0: 242e 2054 6865 2024 6924 2d74 6820 7072  $. The $i$-th pr
│ │ │ │ -000834d0: 6f6a 6563 7469 7665 2064 6567 7265 6520  ojective degree 
│ │ │ │ -000834e0: 6f66 2024 5c50 6869 240a 6973 2064 6566  of $\Phi$.is def
│ │ │ │ -000834f0: 696e 6564 2069 6e20 7465 726d 7320 6f66  ined in terms of
│ │ │ │ -00083500: 2064 696d 656e 7369 6f6e 2061 6e64 2064   dimension and d
│ │ │ │ -00083510: 6567 7265 6520 6f66 2074 6865 2063 6c6f  egree of the clo
│ │ │ │ -00083520: 7375 7265 206f 6620 245c 5068 695e 7b2d  sure of $\Phi^{-
│ │ │ │ -00083530: 317d 284c 2924 2c0a 7768 6572 6520 244c  1}(L)$,.where $L
│ │ │ │ -00083540: 2420 6973 2061 2067 656e 6572 616c 206c  $ is a general l
│ │ │ │ -00083550: 696e 6561 7220 7375 6273 7061 6365 206f  inear subspace o
│ │ │ │ -00083560: 6620 245c 6d61 7468 6262 7b50 7d5e 6d24  f $\mathbb{P}^m$
│ │ │ │ -00083570: 206f 6620 6120 6365 7274 6169 6e0a 6469   of a certain.di
│ │ │ │ -00083580: 6d65 6e73 696f 6e3b 2066 6f72 2074 6865  mension; for the
│ │ │ │ -00083590: 2070 7265 6369 7365 2064 6566 696e 6974   precise definit
│ │ │ │ -000835a0: 696f 6e2c 2073 6565 2048 6172 7269 7327  ion, see Harris'
│ │ │ │ -000835b0: 7320 626f 6f6b 2028 416c 6765 6272 6169  s book (Algebrai
│ │ │ │ -000835c0: 6320 6765 6f6d 6574 7279 3a20 410a 6669  c geometry: A.fi
│ │ │ │ -000835d0: 7273 7420 636f 7572 7365 202d 2056 6f6c  rst course - Vol
│ │ │ │ -000835e0: 2e20 3133 3320 6f66 2047 7261 642e 2054  . 133 of Grad. T
│ │ │ │ -000835f0: 6578 7473 2069 6e20 4d61 7468 2e2c 2070  exts in Math., p
│ │ │ │ -00083600: 2e20 3234 3029 2e20 4966 2024 5c50 6869  . 240). If $\Phi
│ │ │ │ -00083610: 2420 6973 2064 6566 696e 6564 0a62 7920  $ is defined.by 
│ │ │ │ -00083620: 656c 656d 656e 7473 2024 465f 3028 785f  elements $F_0(x_
│ │ │ │ -00083630: 302c 5c6c 646f 7473 2c78 5f6e 292c 5c6c  0,\ldots,x_n),\l
│ │ │ │ -00083640: 646f 7473 2c46 5f6d 2878 5f30 2c5c 6c64  dots,F_m(x_0,\ld
│ │ │ │ -00083650: 6f74 732c 785f 6e29 2420 616e 6420 2449  ots,x_n)$ and $I
│ │ │ │ -00083660: 5f4c 2420 6465 6e6f 7465 730a 7468 6520  _L$ denotes.the 
│ │ │ │ -00083670: 6964 6561 6c20 6f66 2074 6865 2073 7562  ideal of the sub
│ │ │ │ -00083680: 7370 6163 6520 244c 5c73 7562 7365 7465  space $L\subsete
│ │ │ │ -00083690: 7120 5c6d 6174 6862 627b 507d 5e6d 242c  q \mathbb{P}^m$,
│ │ │ │ -000836a0: 2074 6865 6e20 7468 6520 6964 6561 6c20   then the ideal 
│ │ │ │ -000836b0: 6f66 2074 6865 0a63 6c6f 7375 7265 206f  of the.closure o
│ │ │ │ -000836c0: 6620 245c 5068 695e 7b2d 317d 284c 2920  f $\Phi^{-1}(L) 
│ │ │ │ -000836d0: 2420 6973 206e 6f74 6869 6e67 2062 7574  $ is nothing but
│ │ │ │ -000836e0: 2074 6865 2073 6174 7572 6174 696f 6e20   the saturation 
│ │ │ │ -000836f0: 6f66 2074 6865 2069 6465 616c 0a24 285c  of the ideal.$(\
│ │ │ │ -00083700: 7068 6928 495f 4c29 2924 2062 7920 2428  phi(I_L))$ by $(
│ │ │ │ -00083710: 465f 302c 2e2e 2e2e 2c46 5f6d 2924 2069  F_0,....,F_m)$ i
│ │ │ │ -00083720: 6e20 7468 6520 7269 6e67 2024 4b5b 785f  n the ring $K[x_
│ │ │ │ -00083730: 302c 5c6c 646f 7473 2c78 5f6e 5d2f 4924  0,\ldots,x_n]/I$
│ │ │ │ -00083740: 2e20 536f 2c0a 7265 706c 6163 696e 6720  . So,.replacing 
│ │ │ │ -00083750: 696e 2074 6865 2064 6566 696e 6974 696f  in the definitio
│ │ │ │ -00083760: 6e2c 2067 656e 6572 616c 206c 696e 6561  n, general linea
│ │ │ │ -00083770: 7220 7375 6273 7061 6365 2062 7920 7261  r subspace by ra
│ │ │ │ -00083780: 6e64 6f6d 206c 696e 6561 7220 7375 6273  ndom linear subs
│ │ │ │ -00083790: 7061 6365 2c0a 7765 2067 6574 2061 2070  pace,.we get a p
│ │ │ │ -000837a0: 726f 6261 6269 6c69 7374 6963 2061 6c67  robabilistic alg
│ │ │ │ -000837b0: 6f72 6974 686d 2074 6f20 636f 6d70 7574  orithm to comput
│ │ │ │ -000837c0: 6520 616c 6c20 7072 6f6a 6563 7469 7665  e all projective
│ │ │ │ -000837d0: 2064 6567 7265 6573 2e0a 4675 7274 6865   degrees..Furthe
│ │ │ │ -000837e0: 726d 6f72 652c 2077 6520 6361 6e20 636f  rmore, we can co
│ │ │ │ -000837f0: 6e73 6964 6572 6162 6c79 2073 7065 6564  nsiderably speed
│ │ │ │ -00083800: 2075 7020 7468 6973 2061 6c67 6f72 6974   up this algorit
│ │ │ │ -00083810: 686d 2062 7920 7461 6b69 6e67 2069 6e74  hm by taking int
│ │ │ │ -00083820: 6f20 6163 636f 756e 740a 7477 6f20 7369  o account.two si
│ │ │ │ -00083830: 6d70 6c65 2072 656d 6172 6b73 3a20 3129  mple remarks: 1)
│ │ │ │ -00083840: 2074 6865 2073 6174 7572 6174 696f 6e20   the saturation 
│ │ │ │ -00083850: 2428 5c70 6869 2849 5f4c 2929 3a7b 2846  $(\phi(I_L)):{(F
│ │ │ │ -00083860: 5f30 2c5c 6c64 6f74 732c 465f 6d29 7d5e  _0,\ldots,F_m)}^
│ │ │ │ -00083870: 7b5c 696e 6674 797d 240a 6973 2074 6865  {\infty}$.is the
│ │ │ │ -00083880: 2073 616d 6520 6173 2024 285c 7068 6928   same as $(\phi(
│ │ │ │ -00083890: 495f 4c29 293a 7b28 5c6c 616d 6264 615f  I_L)):{(\lambda_
│ │ │ │ -000838a0: 3020 465f 302b 5c63 646f 7473 2b5c 6c61  0 F_0+\cdots+\la
│ │ │ │ -000838b0: 6d62 6461 5f6d 2046 5f6d 297d 5e7b 5c69  mbda_m F_m)}^{\i
│ │ │ │ -000838c0: 6e66 7479 7d24 2c0a 7768 6572 6520 245c  nfty}$,.where $\
│ │ │ │ -000838d0: 6c61 6d62 6461 5f30 2c5c 6c64 6f74 732c  lambda_0,\ldots,
│ │ │ │ -000838e0: 5c6c 616d 6264 615f 6d5c 696e 5c6d 6174  \lambda_m\in\mat
│ │ │ │ -000838f0: 6862 627b 4b7d 2420 6172 6520 6765 6e65  hbb{K}$ are gene
│ │ │ │ -00083900: 7261 6c20 7363 616c 6172 733b 2032 2920  ral scalars; 2) 
│ │ │ │ -00083910: 7468 650a 2469 242d 7468 2070 726f 6a65  the.$i$-th proje
│ │ │ │ -00083920: 6374 6976 6520 6465 6772 6565 206f 6620  ctive degree of 
│ │ │ │ -00083930: 245c 5068 6924 2063 6f69 6e63 6964 6573  $\Phi$ coincides
│ │ │ │ -00083940: 2077 6974 6820 7468 6520 2428 692d 3129   with the $(i-1)
│ │ │ │ -00083950: 242d 7468 2070 726f 6a65 6374 6976 650a  $-th projective.
│ │ │ │ -00083960: 6465 6772 6565 206f 6620 7468 6520 7265  degree of the re
│ │ │ │ -00083970: 7374 7269 6374 696f 6e20 6f66 2024 5c50  striction of $\P
│ │ │ │ -00083980: 6869 2420 746f 2061 2067 656e 6572 616c  hi$ to a general
│ │ │ │ -00083990: 2068 7970 6572 706c 616e 6520 7365 6374   hyperplane sect
│ │ │ │ -000839a0: 696f 6e20 6f66 2024 5824 2028 7365 650a  ion of $X$ (see.
│ │ │ │ -000839b0: 6c6f 632e 2063 6974 2e29 2e20 5468 6973  loc. cit.). This
│ │ │ │ -000839c0: 2069 7320 7768 6174 2074 6865 206d 6574   is what the met
│ │ │ │ -000839d0: 686f 6420 7573 6573 2069 6620 2a6e 6f74  hod uses if *not
│ │ │ │ -000839e0: 6520 4365 7274 6966 793a 2043 6572 7469  e Certify: Certi
│ │ │ │ -000839f0: 6679 2c20 6973 2073 6574 2074 6f0a 6661  fy, is set to.fa
│ │ │ │ -00083a00: 6c73 652e 2049 6620 696e 7374 6561 6420  lse. If instead 
│ │ │ │ -00083a10: 2a6e 6f74 6520 4365 7274 6966 793a 2043  *note Certify: C
│ │ │ │ -00083a20: 6572 7469 6679 2c20 6973 2073 6574 2074  ertify, is set t
│ │ │ │ -00083a30: 6f20 7472 7565 2c20 7468 656e 2074 6865  o true, then the
│ │ │ │ -00083a40: 206d 6574 686f 640a 7369 6d70 6c79 2063   method.simply c
│ │ │ │ -00083a50: 6f6d 7075 7465 7320 7468 6520 2a6e 6f74  omputes the *not
│ │ │ │ -00083a60: 6520 6d75 6c74 6964 6567 7265 653a 2028  e multidegree: (
│ │ │ │ -00083a70: 4d61 6361 756c 6179 3244 6f63 296d 756c  Macaulay2Doc)mul
│ │ │ │ -00083a80: 7469 6465 6772 6565 2c20 6f66 2074 6865  tidegree, of the
│ │ │ │ -00083a90: 202a 6e6f 7465 0a67 7261 7068 3a20 6772   *note.graph: gr
│ │ │ │ -00083aa0: 6170 682c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  aph,...+--------
│ │ │ │ +000830c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ +000830d0: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ +000830e0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +000830f0: 2020 2070 726f 6a65 6374 6976 6544 6567     projectiveDeg
│ │ │ │ +00083100: 7265 6573 2070 6869 0a20 202a 2049 6e70  rees phi.  * Inp
│ │ │ │ +00083110: 7574 733a 0a20 2020 2020 202a 2070 6869  uts:.      * phi
│ │ │ │ +00083120: 2c20 6120 2a6e 6f74 6520 7269 6e67 206d  , a *note ring m
│ │ │ │ +00083130: 6170 3a20 284d 6163 6175 6c61 7932 446f  ap: (Macaulay2Do
│ │ │ │ +00083140: 6329 5269 6e67 4d61 702c 2c20 7768 6963  c)RingMap,, whic
│ │ │ │ +00083150: 6820 7265 7072 6573 656e 7473 2061 0a20  h represents a. 
│ │ │ │ +00083160: 2020 2020 2020 2072 6174 696f 6e61 6c20         rational 
│ │ │ │ +00083170: 6d61 7020 245c 5068 6924 2062 6574 7765  map $\Phi$ betwe
│ │ │ │ +00083180: 656e 2070 726f 6a65 6374 6976 6520 7661  en projective va
│ │ │ │ +00083190: 7269 6574 6965 730a 2020 2a20 2a6e 6f74  rieties.  * *not
│ │ │ │ +000831a0: 6520 4f70 7469 6f6e 616c 2069 6e70 7574  e Optional input
│ │ │ │ +000831b0: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ +000831c0: 2975 7369 6e67 2066 756e 6374 696f 6e73  )using functions
│ │ │ │ +000831d0: 2077 6974 6820 6f70 7469 6f6e 616c 2069   with optional i
│ │ │ │ +000831e0: 6e70 7574 732c 3a0a 2020 2020 2020 2a20  nputs,:.      * 
│ │ │ │ +000831f0: 2a6e 6f74 6520 426c 6f77 5570 5374 7261  *note BlowUpStra
│ │ │ │ +00083200: 7465 6779 3a20 426c 6f77 5570 5374 7261  tegy: BlowUpStra
│ │ │ │ +00083210: 7465 6779 2c20 3d3e 202e 2e2e 2c20 6465  tegy, => ..., de
│ │ │ │ +00083220: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ +00083230: 2020 2020 2245 6c69 6d69 6e61 7465 222c      "Eliminate",
│ │ │ │ +00083240: 0a20 2020 2020 202a 202a 6e6f 7465 2043  .      * *note C
│ │ │ │ +00083250: 6572 7469 6679 3a20 4365 7274 6966 792c  ertify: Certify,
│ │ │ │ +00083260: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +00083270: 2076 616c 7565 2066 616c 7365 2c20 7768   value false, wh
│ │ │ │ +00083280: 6574 6865 7220 746f 2065 6e73 7572 650a  ether to ensure.
│ │ │ │ +00083290: 2020 2020 2020 2020 636f 7272 6563 746e          correctn
│ │ │ │ +000832a0: 6573 7320 6f66 206f 7574 7075 740a 2020  ess of output.  
│ │ │ │ +000832b0: 2020 2020 2a20 2a6e 6f74 6520 4e75 6d44      * *note NumD
│ │ │ │ +000832c0: 6567 7265 6573 3a20 4e75 6d44 6567 7265  egrees: NumDegre
│ │ │ │ +000832d0: 6573 2c20 3d3e 202e 2e2e 2c20 6465 6661  es, => ..., defa
│ │ │ │ +000832e0: 756c 7420 7661 6c75 6520 696e 6669 6e69  ult value infini
│ │ │ │ +000832f0: 7479 2c20 0a20 2020 2020 202a 202a 6e6f  ty, .      * *no
│ │ │ │ +00083300: 7465 2056 6572 626f 7365 3a20 696e 7665  te Verbose: inve
│ │ │ │ +00083310: 7273 654d 6170 5f6c 705f 7064 5f70 645f  rseMap_lp_pd_pd_
│ │ │ │ +00083320: 7064 5f63 6d56 6572 626f 7365 3d3e 5f70  pd_cmVerbose=>_p
│ │ │ │ +00083330: 645f 7064 5f70 645f 7270 2c20 3d3e 202e  d_pd_pd_rp, => .
│ │ │ │ +00083340: 2e2e 2c0a 2020 2020 2020 2020 6465 6661  ..,.        defa
│ │ │ │ +00083350: 756c 7420 7661 6c75 6520 7472 7565 2c0a  ult value true,.
│ │ │ │ +00083360: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +00083370: 2020 202a 2061 202a 6e6f 7465 206c 6973     * a *note lis
│ │ │ │ +00083380: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ +00083390: 294c 6973 742c 2c20 7468 6520 6c69 7374  )List,, the list
│ │ │ │ +000833a0: 206f 6620 7468 6520 7072 6f6a 6563 7469   of the projecti
│ │ │ │ +000833b0: 7665 2064 6567 7265 6573 0a20 2020 2020  ve degrees.     
│ │ │ │ +000833c0: 2020 206f 6620 245c 5068 6924 0a0a 4465     of $\Phi$..De
│ │ │ │ +000833d0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +000833e0: 3d3d 3d3d 3d0a 0a4c 6574 2024 5c70 6869  =====..Let $\phi
│ │ │ │ +000833f0: 3a4b 5b79 5f30 2c5c 6c64 6f74 732c 795f  :K[y_0,\ldots,y_
│ │ │ │ +00083400: 6d5d 2f4a 205c 746f 204b 5b78 5f30 2c5c  m]/J \to K[x_0,\
│ │ │ │ +00083410: 6c64 6f74 732c 785f 6e5d 2f49 2420 6265  ldots,x_n]/I$ be
│ │ │ │ +00083420: 2061 2072 696e 6720 6d61 700a 7265 7072   a ring map.repr
│ │ │ │ +00083430: 6573 656e 7469 6e67 2061 2072 6174 696f  esenting a ratio
│ │ │ │ +00083440: 6e61 6c20 6d61 7020 245c 5068 693a 2056  nal map $\Phi: V
│ │ │ │ +00083450: 2849 2920 5c73 7562 7365 7465 710a 5c6d  (I) \subseteq.\m
│ │ │ │ +00083460: 6174 6862 627b 507d 5e6e 3d50 726f 6a28  athbb{P}^n=Proj(
│ │ │ │ +00083470: 4b5b 785f 302c 5c6c 646f 7473 2c78 5f6e  K[x_0,\ldots,x_n
│ │ │ │ +00083480: 5d29 205c 6461 7368 7269 6768 7461 7272  ]) \dashrightarr
│ │ │ │ +00083490: 6f77 2056 284a 2920 5c73 7562 7365 7465  ow V(J) \subsete
│ │ │ │ +000834a0: 710a 5c6d 6174 6862 627b 507d 5e6d 3d50  q.\mathbb{P}^m=P
│ │ │ │ +000834b0: 726f 6a28 4b5b 795f 302c 5c6c 646f 7473  roj(K[y_0,\ldots
│ │ │ │ +000834c0: 2c79 5f6d 5d29 242e 2054 6865 2024 6924  ,y_m])$. The $i$
│ │ │ │ +000834d0: 2d74 6820 7072 6f6a 6563 7469 7665 2064  -th projective d
│ │ │ │ +000834e0: 6567 7265 6520 6f66 2024 5c50 6869 240a  egree of $\Phi$.
│ │ │ │ +000834f0: 6973 2064 6566 696e 6564 2069 6e20 7465  is defined in te
│ │ │ │ +00083500: 726d 7320 6f66 2064 696d 656e 7369 6f6e  rms of dimension
│ │ │ │ +00083510: 2061 6e64 2064 6567 7265 6520 6f66 2074   and degree of t
│ │ │ │ +00083520: 6865 2063 6c6f 7375 7265 206f 6620 245c  he closure of $\
│ │ │ │ +00083530: 5068 695e 7b2d 317d 284c 2924 2c0a 7768  Phi^{-1}(L)$,.wh
│ │ │ │ +00083540: 6572 6520 244c 2420 6973 2061 2067 656e  ere $L$ is a gen
│ │ │ │ +00083550: 6572 616c 206c 696e 6561 7220 7375 6273  eral linear subs
│ │ │ │ +00083560: 7061 6365 206f 6620 245c 6d61 7468 6262  pace of $\mathbb
│ │ │ │ +00083570: 7b50 7d5e 6d24 206f 6620 6120 6365 7274  {P}^m$ of a cert
│ │ │ │ +00083580: 6169 6e0a 6469 6d65 6e73 696f 6e3b 2066  ain.dimension; f
│ │ │ │ +00083590: 6f72 2074 6865 2070 7265 6369 7365 2064  or the precise d
│ │ │ │ +000835a0: 6566 696e 6974 696f 6e2c 2073 6565 2048  efinition, see H
│ │ │ │ +000835b0: 6172 7269 7327 7320 626f 6f6b 2028 416c  arris's book (Al
│ │ │ │ +000835c0: 6765 6272 6169 6320 6765 6f6d 6574 7279  gebraic geometry
│ │ │ │ +000835d0: 3a20 410a 6669 7273 7420 636f 7572 7365  : A.first course
│ │ │ │ +000835e0: 202d 2056 6f6c 2e20 3133 3320 6f66 2047   - Vol. 133 of G
│ │ │ │ +000835f0: 7261 642e 2054 6578 7473 2069 6e20 4d61  rad. Texts in Ma
│ │ │ │ +00083600: 7468 2e2c 2070 2e20 3234 3029 2e20 4966  th., p. 240). If
│ │ │ │ +00083610: 2024 5c50 6869 2420 6973 2064 6566 696e   $\Phi$ is defin
│ │ │ │ +00083620: 6564 0a62 7920 656c 656d 656e 7473 2024  ed.by elements $
│ │ │ │ +00083630: 465f 3028 785f 302c 5c6c 646f 7473 2c78  F_0(x_0,\ldots,x
│ │ │ │ +00083640: 5f6e 292c 5c6c 646f 7473 2c46 5f6d 2878  _n),\ldots,F_m(x
│ │ │ │ +00083650: 5f30 2c5c 6c64 6f74 732c 785f 6e29 2420  _0,\ldots,x_n)$ 
│ │ │ │ +00083660: 616e 6420 2449 5f4c 2420 6465 6e6f 7465  and $I_L$ denote
│ │ │ │ +00083670: 730a 7468 6520 6964 6561 6c20 6f66 2074  s.the ideal of t
│ │ │ │ +00083680: 6865 2073 7562 7370 6163 6520 244c 5c73  he subspace $L\s
│ │ │ │ +00083690: 7562 7365 7465 7120 5c6d 6174 6862 627b  ubseteq \mathbb{
│ │ │ │ +000836a0: 507d 5e6d 242c 2074 6865 6e20 7468 6520  P}^m$, then the 
│ │ │ │ +000836b0: 6964 6561 6c20 6f66 2074 6865 0a63 6c6f  ideal of the.clo
│ │ │ │ +000836c0: 7375 7265 206f 6620 245c 5068 695e 7b2d  sure of $\Phi^{-
│ │ │ │ +000836d0: 317d 284c 2920 2420 6973 206e 6f74 6869  1}(L) $ is nothi
│ │ │ │ +000836e0: 6e67 2062 7574 2074 6865 2073 6174 7572  ng but the satur
│ │ │ │ +000836f0: 6174 696f 6e20 6f66 2074 6865 2069 6465  ation of the ide
│ │ │ │ +00083700: 616c 0a24 285c 7068 6928 495f 4c29 2924  al.$(\phi(I_L))$
│ │ │ │ +00083710: 2062 7920 2428 465f 302c 2e2e 2e2e 2c46   by $(F_0,....,F
│ │ │ │ +00083720: 5f6d 2924 2069 6e20 7468 6520 7269 6e67  _m)$ in the ring
│ │ │ │ +00083730: 2024 4b5b 785f 302c 5c6c 646f 7473 2c78   $K[x_0,\ldots,x
│ │ │ │ +00083740: 5f6e 5d2f 4924 2e20 536f 2c0a 7265 706c  _n]/I$. So,.repl
│ │ │ │ +00083750: 6163 696e 6720 696e 2074 6865 2064 6566  acing in the def
│ │ │ │ +00083760: 696e 6974 696f 6e2c 2067 656e 6572 616c  inition, general
│ │ │ │ +00083770: 206c 696e 6561 7220 7375 6273 7061 6365   linear subspace
│ │ │ │ +00083780: 2062 7920 7261 6e64 6f6d 206c 696e 6561   by random linea
│ │ │ │ +00083790: 7220 7375 6273 7061 6365 2c0a 7765 2067  r subspace,.we g
│ │ │ │ +000837a0: 6574 2061 2070 726f 6261 6269 6c69 7374  et a probabilist
│ │ │ │ +000837b0: 6963 2061 6c67 6f72 6974 686d 2074 6f20  ic algorithm to 
│ │ │ │ +000837c0: 636f 6d70 7574 6520 616c 6c20 7072 6f6a  compute all proj
│ │ │ │ +000837d0: 6563 7469 7665 2064 6567 7265 6573 2e0a  ective degrees..
│ │ │ │ +000837e0: 4675 7274 6865 726d 6f72 652c 2077 6520  Furthermore, we 
│ │ │ │ +000837f0: 6361 6e20 636f 6e73 6964 6572 6162 6c79  can considerably
│ │ │ │ +00083800: 2073 7065 6564 2075 7020 7468 6973 2061   speed up this a
│ │ │ │ +00083810: 6c67 6f72 6974 686d 2062 7920 7461 6b69  lgorithm by taki
│ │ │ │ +00083820: 6e67 2069 6e74 6f20 6163 636f 756e 740a  ng into account.
│ │ │ │ +00083830: 7477 6f20 7369 6d70 6c65 2072 656d 6172  two simple remar
│ │ │ │ +00083840: 6b73 3a20 3129 2074 6865 2073 6174 7572  ks: 1) the satur
│ │ │ │ +00083850: 6174 696f 6e20 2428 5c70 6869 2849 5f4c  ation $(\phi(I_L
│ │ │ │ +00083860: 2929 3a7b 2846 5f30 2c5c 6c64 6f74 732c  )):{(F_0,\ldots,
│ │ │ │ +00083870: 465f 6d29 7d5e 7b5c 696e 6674 797d 240a  F_m)}^{\infty}$.
│ │ │ │ +00083880: 6973 2074 6865 2073 616d 6520 6173 2024  is the same as $
│ │ │ │ +00083890: 285c 7068 6928 495f 4c29 293a 7b28 5c6c  (\phi(I_L)):{(\l
│ │ │ │ +000838a0: 616d 6264 615f 3020 465f 302b 5c63 646f  ambda_0 F_0+\cdo
│ │ │ │ +000838b0: 7473 2b5c 6c61 6d62 6461 5f6d 2046 5f6d  ts+\lambda_m F_m
│ │ │ │ +000838c0: 297d 5e7b 5c69 6e66 7479 7d24 2c0a 7768  )}^{\infty}$,.wh
│ │ │ │ +000838d0: 6572 6520 245c 6c61 6d62 6461 5f30 2c5c  ere $\lambda_0,\
│ │ │ │ +000838e0: 6c64 6f74 732c 5c6c 616d 6264 615f 6d5c  ldots,\lambda_m\
│ │ │ │ +000838f0: 696e 5c6d 6174 6862 627b 4b7d 2420 6172  in\mathbb{K}$ ar
│ │ │ │ +00083900: 6520 6765 6e65 7261 6c20 7363 616c 6172  e general scalar
│ │ │ │ +00083910: 733b 2032 2920 7468 650a 2469 242d 7468  s; 2) the.$i$-th
│ │ │ │ +00083920: 2070 726f 6a65 6374 6976 6520 6465 6772   projective degr
│ │ │ │ +00083930: 6565 206f 6620 245c 5068 6924 2063 6f69  ee of $\Phi$ coi
│ │ │ │ +00083940: 6e63 6964 6573 2077 6974 6820 7468 6520  ncides with the 
│ │ │ │ +00083950: 2428 692d 3129 242d 7468 2070 726f 6a65  $(i-1)$-th proje
│ │ │ │ +00083960: 6374 6976 650a 6465 6772 6565 206f 6620  ctive.degree of 
│ │ │ │ +00083970: 7468 6520 7265 7374 7269 6374 696f 6e20  the restriction 
│ │ │ │ +00083980: 6f66 2024 5c50 6869 2420 746f 2061 2067  of $\Phi$ to a g
│ │ │ │ +00083990: 656e 6572 616c 2068 7970 6572 706c 616e  eneral hyperplan
│ │ │ │ +000839a0: 6520 7365 6374 696f 6e20 6f66 2024 5824  e section of $X$
│ │ │ │ +000839b0: 2028 7365 650a 6c6f 632e 2063 6974 2e29   (see.loc. cit.)
│ │ │ │ +000839c0: 2e20 5468 6973 2069 7320 7768 6174 2074  . This is what t
│ │ │ │ +000839d0: 6865 206d 6574 686f 6420 7573 6573 2069  he method uses i
│ │ │ │ +000839e0: 6620 2a6e 6f74 6520 4365 7274 6966 793a  f *note Certify:
│ │ │ │ +000839f0: 2043 6572 7469 6679 2c20 6973 2073 6574   Certify, is set
│ │ │ │ +00083a00: 2074 6f0a 6661 6c73 652e 2049 6620 696e   to.false. If in
│ │ │ │ +00083a10: 7374 6561 6420 2a6e 6f74 6520 4365 7274  stead *note Cert
│ │ │ │ +00083a20: 6966 793a 2043 6572 7469 6679 2c20 6973  ify: Certify, is
│ │ │ │ +00083a30: 2073 6574 2074 6f20 7472 7565 2c20 7468   set to true, th
│ │ │ │ +00083a40: 656e 2074 6865 206d 6574 686f 640a 7369  en the method.si
│ │ │ │ +00083a50: 6d70 6c79 2063 6f6d 7075 7465 7320 7468  mply computes th
│ │ │ │ +00083a60: 6520 2a6e 6f74 6520 6d75 6c74 6964 6567  e *note multideg
│ │ │ │ +00083a70: 7265 653a 2028 4d61 6361 756c 6179 3244  ree: (Macaulay2D
│ │ │ │ +00083a80: 6f63 296d 756c 7469 6465 6772 6565 2c20  oc)multidegree, 
│ │ │ │ +00083a90: 6f66 2074 6865 202a 6e6f 7465 0a67 7261  of the *note.gra
│ │ │ │ +00083aa0: 7068 3a20 6772 6170 682c 2e0a 0a2b 2d2d  ph: graph,...+--
│ │ │ │  00083ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00083ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00083ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00083ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00083af0: 2d2d 2d2d 2d2b 0a7c 6931 203a 202d 2d20  -----+.|i1 : -- 
│ │ │ │ -00083b00: 6d61 7020 6672 6f6d 2050 5e34 2074 6f20  map from P^4 to 
│ │ │ │ -00083b10: 4728 312c 3329 2067 6976 656e 2062 7920  G(1,3) given by 
│ │ │ │ -00083b20: 7468 6520 7175 6164 7269 6373 2074 6872  the quadrics thr
│ │ │ │ -00083b30: 6f75 6768 2061 2072 6174 696f 6e61 6c20  ough a rational 
│ │ │ │ -00083b40: 2020 2020 207c 0a7c 2020 2020 2047 4628       |.|     GF(
│ │ │ │ -00083b50: 3333 315e 3229 5b74 5f30 2e2e 745f 345d  331^2)[t_0..t_4]
│ │ │ │ -00083b60: 3b20 7068 693d 746f 4d61 7020 6d69 6e6f  ; phi=toMap mino
│ │ │ │ -00083b70: 7273 2832 2c6d 6174 7269 787b 7b74 5f30  rs(2,matrix{{t_0
│ │ │ │ -00083b80: 2e2e 745f 337d 2c7b 745f 312e 2e74 5f34  ..t_3},{t_1..t_4
│ │ │ │ -00083b90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00083af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00083b00: 203a 202d 2d20 6d61 7020 6672 6f6d 2050   : -- map from P
│ │ │ │ +00083b10: 5e34 2074 6f20 4728 312c 3329 2067 6976  ^4 to G(1,3) giv
│ │ │ │ +00083b20: 656e 2062 7920 7468 6520 7175 6164 7269  en by the quadri
│ │ │ │ +00083b30: 6373 2074 6872 6f75 6768 2061 2072 6174  cs through a rat
│ │ │ │ +00083b40: 696f 6e61 6c20 2020 2020 207c 0a7c 2020  ional      |.|  
│ │ │ │ +00083b50: 2020 2047 4628 3333 315e 3229 5b74 5f30     GF(331^2)[t_0
│ │ │ │ +00083b60: 2e2e 745f 345d 3b20 7068 693d 746f 4d61  ..t_4]; phi=toMa
│ │ │ │ +00083b70: 7020 6d69 6e6f 7273 2832 2c6d 6174 7269  p minors(2,matri
│ │ │ │ +00083b80: 787b 7b74 5f30 2e2e 745f 337d 2c7b 745f  x{{t_0..t_3},{t_
│ │ │ │ +00083b90: 312e 2e74 5f34 2020 2020 207c 0a7c 2020  1..t_4     |.|  
│ │ │ │  00083ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083be0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00083be0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00083bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083c10: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00083c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083c30: 2020 2020 207c 0a7c 6f32 203d 206d 6170       |.|o2 = map
│ │ │ │ -00083c40: 2028 4746 2031 3039 3536 315b 7420 2e2e   (GF 109561[t ..
│ │ │ │ -00083c50: 7420 5d2c 2047 4620 3130 3935 3631 5b78  t ], GF 109561[x
│ │ │ │ -00083c60: 202e 2e78 205d 2c20 7b2d 2074 2020 2b20   ..x ], {- t  + 
│ │ │ │ -00083c70: 7420 7420 2c20 2d20 7420 7420 202b 2074  t t , - t t  + t
│ │ │ │ -00083c80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00083c90: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00083ca0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -00083cb0: 3020 2020 3520 2020 2020 2020 3120 2020  0   5       1   
│ │ │ │ -00083cc0: 2030 2032 2020 2020 2031 2032 2020 2020   0 2     1 2    
│ │ │ │ -00083cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00083c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00083c20: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00083c30: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00083c40: 203d 206d 6170 2028 4746 2031 3039 3536   = map (GF 10956
│ │ │ │ +00083c50: 315b 7420 2e2e 7420 5d2c 2047 4620 3130  1[t ..t ], GF 10
│ │ │ │ +00083c60: 3935 3631 5b78 202e 2e78 205d 2c20 7b2d  9561[x ..x ], {-
│ │ │ │ +00083c70: 2074 2020 2b20 7420 7420 2c20 2d20 7420   t  + t t , - t 
│ │ │ │ +00083c80: 7420 202b 2074 2020 2020 207c 0a7c 2020  t  + t     |.|  
│ │ │ │ +00083c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00083ca0: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │ +00083cb0: 2020 2020 2020 3020 2020 3520 2020 2020        0   5     
│ │ │ │ +00083cc0: 2020 3120 2020 2030 2032 2020 2020 2031    1    0 2     1
│ │ │ │ +00083cd0: 2032 2020 2020 2020 2020 207c 0a7c 2020   2         |.|  
│ │ │ │  00083ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083d20: 2020 2020 207c 0a7c 6f32 203a 2052 696e       |.|o2 : Rin
│ │ │ │ -00083d30: 674d 6170 2047 4620 3130 3935 3631 5b74  gMap GF 109561[t
│ │ │ │ -00083d40: 202e 2e74 205d 203c 2d2d 2047 4620 3130   ..t ] <-- GF 10
│ │ │ │ -00083d50: 3935 3631 5b78 202e 2e78 205d 2020 2020  9561[x ..x ]    
│ │ │ │ -00083d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083d70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00083d20: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
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│ │ │ │ +00083d60: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ +00083d70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00083d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083d90: 3020 2020 3420 2020 2020 2020 2020 2020  0   4           
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│ │ │ │  00083e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00083e70: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00083e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083eb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00083eb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00083ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00083f10: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00083f00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00083f10: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00083f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083f30: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00083f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083f50: 2020 2020 207c 0a7c 2074 202c 202d 2074       |.| t , - t
│ │ │ │ -00083f60: 2020 2b20 7420 7420 2c20 2d20 7420 7420    + t t , - t t 
│ │ │ │ -00083f70: 202b 2074 2074 202c 202d 2074 2074 2020   + t t , - t t  
│ │ │ │ -00083f80: 2b20 7420 7420 2c20 2d20 7420 202b 2074  + t t , - t  + t
│ │ │ │ -00083f90: 2074 202c 2061 7d29 2020 2020 2020 2020   t , a})        
│ │ │ │ -00083fa0: 2020 2020 207c 0a7c 3020 3320 2020 2020       |.|0 3     
│ │ │ │ -00083fb0: 3220 2020 2031 2033 2020 2020 2031 2033  2    1 3     1 3
│ │ │ │ -00083fc0: 2020 2020 3020 3420 2020 2020 3220 3320      0 4     2 3 
│ │ │ │ -00083fd0: 2020 2031 2034 2020 2020 2033 2020 2020     1 4     3    
│ │ │ │ -00083fe0: 3220 3420 2020 2020 2020 2020 2020 2020  2 4             
│ │ │ │ -00083ff0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00083f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00083f40: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00083f50: 2020 2020 2020 2020 2020 207c 0a7c 2074             |.| t
│ │ │ │ +00083f60: 202c 202d 2074 2020 2b20 7420 7420 2c20   , - t  + t t , 
│ │ │ │ +00083f70: 2d20 7420 7420 202b 2074 2074 202c 202d  - t t  + t t , -
│ │ │ │ +00083f80: 2074 2074 2020 2b20 7420 7420 2c20 2d20   t t  + t t , - 
│ │ │ │ +00083f90: 7420 202b 2074 2074 202c 2061 7d29 2020  t  + t t , a})  
│ │ │ │ +00083fa0: 2020 2020 2020 2020 2020 207c 0a7c 3020             |.|0 
│ │ │ │ +00083fb0: 3320 2020 2020 3220 2020 2031 2033 2020  3     2    1 3  
│ │ │ │ +00083fc0: 2020 2031 2033 2020 2020 3020 3420 2020     1 3    0 4   
│ │ │ │ +00083fd0: 2020 3220 3320 2020 2031 2034 2020 2020    2 3    1 4    
│ │ │ │ +00083fe0: 2033 2020 2020 3220 3420 2020 2020 2020   3    2 4       
│ │ │ │ +00083ff0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │  00084010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00084020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00084030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00084040: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d  -----+.|i3 : tim
│ │ │ │ -00084050: 6520 7072 6f6a 6563 7469 7665 4465 6772  e projectiveDegr
│ │ │ │ -00084060: 6565 7328 7068 692c 4365 7274 6966 793d  ees(phi,Certify=
│ │ │ │ -00084070: 3e74 7275 6529 2020 2020 2020 2020 2020  >true)          
│ │ │ │ +00084040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +00084050: 203a 2074 696d 6520 7072 6f6a 6563 7469   : time projecti
│ │ │ │ +00084060: 7665 4465 6772 6565 7328 7068 692c 4365  veDegrees(phi,Ce
│ │ │ │ +00084070: 7274 6966 793d 3e74 7275 6529 2020 2020  rtify=>true)    
│ │ │ │  00084080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084090: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000840a0: 2030 2e30 3138 3235 3832 7320 2863 7075   0.0182582s (cpu
│ │ │ │ -000840b0: 293b 2030 2e30 3135 3530 3238 7320 2874  ); 0.0155028s (t
│ │ │ │ -000840c0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -000840d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000840e0: 2020 2020 207c 0a7c 4365 7274 6966 793a       |.|Certify:
│ │ │ │ -000840f0: 206f 7574 7075 7420 6365 7274 6966 6965   output certifie
│ │ │ │ -00084100: 6421 2020 2020 2020 2020 2020 2020 2020  d!              
│ │ │ │ +00084090: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +000840a0: 2d20 7573 6564 2030 2e30 3330 3137 3238  - used 0.0301728
│ │ │ │ +000840b0: 7320 2863 7075 293b 2030 2e30 3136 3435  s (cpu); 0.01645
│ │ │ │ +000840c0: 3131 7320 2874 6872 6561 6429 3b20 3073  11s (thread); 0s
│ │ │ │ +000840d0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +000840e0: 2020 2020 2020 2020 2020 207c 0a7c 4365             |.|Ce
│ │ │ │ +000840f0: 7274 6966 793a 206f 7574 7075 7420 6365  rtify: output ce
│ │ │ │ +00084100: 7274 6966 6965 6421 2020 2020 2020 2020  rtified!        
│ │ │ │  00084110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084130: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00084130: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00084140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084180: 2020 2020 207c 0a7c 6f33 203d 207b 312c       |.|o3 = {1,
│ │ │ │ -00084190: 2032 2c20 342c 2034 2c20 327d 2020 2020   2, 4, 4, 2}    
│ │ │ │ -000841a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00084180: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00084190: 203d 207b 312c 2032 2c20 342c 2034 2c20   = {1, 2, 4, 4, 
│ │ │ │ +000841a0: 327d 2020 2020 2020 2020 2020 2020 2020  2}              
│ │ │ │  000841b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000841c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000841d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000841d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  000841f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084210: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00084220: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00084230: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  00084240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084250: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00084280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00084290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000842b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000842c0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2070 7369  -----+.|i4 : psi
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│ │ │ │ +000842c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +000842d0: 203a 2070 7369 3d69 6e76 6572 7365 4d61   : psi=inverseMa
│ │ │ │ +000842e0: 7028 746f 4d61 7028 7068 692c 446f 6d69  p(toMap(phi,Domi
│ │ │ │ +000842f0: 6e61 6e74 3d3e 696e 6669 6e69 7479 2929  nant=>infinity))
│ │ │ │  00084300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084310: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00084320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084330: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00084350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00084370: 2020 2047 4620 3130 3935 3631 5b78 202e     GF 109561[x .
│ │ │ │ -00084380: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │ +00084360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00084370: 2020 2020 2020 2020 2047 4620 3130 3935           GF 1095
│ │ │ │ +00084380: 3631 5b78 202e 2e78 205d 2020 2020 2020  61[x ..x ]      
│ │ │ │  00084390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000843a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000843b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -000843e0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
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│ │ │ │ -00084460: 2020 7820 7820 202d 2078 2078 2020 2b20    x x  - x x  + 
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│ │ │ │ -00084490: 3020 3220 2020 2030 2033 2020 2031 2032  0 2    0 3   1 2
│ │ │ │ -000844a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000844b0: 2020 2032 2033 2020 2020 3120 3420 2020     2 3    1 4   
│ │ │ │ -000844c0: 2030 2035 2020 2020 2020 2020 2020 2020   0 5            
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│ │ │ │ +00084460: 2020 2020 2020 2020 7820 7820 202d 2078          x x  - x
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│ │ │ │ +00084480: 2020 2020 2020 2030 2020 2034 2020 2020         0   4    
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│ │ │ │ +000844a0: 2020 2031 2032 2020 2020 207c 0a7c 2020     1 2     |.|  
│ │ │ │ +000844b0: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
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│ │ │ │  000844d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000844e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00084540: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  00084580: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000845a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000845b0: 2030 2020 2035 2020 2020 2020 2020 2020   0   5          
│ │ │ │ +000845b0: 2020 2020 2020 2030 2020 2035 2020 2020         0   5    
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│ │ │ │  000845d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00084630: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00084640: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
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│ │ │ │ -00084670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -000846a0: 2020 2020 3020 3520 2020 2020 2020 2020      0 5         
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│ │ │ │ +00084640: 2020 2020 2020 2020 2020 2078 2078 2020             x x  
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│ │ │ │ -000846d0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000846d0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  000846e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000846f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00084700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00084730: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00084740: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  00084750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084760: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000851a0: 2035 2036 2020 2032 2033 2020 2020 3020   5 6   2 3    0 
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│ │ │ │ +00085190: 2036 2020 2020 2020 2020 2034 2036 2020   6         4 6  
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│ │ │ │ +000859e0: 2020 2020 2020 2020 2020 207c 0a7c 3732             |.|72
│ │ │ │ +000859f0: 3039 3578 2078 2020 2d20 3931 3938 3978  095x x  - 91989x
│ │ │ │ +00085a00: 2078 2020 2d20 3238 3739 3078 2020 2b20   x  - 28790x  + 
│ │ │ │ +00085a10: 3732 3338 3378 2078 2020 2b20 3133 3430  72383x x  + 1340
│ │ │ │ +00085a20: 3732 7820 202d 2039 3635 3438 7820 7820  72x  - 96548x x 
│ │ │ │ +00085a30: 202b 2020 2020 2020 2020 207c 0a7c 2020   +         |.|  
│ │ │ │ +00085a40: 2020 2020 3020 3220 2020 2020 2020 2020      0 2         
│ │ │ │ +00085a50: 3120 3220 2020 2020 2020 2020 3220 2020  1 2         2   
│ │ │ │ +00085a60: 2020 2020 2020 3120 3420 2020 2020 2020        1 4       
│ │ │ │ +00085a70: 2020 2034 2020 2020 2020 2020 2030 2035     4         0 5
│ │ │ │ +00085a80: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │  00085a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00085ad0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ +00085ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │  00085ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085b00: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00085b00: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  00085b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085b20: 2020 2020 207c 0a7c 3334 3739 3778 2078       |.|34797x x
│ │ │ │ -00085b30: 2020 2d20 3132 3837 3237 7820 7820 202b    - 128727x x  +
│ │ │ │ -00085b40: 2031 3234 3932 3578 2078 2020 2d20 3234   124925x x  - 24
│ │ │ │ -00085b50: 3638 3978 2020 2d20 3135 3938 3478 2078  689x  - 15984x x
│ │ │ │ -00085b60: 2020 2b20 3338 3537 3878 2078 2020 2d20    + 38578x x  - 
│ │ │ │ -00085b70: 2020 2020 207c 0a7c 2020 2020 2020 3120       |.|      1 
│ │ │ │ -00085b80: 3520 2020 2020 2020 2020 2032 2035 2020  5          2 5  
│ │ │ │ -00085b90: 2020 2020 2020 2020 3420 3520 2020 2020          4 5     
│ │ │ │ -00085ba0: 2020 2020 3520 2020 2020 2020 2020 3020      5         0 
│ │ │ │ -00085bb0: 3620 2020 2020 2020 2020 3120 3620 2020  6         1 6   
│ │ │ │ -00085bc0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00085b20: 2020 2020 2020 2020 2020 207c 0a7c 3334             |.|34
│ │ │ │ +00085b30: 3739 3778 2078 2020 2d20 3132 3837 3237  797x x  - 128727
│ │ │ │ +00085b40: 7820 7820 202b 2031 3234 3932 3578 2078  x x  + 124925x x
│ │ │ │ +00085b50: 2020 2d20 3234 3638 3978 2020 2d20 3135    - 24689x  - 15
│ │ │ │ +00085b60: 3938 3478 2078 2020 2b20 3338 3537 3878  984x x  + 38578x
│ │ │ │ +00085b70: 2078 2020 2d20 2020 2020 207c 0a7c 2020   x  -      |.|  
│ │ │ │ +00085b80: 2020 2020 3120 3520 2020 2020 2020 2020      1 5         
│ │ │ │ +00085b90: 2032 2035 2020 2020 2020 2020 2020 3420   2 5          4 
│ │ │ │ +00085ba0: 3520 2020 2020 2020 2020 3520 2020 2020  5         5     
│ │ │ │ +00085bb0: 2020 2020 3020 3620 2020 2020 2020 2020      0 6         
│ │ │ │ +00085bc0: 3120 3620 2020 2020 2020 207c 0a7c 2d2d  1 6        |.|--
│ │ │ │  00085bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00085c10: 2d2d 2d2d 2d7c 0a7c 3132 3831 3533 7820  -----|.|128153x 
│ │ │ │ -00085c20: 7820 202b 2031 3330 3536 3778 2078 2020  x  + 130567x x  
│ │ │ │ -00085c30: 2b20 3239 3730 3178 2078 2020 2d20 3439  + 29701x x  - 49
│ │ │ │ -00085c40: 3331 3878 2078 207d 2920 2020 2020 2020  318x x })       
│ │ │ │ +00085c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3132  -----------|.|12
│ │ │ │ +00085c20: 3831 3533 7820 7820 202b 2031 3330 3536  8153x x  + 13056
│ │ │ │ +00085c30: 3778 2078 2020 2b20 3239 3730 3178 2078  7x x  + 29701x x
│ │ │ │ +00085c40: 2020 2d20 3439 3331 3878 2078 207d 2920    - 49318x x }) 
│ │ │ │  00085c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085c60: 2020 2020 207c 0a7c 2020 2020 2020 2032       |.|       2
│ │ │ │ -00085c70: 2036 2020 2020 2020 2020 2020 3320 3620   6          3 6 
│ │ │ │ -00085c80: 2020 2020 2020 2020 3420 3620 2020 2020          4 6     
│ │ │ │ -00085c90: 2020 2020 3520 3620 2020 2020 2020 2020      5 6         
│ │ │ │ +00085c60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00085c70: 2020 2020 2032 2036 2020 2020 2020 2020       2 6        
│ │ │ │ +00085c80: 2020 3320 3620 2020 2020 2020 2020 3420    3 6         4 
│ │ │ │ +00085c90: 3620 2020 2020 2020 2020 3520 3620 2020  6         5 6   
│ │ │ │  00085ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085cb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00085cb0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00085cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00085d00: 2d2d 2d2d 2d2b 0a7c 6937 203a 2074 696d  -----+.|i7 : tim
│ │ │ │ -00085d10: 6520 7072 6f6a 6563 7469 7665 4465 6772  e projectiveDegr
│ │ │ │ -00085d20: 6565 7320 7068 6920 2020 2020 2020 2020  ees phi         
│ │ │ │ +00085d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ +00085d10: 203a 2074 696d 6520 7072 6f6a 6563 7469   : time projecti
│ │ │ │ +00085d20: 7665 4465 6772 6565 7320 7068 6920 2020  veDegrees phi   
│ │ │ │  00085d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085d50: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00085d60: 2030 2e30 3033 3337 3035 3973 2028 6370   0.00337059s (cp
│ │ │ │ -00085d70: 7529 3b20 342e 3132 3038 652d 3035 7320  u); 4.1208e-05s 
│ │ │ │ -00085d80: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -00085d90: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00085da0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00085d50: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00085d60: 2d20 7573 6564 2030 2e30 3032 3631 3230  - used 0.0026120
│ │ │ │ +00085d70: 3473 2028 6370 7529 3b20 332e 3831 3337  4s (cpu); 3.8137
│ │ │ │ +00085d80: 652d 3035 7320 2874 6872 6561 6429 3b20  e-05s (thread); 
│ │ │ │ +00085d90: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +00085da0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00085db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085df0: 2020 2020 207c 0a7c 6f37 203d 207b 312c       |.|o7 = {1,
│ │ │ │ -00085e00: 2032 2c20 342c 2038 2c20 382c 2034 2c20   2, 4, 8, 8, 4, 
│ │ │ │ -00085e10: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │ +00085df0: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +00085e00: 203d 207b 312c 2032 2c20 342c 2038 2c20   = {1, 2, 4, 8, 
│ │ │ │ +00085e10: 382c 2034 2c20 317d 2020 2020 2020 2020  8, 4, 1}        
│ │ │ │  00085e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085e40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00085e40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00085e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085e90: 2020 2020 207c 0a7c 6f37 203a 204c 6973       |.|o7 : Lis
│ │ │ │ -00085ea0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00085e90: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +00085ea0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  00085eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085ee0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00085ee0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00085ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00085f30: 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 696d  -----+.|i8 : tim
│ │ │ │ -00085f40: 6520 7072 6f6a 6563 7469 7665 4465 6772  e projectiveDegr
│ │ │ │ -00085f50: 6565 7328 7068 692c 4e75 6d44 6567 7265  ees(phi,NumDegre
│ │ │ │ -00085f60: 6573 3d3e 3129 2020 2020 2020 2020 2020  es=>1)          
│ │ │ │ +00085f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ +00085f40: 203a 2074 696d 6520 7072 6f6a 6563 7469   : time projecti
│ │ │ │ +00085f50: 7665 4465 6772 6565 7328 7068 692c 4e75  veDegrees(phi,Nu
│ │ │ │ +00085f60: 6d44 6567 7265 6573 3d3e 3129 2020 2020  mDegrees=>1)    
│ │ │ │  00085f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085f80: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00085f90: 2039 2e39 3835 3765 2d30 3573 2028 6370   9.9857e-05s (cp
│ │ │ │ -00085fa0: 7529 3b20 322e 3235 3132 652d 3035 7320  u); 2.2512e-05s 
│ │ │ │ -00085fb0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -00085fc0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00085fd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00085f80: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00085f90: 2d20 7573 6564 2039 2e34 3431 3265 2d30  - used 9.4412e-0
│ │ │ │ +00085fa0: 3573 2028 6370 7529 3b20 312e 3931 3631  5s (cpu); 1.9161
│ │ │ │ +00085fb0: 652d 3035 7320 2874 6872 6561 6429 3b20  e-05s (thread); 
│ │ │ │ +00085fc0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +00085fd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00085fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00086020: 2020 2020 207c 0a7c 6f38 203d 207b 342c       |.|o8 = {4,
│ │ │ │ -00086030: 2031 7d20 2020 2020 2020 2020 2020 2020   1}             
│ │ │ │ +00086020: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +00086030: 203d 207b 342c 2031 7d20 2020 2020 2020   = {4, 1}       
│ │ │ │  00086040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00086070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00086070: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00086080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000860a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000860b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000860c0: 2020 2020 207c 0a7c 6f38 203a 204c 6973       |.|o8 : Lis
│ │ │ │ -000860d0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +000860c0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +000860d0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  000860e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000860f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00086110: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00086110: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00086120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00086130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00086140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00086150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00086160: 2d2d 2d2d 2d2b 0a0a 416e 6f74 6865 7220  -----+..Another 
│ │ │ │ -00086170: 7761 7920 746f 2075 7365 2074 6869 7320  way to use this 
│ │ │ │ -00086180: 6d65 7468 6f64 2069 7320 6279 2070 6173  method is by pas
│ │ │ │ -00086190: 7369 6e67 2061 6e20 696e 7465 6765 7220  sing an integer 
│ │ │ │ -000861a0: 6920 6173 2073 6563 6f6e 6420 6172 6775  i as second argu
│ │ │ │ -000861b0: 6d65 6e74 2e0a 486f 7765 7665 722c 2074  ment..However, t
│ │ │ │ -000861c0: 6869 7320 6973 2065 7175 6976 616c 656e  his is equivalen
│ │ │ │ -000861d0: 7420 746f 2066 6972 7374 2070 726f 6a65  t to first proje
│ │ │ │ -000861e0: 6374 6976 6544 6567 7265 6573 2870 6869  ctiveDegrees(phi
│ │ │ │ -000861f0: 2c4e 756d 4465 6772 6565 733d 3e69 2920  ,NumDegrees=>i) 
│ │ │ │ -00086200: 616e 640a 6765 6e65 7261 6c6c 7920 6974  and.generally it
│ │ │ │ -00086210: 2069 7320 6e6f 7420 6661 7374 6572 2e0a   is not faster..
│ │ │ │ -00086220: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -00086230: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6465  ==..  * *note de
│ │ │ │ -00086240: 6772 6565 7328 5261 7469 6f6e 616c 4d61  grees(RationalMa
│ │ │ │ -00086250: 7029 3a20 6465 6772 6565 735f 6c70 5261  p): degrees_lpRa
│ │ │ │ -00086260: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -00086270: 2070 726f 6a65 6374 6976 6520 6465 6772   projective degr
│ │ │ │ -00086280: 6565 730a 2020 2020 6f66 2061 2072 6174  ees.    of a rat
│ │ │ │ -00086290: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ -000862a0: 6f74 6520 6465 6772 6565 4d61 703a 2064  ote degreeMap: d
│ │ │ │ -000862b0: 6567 7265 654d 6170 2c20 2d2d 2064 6567  egreeMap, -- deg
│ │ │ │ -000862c0: 7265 6520 6f66 2061 2072 6174 696f 6e61  ree of a rationa
│ │ │ │ -000862d0: 6c20 6d61 7020 6265 7477 6565 6e20 7072  l map between pr
│ │ │ │ -000862e0: 6f6a 6563 7469 7665 0a20 2020 2076 6172  ojective.    var
│ │ │ │ -000862f0: 6965 7469 6573 0a20 202a 202a 6e6f 7465  ieties.  * *note
│ │ │ │ -00086300: 2053 6567 7265 436c 6173 733a 2053 6567   SegreClass: Seg
│ │ │ │ -00086310: 7265 436c 6173 732c 202d 2d20 5365 6772  reClass, -- Segr
│ │ │ │ -00086320: 6520 636c 6173 7320 6f66 2061 2063 6c6f  e class of a clo
│ │ │ │ -00086330: 7365 6420 7375 6273 6368 656d 6520 6f66  sed subscheme of
│ │ │ │ -00086340: 2061 0a20 2020 2070 726f 6a65 6374 6976   a.    projectiv
│ │ │ │ -00086350: 6520 7661 7269 6574 790a 0a57 6179 7320  e variety..Ways 
│ │ │ │ -00086360: 746f 2075 7365 2070 726f 6a65 6374 6976  to use projectiv
│ │ │ │ -00086370: 6544 6567 7265 6573 3a0a 3d3d 3d3d 3d3d  eDegrees:.======
│ │ │ │ +00086160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 416e  -----------+..An
│ │ │ │ +00086170: 6f74 6865 7220 7761 7920 746f 2075 7365  other way to use
│ │ │ │ +00086180: 2074 6869 7320 6d65 7468 6f64 2069 7320   this method is 
│ │ │ │ +00086190: 6279 2070 6173 7369 6e67 2061 6e20 696e  by passing an in
│ │ │ │ +000861a0: 7465 6765 7220 6920 6173 2073 6563 6f6e  teger i as secon
│ │ │ │ +000861b0: 6420 6172 6775 6d65 6e74 2e0a 486f 7765  d argument..Howe
│ │ │ │ +000861c0: 7665 722c 2074 6869 7320 6973 2065 7175  ver, this is equ
│ │ │ │ +000861d0: 6976 616c 656e 7420 746f 2066 6972 7374  ivalent to first
│ │ │ │ +000861e0: 2070 726f 6a65 6374 6976 6544 6567 7265   projectiveDegre
│ │ │ │ +000861f0: 6573 2870 6869 2c4e 756d 4465 6772 6565  es(phi,NumDegree
│ │ │ │ +00086200: 733d 3e69 2920 616e 640a 6765 6e65 7261  s=>i) and.genera
│ │ │ │ +00086210: 6c6c 7920 6974 2069 7320 6e6f 7420 6661  lly it is not fa
│ │ │ │ +00086220: 7374 6572 2e0a 0a53 6565 2061 6c73 6f0a  ster...See also.
│ │ │ │ +00086230: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00086240: 6f74 6520 6465 6772 6565 7328 5261 7469  ote degrees(Rati
│ │ │ │ +00086250: 6f6e 616c 4d61 7029 3a20 6465 6772 6565  onalMap): degree
│ │ │ │ +00086260: 735f 6c70 5261 7469 6f6e 616c 4d61 705f  s_lpRationalMap_
│ │ │ │ +00086270: 7270 2c20 2d2d 2070 726f 6a65 6374 6976  rp, -- projectiv
│ │ │ │ +00086280: 6520 6465 6772 6565 730a 2020 2020 6f66  e degrees.    of
│ │ │ │ +00086290: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +000862a0: 2020 2a20 2a6e 6f74 6520 6465 6772 6565    * *note degree
│ │ │ │ +000862b0: 4d61 703a 2064 6567 7265 654d 6170 2c20  Map: degreeMap, 
│ │ │ │ +000862c0: 2d2d 2064 6567 7265 6520 6f66 2061 2072  -- degree of a r
│ │ │ │ +000862d0: 6174 696f 6e61 6c20 6d61 7020 6265 7477  ational map betw
│ │ │ │ +000862e0: 6565 6e20 7072 6f6a 6563 7469 7665 0a20  een projective. 
│ │ │ │ +000862f0: 2020 2076 6172 6965 7469 6573 0a20 202a     varieties.  *
│ │ │ │ +00086300: 202a 6e6f 7465 2053 6567 7265 436c 6173   *note SegreClas
│ │ │ │ +00086310: 733a 2053 6567 7265 436c 6173 732c 202d  s: SegreClass, -
│ │ │ │ +00086320: 2d20 5365 6772 6520 636c 6173 7320 6f66  - Segre class of
│ │ │ │ +00086330: 2061 2063 6c6f 7365 6420 7375 6273 6368   a closed subsch
│ │ │ │ +00086340: 656d 6520 6f66 2061 0a20 2020 2070 726f  eme of a.    pro
│ │ │ │ +00086350: 6a65 6374 6976 6520 7661 7269 6574 790a  jective variety.
│ │ │ │ +00086360: 0a57 6179 7320 746f 2075 7365 2070 726f  .Ways to use pro
│ │ │ │ +00086370: 6a65 6374 6976 6544 6567 7265 6573 3a0a  jectiveDegrees:.
│ │ │ │  00086380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00086390: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2270  ========..  * "p
│ │ │ │ -000863a0: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ -000863b0: 2852 696e 674d 6170 2922 0a20 202a 202a  (RingMap)".  * *
│ │ │ │ -000863c0: 6e6f 7465 2070 726f 6a65 6374 6976 6544  note projectiveD
│ │ │ │ -000863d0: 6567 7265 6573 2852 6174 696f 6e61 6c4d  egrees(RationalM
│ │ │ │ -000863e0: 6170 293a 2070 726f 6a65 6374 6976 6544  ap): projectiveD
│ │ │ │ -000863f0: 6567 7265 6573 5f6c 7052 6174 696f 6e61  egrees_lpRationa
│ │ │ │ -00086400: 6c4d 6170 5f72 702c 0a20 2020 202d 2d20  lMap_rp,.    -- 
│ │ │ │ -00086410: 7072 6f6a 6563 7469 7665 2064 6567 7265  projective degre
│ │ │ │ -00086420: 6573 206f 6620 6120 7261 7469 6f6e 616c  es of a rational
│ │ │ │ -00086430: 206d 6170 0a0a 466f 7220 7468 6520 7072   map..For the pr
│ │ │ │ -00086440: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -00086450: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -00086460: 206f 626a 6563 7420 2a6e 6f74 6520 7072   object *note pr
│ │ │ │ -00086470: 6f6a 6563 7469 7665 4465 6772 6565 733a  ojectiveDegrees:
│ │ │ │ -00086480: 2070 726f 6a65 6374 6976 6544 6567 7265   projectiveDegre
│ │ │ │ -00086490: 6573 2c20 6973 2061 202a 6e6f 7465 206d  es, is a *note m
│ │ │ │ -000864a0: 6574 686f 640a 6675 6e63 7469 6f6e 2077  ethod.function w
│ │ │ │ -000864b0: 6974 6820 6f70 7469 6f6e 733a 2028 4d61  ith options: (Ma
│ │ │ │ -000864c0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -000864d0: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
│ │ │ │ -000864e0: 696f 6e73 2c2e 0a1f 0a46 696c 653a 2043  ions,....File: C
│ │ │ │ -000864f0: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ -00086500: 653a 2070 726f 6a65 6374 6976 6544 6567  e: projectiveDeg
│ │ │ │ -00086510: 7265 6573 5f6c 7052 6174 696f 6e61 6c4d  rees_lpRationalM
│ │ │ │ -00086520: 6170 5f72 702c 204e 6578 743a 2071 7561  ap_rp, Next: qua
│ │ │ │ -00086530: 6472 6f51 7561 6472 6963 4372 656d 6f6e  droQuadricCremon
│ │ │ │ -00086540: 6154 7261 6e73 666f 726d 6174 696f 6e2c  aTransformation,
│ │ │ │ -00086550: 2050 7265 763a 2070 726f 6a65 6374 6976   Prev: projectiv
│ │ │ │ -00086560: 6544 6567 7265 6573 2c20 5570 3a20 546f  eDegrees, Up: To
│ │ │ │ -00086570: 700a 0a70 726f 6a65 6374 6976 6544 6567  p..projectiveDeg
│ │ │ │ -00086580: 7265 6573 2852 6174 696f 6e61 6c4d 6170  rees(RationalMap
│ │ │ │ -00086590: 2920 2d2d 2070 726f 6a65 6374 6976 6520  ) -- projective 
│ │ │ │ -000865a0: 6465 6772 6565 7320 6f66 2061 2072 6174  degrees of a rat
│ │ │ │ -000865b0: 696f 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a  ional map.******
│ │ │ │ +00086390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +000863a0: 2020 2a20 2270 726f 6a65 6374 6976 6544    * "projectiveD
│ │ │ │ +000863b0: 6567 7265 6573 2852 696e 674d 6170 2922  egrees(RingMap)"
│ │ │ │ +000863c0: 0a20 202a 202a 6e6f 7465 2070 726f 6a65  .  * *note proje
│ │ │ │ +000863d0: 6374 6976 6544 6567 7265 6573 2852 6174  ctiveDegrees(Rat
│ │ │ │ +000863e0: 696f 6e61 6c4d 6170 293a 2070 726f 6a65  ionalMap): proje
│ │ │ │ +000863f0: 6374 6976 6544 6567 7265 6573 5f6c 7052  ctiveDegrees_lpR
│ │ │ │ +00086400: 6174 696f 6e61 6c4d 6170 5f72 702c 0a20  ationalMap_rp,. 
│ │ │ │ +00086410: 2020 202d 2d20 7072 6f6a 6563 7469 7665     -- projective
│ │ │ │ +00086420: 2064 6567 7265 6573 206f 6620 6120 7261   degrees of a ra
│ │ │ │ +00086430: 7469 6f6e 616c 206d 6170 0a0a 466f 7220  tional map..For 
│ │ │ │ +00086440: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +00086450: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00086460: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +00086470: 6f74 6520 7072 6f6a 6563 7469 7665 4465  ote projectiveDe
│ │ │ │ +00086480: 6772 6565 733a 2070 726f 6a65 6374 6976  grees: projectiv
│ │ │ │ +00086490: 6544 6567 7265 6573 2c20 6973 2061 202a  eDegrees, is a *
│ │ │ │ +000864a0: 6e6f 7465 206d 6574 686f 640a 6675 6e63  note method.func
│ │ │ │ +000864b0: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e  tion with option
│ │ │ │ +000864c0: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ +000864d0: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ +000864e0: 6974 684f 7074 696f 6e73 2c2e 0a1f 0a46  ithOptions,....F
│ │ │ │ +000864f0: 696c 653a 2043 7265 6d6f 6e61 2e69 6e66  ile: Cremona.inf
│ │ │ │ +00086500: 6f2c 204e 6f64 653a 2070 726f 6a65 6374  o, Node: project
│ │ │ │ +00086510: 6976 6544 6567 7265 6573 5f6c 7052 6174  iveDegrees_lpRat
│ │ │ │ +00086520: 696f 6e61 6c4d 6170 5f72 702c 204e 6578  ionalMap_rp, Nex
│ │ │ │ +00086530: 743a 2071 7561 6472 6f51 7561 6472 6963  t: quadroQuadric
│ │ │ │ +00086540: 4372 656d 6f6e 6154 7261 6e73 666f 726d  CremonaTransform
│ │ │ │ +00086550: 6174 696f 6e2c 2050 7265 763a 2070 726f  ation, Prev: pro
│ │ │ │ +00086560: 6a65 6374 6976 6544 6567 7265 6573 2c20  jectiveDegrees, 
│ │ │ │ +00086570: 5570 3a20 546f 700a 0a70 726f 6a65 6374  Up: Top..project
│ │ │ │ +00086580: 6976 6544 6567 7265 6573 2852 6174 696f  iveDegrees(Ratio
│ │ │ │ +00086590: 6e61 6c4d 6170 2920 2d2d 2070 726f 6a65  nalMap) -- proje
│ │ │ │ +000865a0: 6374 6976 6520 6465 6772 6565 7320 6f66  ctive degrees of
│ │ │ │ +000865b0: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │  000865c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000865d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000865e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000865f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00086600: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ -00086610: 3d3d 3d0a 0a20 202a 2046 756e 6374 696f  ===..  * Functio
│ │ │ │ -00086620: 6e3a 202a 6e6f 7465 2070 726f 6a65 6374  n: *note project
│ │ │ │ -00086630: 6976 6544 6567 7265 6573 3a20 7072 6f6a  iveDegrees: proj
│ │ │ │ -00086640: 6563 7469 7665 4465 6772 6565 732c 0a20  ectiveDegrees,. 
│ │ │ │ -00086650: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -00086660: 2020 2070 726f 6a65 6374 6976 6544 6567     projectiveDeg
│ │ │ │ -00086670: 7265 6573 2050 6869 0a20 202a 2049 6e70  rees Phi.  * Inp
│ │ │ │ -00086680: 7574 733a 0a20 2020 2020 202a 2050 6869  uts:.      * Phi
│ │ │ │ -00086690: 2c20 6120 2a6e 6f74 6520 7261 7469 6f6e  , a *note ration
│ │ │ │ -000866a0: 616c 206d 6170 3a20 5261 7469 6f6e 616c  al map: Rational
│ │ │ │ -000866b0: 4d61 702c 0a20 202a 202a 6e6f 7465 204f  Map,.  * *note O
│ │ │ │ -000866c0: 7074 696f 6e61 6c20 696e 7075 7473 3a20  ptional inputs: 
│ │ │ │ -000866d0: 284d 6163 6175 6c61 7932 446f 6329 7573  (Macaulay2Doc)us
│ │ │ │ -000866e0: 696e 6720 6675 6e63 7469 6f6e 7320 7769  ing functions wi
│ │ │ │ -000866f0: 7468 206f 7074 696f 6e61 6c20 696e 7075  th optional inpu
│ │ │ │ -00086700: 7473 2c3a 0a20 2020 2020 202a 202a 6e6f  ts,:.      * *no
│ │ │ │ -00086710: 7465 2042 6c6f 7755 7053 7472 6174 6567  te BlowUpStrateg
│ │ │ │ -00086720: 793a 2042 6c6f 7755 7053 7472 6174 6567  y: BlowUpStrateg
│ │ │ │ -00086730: 792c 203d 3e20 2e2e 2e2c 2064 6566 6175  y, => ..., defau
│ │ │ │ -00086740: 6c74 2076 616c 7565 0a20 2020 2020 2020  lt value.       
│ │ │ │ -00086750: 2022 456c 696d 696e 6174 6522 2c0a 2020   "Eliminate",.  
│ │ │ │ -00086760: 2020 2020 2a20 2a6e 6f74 6520 4365 7274      * *note Cert
│ │ │ │ -00086770: 6966 793a 2043 6572 7469 6679 2c20 3d3e  ify: Certify, =>
│ │ │ │ -00086780: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -00086790: 6c75 6520 6661 6c73 652c 2077 6865 7468  lue false, wheth
│ │ │ │ -000867a0: 6572 2074 6f20 656e 7375 7265 0a20 2020  er to ensure.   
│ │ │ │ -000867b0: 2020 2020 2063 6f72 7265 6374 6e65 7373       correctness
│ │ │ │ -000867c0: 206f 6620 6f75 7470 7574 0a20 2020 2020   of output.     
│ │ │ │ -000867d0: 202a 202a 6e6f 7465 204e 756d 4465 6772   * *note NumDegr
│ │ │ │ -000867e0: 6565 733a 204e 756d 4465 6772 6565 732c  ees: NumDegrees,
│ │ │ │ -000867f0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -00086800: 2076 616c 7565 2069 6e66 696e 6974 792c   value infinity,
│ │ │ │ -00086810: 200a 2020 2020 2020 2a20 2a6e 6f74 6520   .      * *note 
│ │ │ │ -00086820: 5665 7262 6f73 653a 2069 6e76 6572 7365  Verbose: inverse
│ │ │ │ -00086830: 4d61 705f 6c70 5f70 645f 7064 5f70 645f  Map_lp_pd_pd_pd_
│ │ │ │ -00086840: 636d 5665 7262 6f73 653d 3e5f 7064 5f70  cmVerbose=>_pd_p
│ │ │ │ -00086850: 645f 7064 5f72 702c 203d 3e20 2e2e 2e2c  d_pd_rp, => ...,
│ │ │ │ -00086860: 0a20 2020 2020 2020 2064 6566 6175 6c74  .        default
│ │ │ │ -00086870: 2076 616c 7565 2074 7275 652c 0a20 202a   value true,.  *
│ │ │ │ -00086880: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -00086890: 2a20 6120 2a6e 6f74 6520 6c69 7374 3a20  * a *note list: 
│ │ │ │ -000868a0: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -000868b0: 7374 2c2c 2074 6865 206c 6973 7420 6f66  st,, the list of
│ │ │ │ -000868c0: 2070 726f 6a65 6374 6976 6520 6465 6772   projective degr
│ │ │ │ -000868d0: 6565 7320 6f66 0a20 2020 2020 2020 2050  ees of.        P
│ │ │ │ -000868e0: 6869 0a0a 4465 7363 7269 7074 696f 6e0a  hi..Description.
│ │ │ │ -000868f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ -00086900: 7320 636f 6d70 7574 6174 696f 6e20 6973  s computation is
│ │ │ │ -00086910: 2064 6f6e 6520 7468 726f 7567 6820 7468   done through th
│ │ │ │ -00086920: 6520 636f 7272 6573 706f 6e64 696e 6720  e corresponding 
│ │ │ │ -00086930: 6d65 7468 6f64 2066 6f72 2072 696e 6720  method for ring 
│ │ │ │ -00086940: 6d61 7073 2e20 5365 650a 2a6e 6f74 6520  maps. See.*note 
│ │ │ │ -00086950: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -00086960: 7328 5269 6e67 4d61 7029 3a20 7072 6f6a  s(RingMap): proj
│ │ │ │ -00086970: 6563 7469 7665 4465 6772 6565 732c 2066  ectiveDegrees, f
│ │ │ │ -00086980: 6f72 206d 6f72 6520 6465 7461 696c 7320  or more details 
│ │ │ │ -00086990: 616e 640a 6578 616d 706c 6573 2e0a 0a53  and.examples...S
│ │ │ │ -000869a0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -000869b0: 0a0a 2020 2a20 2a6e 6f74 6520 7072 6f6a  ..  * *note proj
│ │ │ │ -000869c0: 6563 7469 7665 4465 6772 6565 7328 5269  ectiveDegrees(Ri
│ │ │ │ -000869d0: 6e67 4d61 7029 3a20 7072 6f6a 6563 7469  ngMap): projecti
│ │ │ │ -000869e0: 7665 4465 6772 6565 732c 202d 2d20 7072  veDegrees, -- pr
│ │ │ │ -000869f0: 6f6a 6563 7469 7665 2064 6567 7265 6573  ojective degrees
│ │ │ │ -00086a00: 0a20 2020 206f 6620 6120 7261 7469 6f6e  .    of a ration
│ │ │ │ -00086a10: 616c 206d 6170 2062 6574 7765 656e 2070  al map between p
│ │ │ │ -00086a20: 726f 6a65 6374 6976 6520 7661 7269 6574  rojective variet
│ │ │ │ -00086a30: 6965 730a 2020 2a20 2a6e 6f74 6520 6465  ies.  * *note de
│ │ │ │ -00086a40: 6772 6565 7328 5261 7469 6f6e 616c 4d61  grees(RationalMa
│ │ │ │ -00086a50: 7029 3a20 6465 6772 6565 735f 6c70 5261  p): degrees_lpRa
│ │ │ │ -00086a60: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -00086a70: 2070 726f 6a65 6374 6976 6520 6465 6772   projective degr
│ │ │ │ -00086a80: 6565 730a 2020 2020 6f66 2061 2072 6174  ees.    of a rat
│ │ │ │ -00086a90: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ -00086aa0: 6f74 6520 6465 6772 6565 2852 6174 696f  ote degree(Ratio
│ │ │ │ -00086ab0: 6e61 6c4d 6170 293a 2064 6567 7265 655f  nalMap): degree_
│ │ │ │ -00086ac0: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00086ad0: 2c20 2d2d 2064 6567 7265 6520 6f66 2061  , -- degree of a
│ │ │ │ -00086ae0: 2072 6174 696f 6e61 6c0a 2020 2020 6d61   rational.    ma
│ │ │ │ -00086af0: 700a 0a57 6179 7320 746f 2075 7365 2074  p..Ways to use t
│ │ │ │ -00086b00: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d  his method:.====
│ │ │ │ -00086b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00086b20: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -00086b30: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -00086b40: 7328 5261 7469 6f6e 616c 4d61 7029 3a20  s(RationalMap): 
│ │ │ │ -00086b50: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -00086b60: 735f 6c70 5261 7469 6f6e 616c 4d61 705f  s_lpRationalMap_
│ │ │ │ -00086b70: 7270 2c0a 2020 2020 2d2d 2070 726f 6a65  rp,.    -- proje
│ │ │ │ -00086b80: 6374 6976 6520 6465 6772 6565 7320 6f66  ctive degrees of
│ │ │ │ -00086b90: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ -00086ba0: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -00086bb0: 696e 666f 2c20 4e6f 6465 3a20 7175 6164  info, Node: quad
│ │ │ │ -00086bc0: 726f 5175 6164 7269 6343 7265 6d6f 6e61  roQuadricCremona
│ │ │ │ -00086bd0: 5472 616e 7366 6f72 6d61 7469 6f6e 2c20  Transformation, 
│ │ │ │ -00086be0: 4e65 7874 3a20 5261 7469 6f6e 616c 4d61  Next: RationalMa
│ │ │ │ -00086bf0: 702c 2050 7265 763a 2070 726f 6a65 6374  p, Prev: project
│ │ │ │ -00086c00: 6976 6544 6567 7265 6573 5f6c 7052 6174  iveDegrees_lpRat
│ │ │ │ -00086c10: 696f 6e61 6c4d 6170 5f72 702c 2055 703a  ionalMap_rp, Up:
│ │ │ │ -00086c20: 2054 6f70 0a0a 7175 6164 726f 5175 6164   Top..quadroQuad
│ │ │ │ -00086c30: 7269 6343 7265 6d6f 6e61 5472 616e 7366  ricCremonaTransf
│ │ │ │ -00086c40: 6f72 6d61 7469 6f6e 202d 2d20 7175 6164  ormation -- quad
│ │ │ │ -00086c50: 726f 2d71 7561 6472 6963 2043 7265 6d6f  ro-quadric Cremo
│ │ │ │ -00086c60: 6e61 2074 7261 6e73 666f 726d 6174 696f  na transformatio
│ │ │ │ -00086c70: 6e73 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ns.*************
│ │ │ │ +00086600: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +00086610: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2046  .========..  * F
│ │ │ │ +00086620: 756e 6374 696f 6e3a 202a 6e6f 7465 2070  unction: *note p
│ │ │ │ +00086630: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ +00086640: 3a20 7072 6f6a 6563 7469 7665 4465 6772  : projectiveDegr
│ │ │ │ +00086650: 6565 732c 0a20 202a 2055 7361 6765 3a20  ees,.  * Usage: 
│ │ │ │ +00086660: 0a20 2020 2020 2020 2070 726f 6a65 6374  .        project
│ │ │ │ +00086670: 6976 6544 6567 7265 6573 2050 6869 0a20  iveDegrees Phi. 
│ │ │ │ +00086680: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +00086690: 202a 2050 6869 2c20 6120 2a6e 6f74 6520   * Phi, a *note 
│ │ │ │ +000866a0: 7261 7469 6f6e 616c 206d 6170 3a20 5261  rational map: Ra
│ │ │ │ +000866b0: 7469 6f6e 616c 4d61 702c 0a20 202a 202a  tionalMap,.  * *
│ │ │ │ +000866c0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ +000866d0: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ +000866e0: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ +000866f0: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ +00086700: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ +00086710: 202a 202a 6e6f 7465 2042 6c6f 7755 7053   * *note BlowUpS
│ │ │ │ +00086720: 7472 6174 6567 793a 2042 6c6f 7755 7053  trategy: BlowUpS
│ │ │ │ +00086730: 7472 6174 6567 792c 203d 3e20 2e2e 2e2c  trategy, => ...,
│ │ │ │ +00086740: 2064 6566 6175 6c74 2076 616c 7565 0a20   default value. 
│ │ │ │ +00086750: 2020 2020 2020 2022 456c 696d 696e 6174         "Eliminat
│ │ │ │ +00086760: 6522 2c0a 2020 2020 2020 2a20 2a6e 6f74  e",.      * *not
│ │ │ │ +00086770: 6520 4365 7274 6966 793a 2043 6572 7469  e Certify: Certi
│ │ │ │ +00086780: 6679 2c20 3d3e 202e 2e2e 2c20 6465 6661  fy, => ..., defa
│ │ │ │ +00086790: 756c 7420 7661 6c75 6520 6661 6c73 652c  ult value false,
│ │ │ │ +000867a0: 2077 6865 7468 6572 2074 6f20 656e 7375   whether to ensu
│ │ │ │ +000867b0: 7265 0a20 2020 2020 2020 2063 6f72 7265  re.        corre
│ │ │ │ +000867c0: 6374 6e65 7373 206f 6620 6f75 7470 7574  ctness of output
│ │ │ │ +000867d0: 0a20 2020 2020 202a 202a 6e6f 7465 204e  .      * *note N
│ │ │ │ +000867e0: 756d 4465 6772 6565 733a 204e 756d 4465  umDegrees: NumDe
│ │ │ │ +000867f0: 6772 6565 732c 203d 3e20 2e2e 2e2c 2064  grees, => ..., d
│ │ │ │ +00086800: 6566 6175 6c74 2076 616c 7565 2069 6e66  efault value inf
│ │ │ │ +00086810: 696e 6974 792c 200a 2020 2020 2020 2a20  inity, .      * 
│ │ │ │ +00086820: 2a6e 6f74 6520 5665 7262 6f73 653a 2069  *note Verbose: i
│ │ │ │ +00086830: 6e76 6572 7365 4d61 705f 6c70 5f70 645f  nverseMap_lp_pd_
│ │ │ │ +00086840: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ +00086850: 3e5f 7064 5f70 645f 7064 5f72 702c 203d  >_pd_pd_pd_rp, =
│ │ │ │ +00086860: 3e20 2e2e 2e2c 0a20 2020 2020 2020 2064  > ...,.        d
│ │ │ │ +00086870: 6566 6175 6c74 2076 616c 7565 2074 7275  efault value tru
│ │ │ │ +00086880: 652c 0a20 202a 204f 7574 7075 7473 3a0a  e,.  * Outputs:.
│ │ │ │ +00086890: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ +000868a0: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ +000868b0: 446f 6329 4c69 7374 2c2c 2074 6865 206c  Doc)List,, the l
│ │ │ │ +000868c0: 6973 7420 6f66 2070 726f 6a65 6374 6976  ist of projectiv
│ │ │ │ +000868d0: 6520 6465 6772 6565 7320 6f66 0a20 2020  e degrees of.   
│ │ │ │ +000868e0: 2020 2020 2050 6869 0a0a 4465 7363 7269       Phi..Descri
│ │ │ │ +000868f0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +00086900: 3d0a 0a54 6869 7320 636f 6d70 7574 6174  =..This computat
│ │ │ │ +00086910: 696f 6e20 6973 2064 6f6e 6520 7468 726f  ion is done thro
│ │ │ │ +00086920: 7567 6820 7468 6520 636f 7272 6573 706f  ugh the correspo
│ │ │ │ +00086930: 6e64 696e 6720 6d65 7468 6f64 2066 6f72  nding method for
│ │ │ │ +00086940: 2072 696e 6720 6d61 7073 2e20 5365 650a   ring maps. See.
│ │ │ │ +00086950: 2a6e 6f74 6520 7072 6f6a 6563 7469 7665  *note projective
│ │ │ │ +00086960: 4465 6772 6565 7328 5269 6e67 4d61 7029  Degrees(RingMap)
│ │ │ │ +00086970: 3a20 7072 6f6a 6563 7469 7665 4465 6772  : projectiveDegr
│ │ │ │ +00086980: 6565 732c 2066 6f72 206d 6f72 6520 6465  ees, for more de
│ │ │ │ +00086990: 7461 696c 7320 616e 640a 6578 616d 706c  tails and.exampl
│ │ │ │ +000869a0: 6573 2e0a 0a53 6565 2061 6c73 6f0a 3d3d  es...See also.==
│ │ │ │ +000869b0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +000869c0: 6520 7072 6f6a 6563 7469 7665 4465 6772  e projectiveDegr
│ │ │ │ +000869d0: 6565 7328 5269 6e67 4d61 7029 3a20 7072  ees(RingMap): pr
│ │ │ │ +000869e0: 6f6a 6563 7469 7665 4465 6772 6565 732c  ojectiveDegrees,
│ │ │ │ +000869f0: 202d 2d20 7072 6f6a 6563 7469 7665 2064   -- projective d
│ │ │ │ +00086a00: 6567 7265 6573 0a20 2020 206f 6620 6120  egrees.    of a 
│ │ │ │ +00086a10: 7261 7469 6f6e 616c 206d 6170 2062 6574  rational map bet
│ │ │ │ +00086a20: 7765 656e 2070 726f 6a65 6374 6976 6520  ween projective 
│ │ │ │ +00086a30: 7661 7269 6574 6965 730a 2020 2a20 2a6e  varieties.  * *n
│ │ │ │ +00086a40: 6f74 6520 6465 6772 6565 7328 5261 7469  ote degrees(Rati
│ │ │ │ +00086a50: 6f6e 616c 4d61 7029 3a20 6465 6772 6565  onalMap): degree
│ │ │ │ +00086a60: 735f 6c70 5261 7469 6f6e 616c 4d61 705f  s_lpRationalMap_
│ │ │ │ +00086a70: 7270 2c20 2d2d 2070 726f 6a65 6374 6976  rp, -- projectiv
│ │ │ │ +00086a80: 6520 6465 6772 6565 730a 2020 2020 6f66  e degrees.    of
│ │ │ │ +00086a90: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +00086aa0: 2020 2a20 2a6e 6f74 6520 6465 6772 6565    * *note degree
│ │ │ │ +00086ab0: 2852 6174 696f 6e61 6c4d 6170 293a 2064  (RationalMap): d
│ │ │ │ +00086ac0: 6567 7265 655f 6c70 5261 7469 6f6e 616c  egree_lpRational
│ │ │ │ +00086ad0: 4d61 705f 7270 2c20 2d2d 2064 6567 7265  Map_rp, -- degre
│ │ │ │ +00086ae0: 6520 6f66 2061 2072 6174 696f 6e61 6c0a  e of a rational.
│ │ │ │ +00086af0: 2020 2020 6d61 700a 0a57 6179 7320 746f      map..Ways to
│ │ │ │ +00086b00: 2075 7365 2074 6869 7320 6d65 7468 6f64   use this method
│ │ │ │ +00086b10: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +00086b20: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00086b30: 2a6e 6f74 6520 7072 6f6a 6563 7469 7665  *note projective
│ │ │ │ +00086b40: 4465 6772 6565 7328 5261 7469 6f6e 616c  Degrees(Rational
│ │ │ │ +00086b50: 4d61 7029 3a20 7072 6f6a 6563 7469 7665  Map): projective
│ │ │ │ +00086b60: 4465 6772 6565 735f 6c70 5261 7469 6f6e  Degrees_lpRation
│ │ │ │ +00086b70: 616c 4d61 705f 7270 2c0a 2020 2020 2d2d  alMap_rp,.    --
│ │ │ │ +00086b80: 2070 726f 6a65 6374 6976 6520 6465 6772   projective degr
│ │ │ │ +00086b90: 6565 7320 6f66 2061 2072 6174 696f 6e61  ees of a rationa
│ │ │ │ +00086ba0: 6c20 6d61 700a 1f0a 4669 6c65 3a20 4372  l map...File: Cr
│ │ │ │ +00086bb0: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ +00086bc0: 3a20 7175 6164 726f 5175 6164 7269 6343  : quadroQuadricC
│ │ │ │ +00086bd0: 7265 6d6f 6e61 5472 616e 7366 6f72 6d61  remonaTransforma
│ │ │ │ +00086be0: 7469 6f6e 2c20 4e65 7874 3a20 5261 7469  tion, Next: Rati
│ │ │ │ +00086bf0: 6f6e 616c 4d61 702c 2050 7265 763a 2070  onalMap, Prev: p
│ │ │ │ +00086c00: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ +00086c10: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +00086c20: 702c 2055 703a 2054 6f70 0a0a 7175 6164  p, Up: Top..quad
│ │ │ │ +00086c30: 726f 5175 6164 7269 6343 7265 6d6f 6e61  roQuadricCremona
│ │ │ │ +00086c40: 5472 616e 7366 6f72 6d61 7469 6f6e 202d  Transformation -
│ │ │ │ +00086c50: 2d20 7175 6164 726f 2d71 7561 6472 6963  - quadro-quadric
│ │ │ │ +00086c60: 2043 7265 6d6f 6e61 2074 7261 6e73 666f   Cremona transfo
│ │ │ │ +00086c70: 726d 6174 696f 6e73 0a2a 2a2a 2a2a 2a2a  rmations.*******
│ │ │ │  00086c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00086c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00086ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00086cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00086cc0: 0a53 796e 6f70 7369 730a 3d3d 3d3d 3d3d  .Synopsis.======
│ │ │ │ -00086cd0: 3d3d 0a0a 2020 2a20 5573 6167 653a 200a  ==..  * Usage: .
│ │ │ │ -00086ce0: 2020 2020 2020 2020 7175 6164 726f 5175          quadroQu
│ │ │ │ -00086cf0: 6164 7269 6343 7265 6d6f 6e61 5472 616e  adricCremonaTran
│ │ │ │ -00086d00: 7366 6f72 6d61 7469 6f6e 286e 2c69 2920  sformation(n,i) 
│ │ │ │ -00086d10: 0a20 2020 2020 2020 2071 7561 6472 6f51  .        quadroQ
│ │ │ │ -00086d20: 7561 6472 6963 4372 656d 6f6e 6154 7261  uadricCremonaTra
│ │ │ │ -00086d30: 6e73 666f 726d 6174 696f 6e28 6e2c 692c  nsformation(n,i,
│ │ │ │ -00086d40: 4b29 0a20 202a 2049 6e70 7574 733a 0a20  K).  * Inputs:. 
│ │ │ │ -00086d50: 2020 2020 202a 206e 2c20 616e 202a 6e6f       * n, an *no
│ │ │ │ -00086d60: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ -00086d70: 6175 6c61 7932 446f 6329 5a5a 2c2c 2074  aulay2Doc)ZZ,, t
│ │ │ │ -00086d80: 6865 2064 696d 656e 7369 6f6e 206f 6620  he dimension of 
│ │ │ │ -00086d90: 7468 6520 7072 6f6a 6563 7469 7665 0a20  the projective. 
│ │ │ │ -00086da0: 2020 2020 2020 2073 7061 6365 0a20 2020         space.   
│ │ │ │ -00086db0: 2020 202a 2069 2c20 616e 202a 6e6f 7465     * i, an *note
│ │ │ │ -00086dc0: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ -00086dd0: 6c61 7932 446f 6329 5a5a 2c2c 2074 6865  lay2Doc)ZZ,, the
│ │ │ │ -00086de0: 2069 2d74 6820 6361 7365 2069 6e20 7468   i-th case in th
│ │ │ │ -00086df0: 650a 2020 2020 2020 2020 636c 6173 7369  e.        classi
│ │ │ │ -00086e00: 6669 6361 7469 6f6e 2066 6f72 2050 5e6e  fication for P^n
│ │ │ │ -00086e10: 2028 666f 7220 696e 7374 616e 6365 2c20   (for instance, 
│ │ │ │ -00086e20: 6966 206e 3d35 2074 6865 6e20 313c 3d69  if n=5 then 1<=i
│ │ │ │ -00086e30: 3c3d 3339 290a 2020 2020 2020 2a20 4b2c  <=39).      * K,
│ │ │ │ -00086e40: 2061 202a 6e6f 7465 2072 696e 673a 2028   a *note ring: (
│ │ │ │ -00086e50: 4d61 6361 756c 6179 3244 6f63 2952 696e  Macaulay2Doc)Rin
│ │ │ │ -00086e60: 672c 2c20 7468 6520 6772 6f75 6e64 2066  g,, the ground f
│ │ │ │ -00086e70: 6965 6c64 2028 6f70 7469 6f6e 616c 2c20  ield (optional, 
│ │ │ │ -00086e80: 7468 650a 2020 2020 2020 2020 6465 6661  the.        defa
│ │ │ │ -00086e90: 756c 7420 7661 6c75 6520 6973 202a 6e6f  ult value is *no
│ │ │ │ -00086ea0: 7465 2051 513a 2028 4d61 6361 756c 6179  te QQ: (Macaulay
│ │ │ │ -00086eb0: 3244 6f63 2951 512c 290a 2020 2a20 4f75  2Doc)QQ,).  * Ou
│ │ │ │ -00086ec0: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ -00086ed0: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ -00086ee0: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ -00086ef0: 2c2c 2061 6e20 6578 616d 706c 6520 6f66  ,, an example of
│ │ │ │ -00086f00: 2071 7561 6472 6f2d 7175 6164 7269 630a   quadro-quadric.
│ │ │ │ -00086f10: 2020 2020 2020 2020 4372 656d 6f6e 6120          Cremona 
│ │ │ │ -00086f20: 7472 616e 7366 6f72 6d61 7469 6f6e 206f  transformation o
│ │ │ │ -00086f30: 7665 7220 4b2c 2061 6363 6f72 6469 6e67  ver K, according
│ │ │ │ -00086f40: 2074 6f20 7468 6520 636c 6173 7369 6669   to the classifi
│ │ │ │ -00086f50: 6361 7469 6f6e 7320 6769 7665 6e0a 2020  cations given.  
│ │ │ │ -00086f60: 2020 2020 2020 696e 2074 6865 2070 6170        in the pap
│ │ │ │ -00086f70: 6572 2051 7561 6472 6f2d 7175 6164 7269  er Quadro-quadri
│ │ │ │ -00086f80: 6320 4372 656d 6f6e 6120 7472 616e 7366  c Cremona transf
│ │ │ │ -00086f90: 6f72 6d61 7469 6f6e 7320 696e 206c 6f77  ormations in low
│ │ │ │ -00086fa0: 2064 696d 656e 7369 6f6e 730a 2020 2020   dimensions.    
│ │ │ │ -00086fb0: 2020 2020 7669 6120 7468 6520 4a43 2d63      via the JC-c
│ │ │ │ -00086fc0: 6f72 7265 7370 6f6e 6465 6e63 6520 2873  orrespondence (s
│ │ │ │ -00086fd0: 6565 0a20 2020 2020 2020 2068 7474 7073  ee.        https
│ │ │ │ -00086fe0: 3a2f 2f61 6966 2e63 656e 7472 652d 6d65  ://aif.centre-me
│ │ │ │ -00086ff0: 7273 656e 6e65 2e6f 7267 2f69 7465 6d2f  rsenne.org/item/
│ │ │ │ -00087000: 4149 465f 3230 3134 5f5f 3634 5f31 5f37  AIF_2014__64_1_7
│ │ │ │ -00087010: 315f 302f 2029 2c20 6279 2050 6972 696f  1_0/ ), by Pirio
│ │ │ │ -00087020: 0a20 2020 2020 2020 2061 6e64 2052 7573  .        and Rus
│ │ │ │ -00087030: 736f 2e0a 0a44 6573 6372 6970 7469 6f6e  so...Description
│ │ │ │ -00087040: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ -00087050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00086cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00086cc0: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ +00086cd0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 5573  ========..  * Us
│ │ │ │ +00086ce0: 6167 653a 200a 2020 2020 2020 2020 7175  age: .        qu
│ │ │ │ +00086cf0: 6164 726f 5175 6164 7269 6343 7265 6d6f  adroQuadricCremo
│ │ │ │ +00086d00: 6e61 5472 616e 7366 6f72 6d61 7469 6f6e  naTransformation
│ │ │ │ +00086d10: 286e 2c69 2920 0a20 2020 2020 2020 2071  (n,i) .        q
│ │ │ │ +00086d20: 7561 6472 6f51 7561 6472 6963 4372 656d  uadroQuadricCrem
│ │ │ │ +00086d30: 6f6e 6154 7261 6e73 666f 726d 6174 696f  onaTransformatio
│ │ │ │ +00086d40: 6e28 6e2c 692c 4b29 0a20 202a 2049 6e70  n(n,i,K).  * Inp
│ │ │ │ +00086d50: 7574 733a 0a20 2020 2020 202a 206e 2c20  uts:.      * n, 
│ │ │ │ +00086d60: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ +00086d70: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00086d80: 5a5a 2c2c 2074 6865 2064 696d 656e 7369  ZZ,, the dimensi
│ │ │ │ +00086d90: 6f6e 206f 6620 7468 6520 7072 6f6a 6563  on of the projec
│ │ │ │ +00086da0: 7469 7665 0a20 2020 2020 2020 2073 7061  tive.        spa
│ │ │ │ +00086db0: 6365 0a20 2020 2020 202a 2069 2c20 616e  ce.      * i, an
│ │ │ │ +00086dc0: 202a 6e6f 7465 2069 6e74 6567 6572 3a20   *note integer: 
│ │ │ │ +00086dd0: 284d 6163 6175 6c61 7932 446f 6329 5a5a  (Macaulay2Doc)ZZ
│ │ │ │ +00086de0: 2c2c 2074 6865 2069 2d74 6820 6361 7365  ,, the i-th case
│ │ │ │ +00086df0: 2069 6e20 7468 650a 2020 2020 2020 2020   in the.        
│ │ │ │ +00086e00: 636c 6173 7369 6669 6361 7469 6f6e 2066  classification f
│ │ │ │ +00086e10: 6f72 2050 5e6e 2028 666f 7220 696e 7374  or P^n (for inst
│ │ │ │ +00086e20: 616e 6365 2c20 6966 206e 3d35 2074 6865  ance, if n=5 the
│ │ │ │ +00086e30: 6e20 313c 3d69 3c3d 3339 290a 2020 2020  n 1<=i<=39).    
│ │ │ │ +00086e40: 2020 2a20 4b2c 2061 202a 6e6f 7465 2072    * K, a *note r
│ │ │ │ +00086e50: 696e 673a 2028 4d61 6361 756c 6179 3244  ing: (Macaulay2D
│ │ │ │ +00086e60: 6f63 2952 696e 672c 2c20 7468 6520 6772  oc)Ring,, the gr
│ │ │ │ +00086e70: 6f75 6e64 2066 6965 6c64 2028 6f70 7469  ound field (opti
│ │ │ │ +00086e80: 6f6e 616c 2c20 7468 650a 2020 2020 2020  onal, the.      
│ │ │ │ +00086e90: 2020 6465 6661 756c 7420 7661 6c75 6520    default value 
│ │ │ │ +00086ea0: 6973 202a 6e6f 7465 2051 513a 2028 4d61  is *note QQ: (Ma
│ │ │ │ +00086eb0: 6361 756c 6179 3244 6f63 2951 512c 290a  caulay2Doc)QQ,).
│ │ │ │ +00086ec0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +00086ed0: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ +00086ee0: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +00086ef0: 6e61 6c4d 6170 2c2c 2061 6e20 6578 616d  nalMap,, an exam
│ │ │ │ +00086f00: 706c 6520 6f66 2071 7561 6472 6f2d 7175  ple of quadro-qu
│ │ │ │ +00086f10: 6164 7269 630a 2020 2020 2020 2020 4372  adric.        Cr
│ │ │ │ +00086f20: 656d 6f6e 6120 7472 616e 7366 6f72 6d61  emona transforma
│ │ │ │ +00086f30: 7469 6f6e 206f 7665 7220 4b2c 2061 6363  tion over K, acc
│ │ │ │ +00086f40: 6f72 6469 6e67 2074 6f20 7468 6520 636c  ording to the cl
│ │ │ │ +00086f50: 6173 7369 6669 6361 7469 6f6e 7320 6769  assifications gi
│ │ │ │ +00086f60: 7665 6e0a 2020 2020 2020 2020 696e 2074  ven.        in t
│ │ │ │ +00086f70: 6865 2070 6170 6572 2051 7561 6472 6f2d  he paper Quadro-
│ │ │ │ +00086f80: 7175 6164 7269 6320 4372 656d 6f6e 6120  quadric Cremona 
│ │ │ │ +00086f90: 7472 616e 7366 6f72 6d61 7469 6f6e 7320  transformations 
│ │ │ │ +00086fa0: 696e 206c 6f77 2064 696d 656e 7369 6f6e  in low dimension
│ │ │ │ +00086fb0: 730a 2020 2020 2020 2020 7669 6120 7468  s.        via th
│ │ │ │ +00086fc0: 6520 4a43 2d63 6f72 7265 7370 6f6e 6465  e JC-corresponde
│ │ │ │ +00086fd0: 6e63 6520 2873 6565 0a20 2020 2020 2020  nce (see.       
│ │ │ │ +00086fe0: 2068 7474 7073 3a2f 2f61 6966 2e63 656e   https://aif.cen
│ │ │ │ +00086ff0: 7472 652d 6d65 7273 656e 6e65 2e6f 7267  tre-mersenne.org
│ │ │ │ +00087000: 2f69 7465 6d2f 4149 465f 3230 3134 5f5f  /item/AIF_2014__
│ │ │ │ +00087010: 3634 5f31 5f37 315f 302f 2029 2c20 6279  64_1_71_0/ ), by
│ │ │ │ +00087020: 2050 6972 696f 0a20 2020 2020 2020 2061   Pirio.        a
│ │ │ │ +00087030: 6e64 2052 7573 736f 2e0a 0a44 6573 6372  nd Russo...Descr
│ │ │ │ +00087040: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00087050: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ==..+-----------
│ │ │ │  00087060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00087080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00087090: 7c69 3120 3a20 7175 6164 726f 5175 6164  |i1 : quadroQuad
│ │ │ │ -000870a0: 7269 6343 7265 6d6f 6e61 5472 616e 7366  ricCremonaTransf
│ │ │ │ -000870b0: 6f72 6d61 7469 6f6e 2835 2c32 3329 2020  ormation(5,23)  
│ │ │ │ -000870c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000870d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00087080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00087090: 2d2d 2d2d 2b0a 7c69 3120 3a20 7175 6164  ----+.|i1 : quad
│ │ │ │ +000870a0: 726f 5175 6164 7269 6343 7265 6d6f 6e61  roQuadricCremona
│ │ │ │ +000870b0: 5472 616e 7366 6f72 6d61 7469 6f6e 2835  Transformation(5
│ │ │ │ +000870c0: 2c32 3329 2020 2020 2020 2020 2020 2020  ,23)            
│ │ │ │ +000870d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000870e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000870f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087110: 2020 7c0a 7c6f 3120 3d20 2d2d 2072 6174    |.|o1 = -- rat
│ │ │ │ -00087120: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │ -00087130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087110: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ +00087120: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ +00087130: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  00087140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087150: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ -00087160: 6365 3a20 5072 6f6a 2851 515b 782c 2079  ce: Proj(QQ[x, y
│ │ │ │ -00087170: 2c20 7a2c 2074 2c20 752c 2076 5d29 2020  , z, t, u, v])  
│ │ │ │ -00087180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087190: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ -000871a0: 7267 6574 3a20 5072 6f6a 2851 515b 782c  rget: Proj(QQ[x,
│ │ │ │ -000871b0: 2079 2c20 7a2c 2074 2c20 752c 2076 5d29   y, z, t, u, v])
│ │ │ │ -000871c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000871d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000871e0: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -000871f0: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +00087150: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00087160: 2020 736f 7572 6365 3a20 5072 6f6a 2851    source: Proj(Q
│ │ │ │ +00087170: 515b 782c 2079 2c20 7a2c 2074 2c20 752c  Q[x, y, z, t, u,
│ │ │ │ +00087180: 2076 5d29 2020 2020 2020 2020 2020 2020   v])            
│ │ │ │ +00087190: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000871a0: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ +000871b0: 2851 515b 782c 2079 2c20 7a2c 2074 2c20  (QQ[x, y, z, t, 
│ │ │ │ +000871c0: 752c 2076 5d29 2020 2020 2020 2020 2020  u, v])          
│ │ │ │ +000871d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000871e0: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ +000871f0: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │  00087200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087210: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00087220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087230: 2020 2078 2a79 2c20 2020 2020 2020 2020     x*y,         
│ │ │ │ +00087210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087220: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00087230: 2020 2020 2020 2020 2078 2a79 2c20 2020           x*y,   
│ │ │ │  00087240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00087260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087260: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00087270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000872a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000872b0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00087290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000872a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000872b0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  000872c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000872d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000872e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000872f0: 2020 2020 2020 2020 2078 202c 2020 2020           x ,    
│ │ │ │ -00087300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000872e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000872f0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +00087300: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  00087310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087320: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00087320: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00087330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00087360: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00087370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087380: 2020 2020 2020 3220 2020 2032 2020 2020        2    2    
│ │ │ │ -00087390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000873a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000873b0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -000873c0: 2079 2a7a 202b 2074 2020 2b20 7520 2c20   y*z + t  + u , 
│ │ │ │ -000873d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000873e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000873f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087380: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00087390: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000873a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000873b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000873c0: 2020 2020 202d 2079 2a7a 202b 2074 2020       - y*z + t  
│ │ │ │ +000873d0: 2b20 7520 2c20 2020 2020 2020 2020 2020  + u ,           
│ │ │ │ +000873e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000873f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00087400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087420: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00087430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087440: 2020 202d 782a 742c 2020 2020 2020 2020     -x*t,        
│ │ │ │ +00087420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087430: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00087440: 2020 2020 2020 2020 202d 782a 742c 2020           -x*t,  
│ │ │ │  00087450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087460: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00087470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087470: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00087480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000874a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000874b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000874c0: 2020 2020 2020 202d 782a 752c 2020 2020         -x*u,    
│ │ │ │ -000874d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000874a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000874b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000874c0: 2020 2020 2020 2020 2020 2020 202d 782a               -x*
│ │ │ │ +000874d0: 752c 2020 2020 2020 2020 2020 2020 2020  u,              
│ │ │ │  000874e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000874f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000874f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00087500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087530: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00087540: 2020 2020 2020 2020 2020 202d 782a 7620             -x*v 
│ │ │ │ -00087550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087530: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00087540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087550: 202d 782a 7620 2020 2020 2020 2020 2020   -x*v           
│ │ │ │  00087560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087570: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00087580: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ -00087590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087570: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00087580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087590: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │  000875a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000875b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000875b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000875c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000875d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000875e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000875f0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -00087600: 5261 7469 6f6e 616c 4d61 7020 2843 7265  RationalMap (Cre
│ │ │ │ -00087610: 6d6f 6e61 2074 7261 6e73 666f 726d 6174  mona transformat
│ │ │ │ -00087620: 696f 6e20 6f66 2050 505e 3520 6f66 2074  ion of PP^5 of t
│ │ │ │ -00087630: 7970 6520 2832 2c32 2929 7c0a 2b2d 2d2d  ype (2,2))|.+---
│ │ │ │ -00087640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000875f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00087600: 7c6f 3120 3a20 5261 7469 6f6e 616c 4d61  |o1 : RationalMa
│ │ │ │ +00087610: 7020 2843 7265 6d6f 6e61 2074 7261 6e73  p (Cremona trans
│ │ │ │ +00087620: 666f 726d 6174 696f 6e20 6f66 2050 505e  formation of PP^
│ │ │ │ +00087630: 3520 6f66 2074 7970 6520 2832 2c32 2929  5 of type (2,2))
│ │ │ │ +00087640: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00087650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00087670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00087680: 3220 3a20 6465 7363 7269 6265 206f 6f20  2 : describe oo 
│ │ │ │ -00087690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00087680: 2d2d 2b0a 7c69 3220 3a20 6465 7363 7269  --+.|i2 : descri
│ │ │ │ +00087690: 6265 206f 6f20 2020 2020 2020 2020 2020  be oo           
│ │ │ │  000876a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000876b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000876c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000876b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000876c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000876d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000876e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000876f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087700: 7c0a 7c6f 3220 3d20 7261 7469 6f6e 616c  |.|o2 = rational
│ │ │ │ -00087710: 206d 6170 2064 6566 696e 6564 2062 7920   map defined by 
│ │ │ │ -00087720: 666f 726d 7320 6f66 2064 6567 7265 6520  forms of degree 
│ │ │ │ -00087730: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00087740: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ -00087750: 2076 6172 6965 7479 3a20 5050 5e35 2020   variety: PP^5  
│ │ │ │ -00087760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087700: 2020 2020 2020 7c0a 7c6f 3220 3d20 7261        |.|o2 = ra
│ │ │ │ +00087710: 7469 6f6e 616c 206d 6170 2064 6566 696e  tional map defin
│ │ │ │ +00087720: 6564 2062 7920 666f 726d 7320 6f66 2064  ed by forms of d
│ │ │ │ +00087730: 6567 7265 6520 3220 2020 2020 2020 2020  egree 2         
│ │ │ │ +00087740: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00087750: 736f 7572 6365 2076 6172 6965 7479 3a20  source variety: 
│ │ │ │ +00087760: 5050 5e35 2020 2020 2020 2020 2020 2020  PP^5            
│ │ │ │  00087770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087780: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ -00087790: 6574 2076 6172 6965 7479 3a20 5050 5e35  et variety: PP^5
│ │ │ │ -000877a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087780: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00087790: 2020 7461 7267 6574 2076 6172 6965 7479    target variety
│ │ │ │ +000877a0: 3a20 5050 5e35 2020 2020 2020 2020 2020  : PP^5          
│ │ │ │  000877b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000877c0: 2020 2020 2020 7c0a 7c20 2020 2020 646f        |.|     do
│ │ │ │ -000877d0: 6d69 6e61 6e63 653a 2074 7275 6520 2020  minance: true   
│ │ │ │ -000877e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000877c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000877d0: 2020 2020 646f 6d69 6e61 6e63 653a 2074      dominance: t
│ │ │ │ +000877e0: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │  000877f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00087810: 6269 7261 7469 6f6e 616c 6974 793a 2074  birationality: t
│ │ │ │ -00087820: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00087800: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00087810: 7c20 2020 2020 6269 7261 7469 6f6e 616c  |     birational
│ │ │ │ +00087820: 6974 793a 2074 7275 6520 2020 2020 2020  ity: true       
│ │ │ │  00087830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087840: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00087850: 2020 7072 6f6a 6563 7469 7665 2064 6567    projective deg
│ │ │ │ -00087860: 7265 6573 3a20 7b31 2c20 322c 2032 2c20  rees: {1, 2, 2, 
│ │ │ │ -00087870: 322c 2032 2c20 317d 2020 2020 2020 2020  2, 2, 1}        
│ │ │ │ -00087880: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00087890: 2020 2020 6e75 6d62 6572 206f 6620 6d69      number of mi
│ │ │ │ -000878a0: 6e69 6d61 6c20 7265 7072 6573 656e 7461  nimal representa
│ │ │ │ -000878b0: 7469 7665 733a 2031 2020 2020 2020 2020  tives: 1        
│ │ │ │ -000878c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000878d0: 7c20 2020 2020 6469 6d65 6e73 696f 6e20  |     dimension 
│ │ │ │ -000878e0: 6261 7365 206c 6f63 7573 3a20 3320 2020  base locus: 3   
│ │ │ │ -000878f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087850: 7c0a 7c20 2020 2020 7072 6f6a 6563 7469  |.|     projecti
│ │ │ │ +00087860: 7665 2064 6567 7265 6573 3a20 7b31 2c20  ve degrees: {1, 
│ │ │ │ +00087870: 322c 2032 2c20 322c 2032 2c20 317d 2020  2, 2, 2, 2, 1}  
│ │ │ │ +00087880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087890: 2020 7c0a 7c20 2020 2020 6e75 6d62 6572    |.|     number
│ │ │ │ +000878a0: 206f 6620 6d69 6e69 6d61 6c20 7265 7072   of minimal repr
│ │ │ │ +000878b0: 6573 656e 7461 7469 7665 733a 2031 2020  esentatives: 1  
│ │ │ │ +000878c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000878d0: 2020 2020 7c0a 7c20 2020 2020 6469 6d65      |.|     dime
│ │ │ │ +000878e0: 6e73 696f 6e20 6261 7365 206c 6f63 7573  nsion base locus
│ │ │ │ +000878f0: 3a20 3320 2020 2020 2020 2020 2020 2020  : 3             
│ │ │ │  00087900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087910: 7c0a 7c20 2020 2020 6465 6772 6565 2062  |.|     degree b
│ │ │ │ -00087920: 6173 6520 6c6f 6375 733a 2032 2020 2020  ase locus: 2    
│ │ │ │ -00087930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087910: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +00087920: 6772 6565 2062 6173 6520 6c6f 6375 733a  gree base locus:
│ │ │ │ +00087930: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00087940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087950: 2020 7c0a 7c20 2020 2020 636f 6566 6669    |.|     coeffi
│ │ │ │ -00087960: 6369 656e 7420 7269 6e67 3a20 5151 2020  cient ring: QQ  
│ │ │ │ -00087970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087950: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00087960: 636f 6566 6669 6369 656e 7420 7269 6e67  coefficient ring
│ │ │ │ +00087970: 3a20 5151 2020 2020 2020 2020 2020 2020  : QQ            
│ │ │ │  00087980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087990: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00087990: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  000879a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000879b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000879c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000879d0: 2d2d 2d2d 2d2d 2b0a 0a49 6e20 6164 6469  ------+..In addi
│ │ │ │ -000879e0: 7469 6f6e 2c20 7468 6520 666f 7572 2070  tion, the four p
│ │ │ │ -000879f0: 6169 7273 2028 6e2c 6929 3d28 352c 3129  airs (n,i)=(5,1)
│ │ │ │ -00087a00: 2c28 382c 3129 2c28 3134 2c31 292c 2832  ,(8,1),(14,1),(2
│ │ │ │ -00087a10: 362c 3129 2063 6f72 7265 7370 6f6e 6420  6,1) correspond 
│ │ │ │ -00087a20: 746f 2074 6865 0a66 6f75 7220 6578 616d  to the.four exam
│ │ │ │ -00087a30: 706c 6573 206f 6620 7370 6563 6961 6c20  ples of special 
│ │ │ │ -00087a40: 7175 6164 726f 2d71 7561 6472 6963 2043  quadro-quadric C
│ │ │ │ -00087a50: 7265 6d6f 6e61 2074 7261 6e73 666f 726d  remona transform
│ │ │ │ -00087a60: 6174 696f 6e73 3a0a 0a2b 2d2d 2d2d 2d2d  ations:..+------
│ │ │ │ +000879d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49  ------------+..I
│ │ │ │ +000879e0: 6e20 6164 6469 7469 6f6e 2c20 7468 6520  n addition, the 
│ │ │ │ +000879f0: 666f 7572 2070 6169 7273 2028 6e2c 6929  four pairs (n,i)
│ │ │ │ +00087a00: 3d28 352c 3129 2c28 382c 3129 2c28 3134  =(5,1),(8,1),(14
│ │ │ │ +00087a10: 2c31 292c 2832 362c 3129 2063 6f72 7265  ,1),(26,1) corre
│ │ │ │ +00087a20: 7370 6f6e 6420 746f 2074 6865 0a66 6f75  spond to the.fou
│ │ │ │ +00087a30: 7220 6578 616d 706c 6573 206f 6620 7370  r examples of sp
│ │ │ │ +00087a40: 6563 6961 6c20 7175 6164 726f 2d71 7561  ecial quadro-qua
│ │ │ │ +00087a50: 6472 6963 2043 7265 6d6f 6e61 2074 7261  dric Cremona tra
│ │ │ │ +00087a60: 6e73 666f 726d 6174 696f 6e73 3a0a 0a2b  nsformations:..+
│ │ │ │  00087a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00087ac0: 656d 6f6e 6154 7261 6e73 666f 726d 6174  emonaTransformat
│ │ │ │ -00087ad0: 696f 6e28 352c 3129 207c 0a7c 2020 2020  ion(5,1) |.|    
│ │ │ │ -00087ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087aa0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 6465  ------+.|i3 : de
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│ │ │ │ +00087ac0: 6472 6963 4372 656d 6f6e 6154 7261 6e73  dricCremonaTrans
│ │ │ │ +00087ad0: 666f 726d 6174 696f 6e28 352c 3129 207c  formation(5,1) |
│ │ │ │ +00087ae0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00087af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087b10: 2020 7c0a 7c6f 3320 3d20 7261 7469 6f6e    |.|o3 = ration
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│ │ │ │  00087b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087b80: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
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│ │ │ │ +00087b80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00087b90: 2020 7461 7267 6574 2076 6172 6965 7479    target variety
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│ │ │ │ +00087bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087bc0: 2020 207c 0a7c 2020 2020 2064 6f6d 696e     |.|     domin
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│ │ │ │  00087be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087bf0: 2020 2020 2020 7c0a 7c20 2020 2020 6269        |.|     bi
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│ │ │ │ -00087cd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ -00087cf0: 6375 733a 2034 2020 2020 2020 2020 2020  cus: 4          
│ │ │ │ +00087bf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00087c00: 2020 2020 6269 7261 7469 6f6e 616c 6974      birationalit
│ │ │ │ +00087c10: 793a 2074 7275 6520 2020 2020 2020 2020  y: true         
│ │ │ │ +00087c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087c30: 2020 2020 207c 0a7c 2020 2020 2070 726f       |.|     pro
│ │ │ │ +00087c40: 6a65 6374 6976 6520 6465 6772 6565 733a  jective degrees:
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│ │ │ │ +00087c60: 2031 7d20 2020 2020 2020 2020 2020 7c0a   1}           |.
│ │ │ │ +00087c70: 7c20 2020 2020 6e75 6d62 6572 206f 6620  |     number of 
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│ │ │ │ +00087c90: 7461 7469 7665 733a 2031 2020 2020 2020  tatives: 1      
│ │ │ │ +00087ca0: 2020 2020 2020 207c 0a7c 2020 2020 2064         |.|     d
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│ │ │ │ +00087cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087ce0: 7c0a 7c20 2020 2020 6465 6772 6565 2062  |.|     degree b
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│ │ │ │ -00087d40: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00087d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00087d10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00087d20: 2063 6f65 6666 6963 6965 6e74 2072 696e   coefficient rin
│ │ │ │ +00087d30: 673a 2051 5120 2020 2020 2020 2020 2020  g: QQ           
│ │ │ │ +00087d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087d50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00087d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00087d80: 2d2d 2d2d 2d2b 0a7c 6934 203a 2064 6573  -----+.|i4 : des
│ │ │ │ -00087d90: 6372 6962 6520 7175 6164 726f 5175 6164  cribe quadroQuad
│ │ │ │ -00087da0: 7269 6343 7265 6d6f 6e61 5472 616e 7366  ricCremonaTransf
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│ │ │ │ -00087dc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00087d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
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│ │ │ │ +00087da0: 726f 5175 6164 7269 6343 7265 6d6f 6e61  roQuadricCremona
│ │ │ │ +00087db0: 5472 616e 7366 6f72 6d61 7469 6f6e 2838  Transformation(8
│ │ │ │ +00087dc0: 2c31 2920 7c0a 7c20 2020 2020 2020 2020  ,1) |.|         
│ │ │ │  00087dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00087de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087df0: 2020 2020 2020 207c 0a7c 6f34 203d 2072         |.|o4 = r
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│ │ │ │ -00087e30: 7c0a 7c20 2020 2020 736f 7572 6365 2076  |.|     source v
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│ │ │ │ -00087e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087e60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00087e70: 2074 6172 6765 7420 7661 7269 6574 793a   target variety:
│ │ │ │ -00087e80: 2050 505e 3820 2020 2020 2020 2020 2020   PP^8           
│ │ │ │ +00087df0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00087e00: 6f34 203d 2072 6174 696f 6e61 6c20 6d61  o4 = rational ma
│ │ │ │ +00087e10: 7020 6465 6669 6e65 6420 6279 2066 6f72  p defined by for
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│ │ │ │ +00087e30: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +00087e40: 7572 6365 2076 6172 6965 7479 3a20 5050  urce variety: PP
│ │ │ │ +00087e50: 5e38 2020 2020 2020 2020 2020 2020 2020  ^8              
│ │ │ │ +00087e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00087e70: 0a7c 2020 2020 2074 6172 6765 7420 7661  .|     target va
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│ │ │ │  00087e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087ea0: 2020 7c0a 7c20 2020 2020 646f 6d69 6e61    |.|     domina
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│ │ │ │ +00087ea0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ -00087ed0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00087ee0: 2020 2062 6972 6174 696f 6e61 6c69 7479     birationality
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│ │ │ │ +00087ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087ee0: 207c 0a7c 2020 2020 2062 6972 6174 696f   |.|     biratio
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│ │ │ │  00087f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087f10: 2020 2020 7c0a 7c20 2020 2020 6e75 6d62      |.|     numb
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│ │ │ │ -00087f40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00087f50: 2020 2020 2064 696d 656e 7369 6f6e 2062       dimension b
│ │ │ │ -00087f60: 6173 6520 6c6f 6375 733a 2034 2020 2020  ase locus: 4    
│ │ │ │ -00087f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00087f80: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ -00087f90: 6772 6565 2062 6173 6520 6c6f 6375 733a  gree base locus:
│ │ │ │ -00087fa0: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ -00087fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00087fc0: 0a7c 2020 2020 2063 6f65 6666 6963 6965  .|     coefficie
│ │ │ │ -00087fd0: 6e74 2072 696e 673a 2051 5120 2020 2020  nt ring: QQ     
│ │ │ │ -00087fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00088000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00087f10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ +00087f50: 2020 207c 0a7c 2020 2020 2064 696d 656e     |.|     dimen
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│ │ │ │ +00087f70: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +00087f80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00087f90: 2020 2020 6465 6772 6565 2062 6173 6520      degree base 
│ │ │ │ +00087fa0: 6c6f 6375 733a 2036 2020 2020 2020 2020  locus: 6        
│ │ │ │ +00087fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087fc0: 2020 2020 207c 0a7c 2020 2020 2063 6f65       |.|     coe
│ │ │ │ +00087fd0: 6666 6963 6965 6e74 2072 696e 673a 2051  fficient ring: Q
│ │ │ │ +00087fe0: 5120 2020 2020 2020 2020 2020 2020 2020  Q               
│ │ │ │ +00087ff0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00088000: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00088010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00088030: 2d2b 0a7c 6935 203a 2064 6573 6372 6962  -+.|i5 : describ
│ │ │ │ -00088040: 6520 7175 6164 726f 5175 6164 7269 6343  e quadroQuadricC
│ │ │ │ -00088050: 7265 6d6f 6e61 5472 616e 7366 6f72 6d61  remonaTransforma
│ │ │ │ -00088060: 7469 6f6e 2831 342c 3129 7c0a 7c20 2020  tion(14,1)|.|   
│ │ │ │ -00088070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00088030: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2064  -------+.|i5 : d
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│ │ │ │ +00088050: 6164 7269 6343 7265 6d6f 6e61 5472 616e  adricCremonaTran
│ │ │ │ +00088060: 7366 6f72 6d61 7469 6f6e 2831 342c 3129  sformation(14,1)
│ │ │ │ +00088070: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00088080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000880a0: 2020 207c 0a7c 6f35 203d 2072 6174 696f     |.|o5 = ratio
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│ │ │ │ -000880e0: 2020 2020 736f 7572 6365 2076 6172 6965      source varie
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│ │ │ │ +000880a0: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
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│ │ │ │ +000880e0: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +000880f0: 2076 6172 6965 7479 3a20 5050 5e31 3420   variety: PP^14 
│ │ │ │  00088100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088110: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
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│ │ │ │ -00088130: 3134 2020 2020 2020 2020 2020 2020 2020  14              
│ │ │ │ -00088140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00088150: 7c20 2020 2020 646f 6d69 6e61 6e63 653a  |     dominance:
│ │ │ │ -00088160: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +00088110: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00088120: 2020 2074 6172 6765 7420 7661 7269 6574     target variet
│ │ │ │ +00088130: 793a 2050 505e 3134 2020 2020 2020 2020  y: PP^14        
│ │ │ │ +00088140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00088150: 2020 2020 7c0a 7c20 2020 2020 646f 6d69      |.|     domi
│ │ │ │ +00088160: 6e61 6e63 653a 2074 7275 6520 2020 2020  nance: true     
│ │ │ │  00088170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088180: 2020 2020 2020 207c 0a7c 2020 2020 2062         |.|     b
│ │ │ │ -00088190: 6972 6174 696f 6e61 6c69 7479 3a20 7472  irationality: tr
│ │ │ │ -000881a0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00088180: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088190: 2020 2020 2062 6972 6174 696f 6e61 6c69       birationali
│ │ │ │ +000881a0: 7479 3a20 7472 7565 2020 2020 2020 2020  ty: true        
│ │ │ │  000881b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000881c0: 7c0a 7c20 2020 2020 6e75 6d62 6572 206f  |.|     number o
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│ │ │ │ -000881e0: 656e 7461 7469 7665 733a 2031 2020 2020  entatives: 1    
│ │ │ │ -000881f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00088200: 2064 696d 656e 7369 6f6e 2062 6173 6520   dimension base 
│ │ │ │ -00088210: 6c6f 6375 733a 2038 2020 2020 2020 2020  locus: 8        
│ │ │ │ +000881c0: 2020 2020 2020 7c0a 7c20 2020 2020 6e75        |.|     nu
│ │ │ │ +000881d0: 6d62 6572 206f 6620 6d69 6e69 6d61 6c20  mber of minimal 
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│ │ │ │ +000881f0: 2031 2020 2020 2020 2020 2020 2020 207c   1             |
│ │ │ │ +00088200: 0a7c 2020 2020 2064 696d 656e 7369 6f6e  .|     dimension
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│ │ │ │  00088220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088230: 2020 7c0a 7c20 2020 2020 6465 6772 6565    |.|     degree
│ │ │ │ -00088240: 2062 6173 6520 6c6f 6375 733a 2031 3420   base locus: 14 
│ │ │ │ -00088250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088260: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00088270: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
│ │ │ │ -00088280: 696e 673a 2051 5120 2020 2020 2020 2020  ing: QQ         
│ │ │ │ +00088230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00088240: 6465 6772 6565 2062 6173 6520 6c6f 6375  degree base locu
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│ │ │ │ +00088260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00088270: 207c 0a7c 2020 2020 2063 6f65 6666 6963   |.|     coeffic
│ │ │ │ +00088280: 6965 6e74 2072 696e 673a 2051 5120 2020  ient ring: QQ   
│ │ │ │  00088290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000882a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000882a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  000882b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000882c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000882d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
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│ │ │ │ +000882f0: 6962 6520 7175 6164 726f 5175 6164 7269  ibe quadroQuadri
│ │ │ │ +00088300: 6343 7265 6d6f 6e61 5472 616e 7366 6f72  cCremonaTransfor
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│ │ │ │ -00088340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -00088380: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00088390: 736f 7572 6365 2076 6172 6965 7479 3a20  source variety: 
│ │ │ │ -000883a0: 5050 5e32 3620 2020 2020 2020 2020 2020  PP^26           
│ │ │ │ +00088340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00088350: 2020 2020 207c 0a7c 6f36 203d 2072 6174       |.|o6 = rat
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│ │ │ │ +000883a0: 6965 7479 3a20 5050 5e32 3620 2020 2020  iety: PP^26     
│ │ │ │  000883b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000883c0: 207c 0a7c 2020 2020 2074 6172 6765 7420   |.|     target 
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│ │ │ │ -000883e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000883f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00088400: 2020 646f 6d69 6e61 6e63 653a 2074 7275    dominance: tru
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│ │ │ │ +000883c0: 2020 2020 2020 207c 0a7c 2020 2020 2074         |.|     t
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│ │ │ │ +000883e0: 505e 3236 2020 2020 2020 2020 2020 2020  P^26            
│ │ │ │ +000883f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00088400: 7c0a 7c20 2020 2020 646f 6d69 6e61 6e63  |.|     dominanc
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│ │ │ │  00088420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088430: 2020 207c 0a7c 2020 2020 2062 6972 6174     |.|     birat
│ │ │ │ -00088440: 696f 6e61 6c69 7479 3a20 7472 7565 2020  ionality: true  
│ │ │ │ -00088450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088460: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00088470: 2020 2020 6e75 6d62 6572 206f 6620 6d69      number of mi
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│ │ │ │ -000884a0: 2020 2020 207c 0a7c 2020 2020 2064 696d       |.|     dim
│ │ │ │ -000884b0: 656e 7369 6f6e 2062 6173 6520 6c6f 6375  ension base locu
│ │ │ │ -000884c0: 733a 2031 3620 2020 2020 2020 2020 2020  s: 16           
│ │ │ │ -000884d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000884e0: 7c20 2020 2020 6465 6772 6565 2062 6173  |     degree bas
│ │ │ │ -000884f0: 6520 6c6f 6375 733a 2037 3820 2020 2020  e locus: 78     
│ │ │ │ -00088500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088510: 2020 2020 2020 207c 0a7c 2020 2020 2063         |.|     c
│ │ │ │ -00088520: 6f65 6666 6963 6965 6e74 2072 696e 673a  oefficient ring:
│ │ │ │ -00088530: 2051 5120 2020 2020 2020 2020 2020 2020   QQ             
│ │ │ │ +00088430: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00088440: 2062 6972 6174 696f 6e61 6c69 7479 3a20   birationality: 
│ │ │ │ +00088450: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00088460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00088470: 2020 7c0a 7c20 2020 2020 6e75 6d62 6572    |.|     number
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│ │ │ │ +00088490: 6573 656e 7461 7469 7665 733a 2031 2020  esentatives: 1  
│ │ │ │ +000884a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000884b0: 2020 2064 696d 656e 7369 6f6e 2062 6173     dimension bas
│ │ │ │ +000884c0: 6520 6c6f 6375 733a 2031 3620 2020 2020  e locus: 16     
│ │ │ │ +000884d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000884e0: 2020 2020 7c0a 7c20 2020 2020 6465 6772      |.|     degr
│ │ │ │ +000884f0: 6565 2062 6173 6520 6c6f 6375 733a 2037  ee base locus: 7
│ │ │ │ +00088500: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │ +00088510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088520: 2020 2020 2063 6f65 6666 6963 6965 6e74       coefficient
│ │ │ │ +00088530: 2072 696e 673a 2051 5120 2020 2020 2020   ring: QQ       
│ │ │ │  00088540: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00088560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000885b0: 4372 656d 6f6e 6154 7261 6e73 666f 726d  CremonaTransform
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│ │ │ │ -000885f0: 6961 6c20 4372 656d 6f6e 6120 7472 616e  ial Cremona tran
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│ │ │ │ -000886a0: 7472 616e 7366 6f72 6d61 7469 6f6e 7320  transformations 
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│ │ │ │ -000886e0: 7065 6369 616c 4375 6269 6354 7261 6e73  pecialCubicTrans
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│ │ │ │ -00088780: 7269 6343 7265 6d6f 6e61 5472 616e 7366  ricCremonaTransf
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│ │ │ │ +00088580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00088590: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +000885a0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2073  ===..  * *note s
│ │ │ │ +000885b0: 7065 6369 616c 4372 656d 6f6e 6154 7261  pecialCremonaTra
│ │ │ │ +000885c0: 6e73 666f 726d 6174 696f 6e3a 2073 7065  nsformation: spe
│ │ │ │ +000885d0: 6369 616c 4372 656d 6f6e 6154 7261 6e73  cialCremonaTrans
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│ │ │ │ +000885f0: 2020 7370 6563 6961 6c20 4372 656d 6f6e    special Cremon
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│ │ │ │ +00088690: 2020 2020 7370 6563 6961 6c20 7175 6164      special quad
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│ │ │ │ +000886c0: 206c 6f63 7573 2068 6173 2064 696d 656e   locus has dimen
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│ │ │ │ +00088720: 7370 6563 6961 6c0a 2020 2020 6375 6269  special.    cubi
│ │ │ │ +00088730: 6320 7472 616e 7366 6f72 6d61 7469 6f6e  c transformation
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│ │ │ │ +00088770: 5761 7973 2074 6f20 7573 6520 7175 6164  Ways to use quad
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│ │ │ │ +00088790: 5472 616e 7366 6f72 6d61 7469 6f6e 3a0a  Transformation:.
│ │ │ │  000887a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  000887b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000887c0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -000887d0: 7175 6164 726f 5175 6164 7269 6343 7265  quadroQuadricCre
│ │ │ │ -000887e0: 6d6f 6e61 5472 616e 7366 6f72 6d61 7469  monaTransformati
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│ │ │ │ -000888d0: 7265 6d6f 6e61 5472 616e 7366 6f72 6d61  remonaTransforma
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│ │ │ │ -00088990: 5261 7469 6f6e 616c 4d61 7020 2d2d 2074  RationalMap -- t
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│ │ │ │ -000889b0: 7261 7469 6f6e 616c 206d 6170 7320 6265  rational maps be
│ │ │ │ -000889c0: 7477 6565 6e20 6162 736f 6c75 7465 6c79  tween absolutely
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│ │ │ │ -000889f0: 7320 6f76 6572 2061 2066 6965 6c64 0a2a  s over a field.*
│ │ │ │ -00088a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000887c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +000887d0: 0a20 202a 2022 7175 6164 726f 5175 6164  .  * "quadroQuad
│ │ │ │ +000887e0: 7269 6343 7265 6d6f 6e61 5472 616e 7366  ricCremonaTransf
│ │ │ │ +000887f0: 6f72 6d61 7469 6f6e 2852 696e 672c 5a5a  ormation(Ring,ZZ
│ │ │ │ +00088800: 2c5a 5a29 220a 2020 2a20 2271 7561 6472  ,ZZ)".  * "quadr
│ │ │ │ +00088810: 6f51 7561 6472 6963 4372 656d 6f6e 6154  oQuadricCremonaT
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│ │ │ │ +00088830: 2c5a 5a29 220a 2020 2a20 2271 7561 6472  ,ZZ)".  * "quadr
│ │ │ │ +00088840: 6f51 7561 6472 6963 4372 656d 6f6e 6154  oQuadricCremonaT
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│ │ │ │ +00088860: 2c5a 5a2c 5269 6e67 2922 0a0a 466f 7220  ,ZZ,Ring)"..For 
│ │ │ │ +00088870: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +00088880: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00088950: 4e65 7874 3a20 7261 7469 6f6e 616c 4d61  Next: rationalMa
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│ │ │ │ -00088b00: 6279 2068 6f6d 6f67 656e 656f 7573 2069  by homogeneous i
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│ │ │ │ -00088c50: 4d61 7029 3a20 7261 7469 6f6e 616c 4d61  Map): rationalMa
│ │ │ │ -00088c60: 702c 2c20 2a6e 6f74 6520 7261 7469 6f6e  p,, *note ration
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│ │ │ │ -00088ca0: 5a5f 7270 2c2c 202a 6e6f 7465 2072 6174  Z_rp,, *note rat
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│ │ │ │ -00088cc0: 0a72 6174 696f 6e61 6c4d 6170 5f6c 7052  .rationalMap_lpR
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│ │ │ │ -00088ce0: 2061 6e64 202a 6e6f 7465 2072 6174 696f   and *note ratio
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│ │ │ │ +00088c20: 2053 6565 2069 6e20 7061 7274 6963 756c   See in particul
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│ │ │ │  00088e60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00088e70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00088e80: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
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│ │ │ │ -00088f80: 2020 2020 7370 6563 6961 6c20 4372 656d      special Crem
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│ │ │ │ -00089080: 7469 6354 7261 6e73 666f 726d 6174 696f  ticTransformatio
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│ │ │ │ -000890c0: 6c20 7175 6164 7261 7469 6320 7472 616e  l quadratic tran
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│ │ │ │ -000890e0: 6520 6261 7365 206c 6f63 7573 2068 6173  e base locus has
│ │ │ │ -000890f0: 2064 696d 656e 7369 6f6e 2074 6872 6565   dimension three
│ │ │ │ -00089100: 0a0a 4d65 7468 6f64 7320 7468 6174 2075  ..Methods that u
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│ │ │ │ +00088e80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
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│ │ │ │ +00088eb0: 7366 6f72 6d61 7469 6f6e 3a0a 2020 2020  sformation:.    
│ │ │ │ +00088ec0: 7175 6164 726f 5175 6164 7269 6343 7265  quadroQuadricCre
│ │ │ │ +00088ed0: 6d6f 6e61 5472 616e 7366 6f72 6d61 7469  monaTransformati
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│ │ │ │ +00088ef0: 6164 7269 6320 4372 656d 6f6e 610a 2020  adric Cremona.  
│ │ │ │ +00088f00: 2020 7472 616e 7366 6f72 6d61 7469 6f6e    transformation
│ │ │ │ +00088f10: 730a 2020 2a20 2a6e 6f74 6520 7365 6772  s.  * *note segr
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│ │ │ │ +00088ff0: 6354 7261 6e73 666f 726d 6174 696f 6e3a  cTransformation:
│ │ │ │ +00089000: 2073 7065 6369 616c 4375 6269 6354 7261   specialCubicTra
│ │ │ │ +00089010: 6e73 666f 726d 6174 696f 6e2c 202d 2d20  nsformation, -- 
│ │ │ │ +00089020: 7370 6563 6961 6c0a 2020 2020 6375 6269  special.    cubi
│ │ │ │ +00089030: 6320 7472 616e 7366 6f72 6d61 7469 6f6e  c transformation
│ │ │ │ +00089040: 7320 7768 6f73 6520 6261 7365 206c 6f63  s whose base loc
│ │ │ │ +00089050: 7573 2068 6173 2064 696d 656e 7369 6f6e  us has dimension
│ │ │ │ +00089060: 2061 7420 6d6f 7374 2074 6872 6565 0a20   at most three. 
│ │ │ │ +00089070: 202a 202a 6e6f 7465 2073 7065 6369 616c   * *note special
│ │ │ │ +00089080: 5175 6164 7261 7469 6354 7261 6e73 666f  QuadraticTransfo
│ │ │ │ +00089090: 726d 6174 696f 6e3a 2073 7065 6369 616c  rmation: special
│ │ │ │ +000890a0: 5175 6164 7261 7469 6354 7261 6e73 666f  QuadraticTransfo
│ │ │ │ +000890b0: 726d 6174 696f 6e2c 202d 2d0a 2020 2020  rmation, --.    
│ │ │ │ +000890c0: 7370 6563 6961 6c20 7175 6164 7261 7469  special quadrati
│ │ │ │ +000890d0: 6320 7472 616e 7366 6f72 6d61 7469 6f6e  c transformation
│ │ │ │ +000890e0: 7320 7768 6f73 6520 6261 7365 206c 6f63  s whose base loc
│ │ │ │ +000890f0: 7573 2068 6173 2064 696d 656e 7369 6f6e  us has dimension
│ │ │ │ +00089100: 2074 6872 6565 0a0a 4d65 7468 6f64 7320   three..Methods 
│ │ │ │ +00089110: 7468 6174 2075 7365 2061 2072 6174 696f  that use a ratio
│ │ │ │ +00089120: 6e61 6c20 6d61 703a 0a3d 3d3d 3d3d 3d3d  nal map:.=======
│ │ │ │  00089130: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00089140: 3d3d 3d0a 0a20 202a 2022 6162 7374 7261  ===..  * "abstra
│ │ │ │ -00089150: 6374 5261 7469 6f6e 616c 4d61 7028 5261  ctRationalMap(Ra
│ │ │ │ -00089160: 7469 6f6e 616c 4d61 7029 2220 2d2d 2073  tionalMap)" -- s
│ │ │ │ -00089170: 6565 202a 6e6f 7465 2061 6273 7472 6163  ee *note abstrac
│ │ │ │ -00089180: 7452 6174 696f 6e61 6c4d 6170 3a0a 2020  tRationalMap:.  
│ │ │ │ -00089190: 2020 6162 7374 7261 6374 5261 7469 6f6e    abstractRation
│ │ │ │ -000891a0: 616c 4d61 702c 202d 2d20 6d61 6b65 2061  alMap, -- make a
│ │ │ │ -000891b0: 6e20 6162 7374 7261 6374 2072 6174 696f  n abstract ratio
│ │ │ │ -000891c0: 6e61 6c20 6d61 700a 2020 2a20 2261 7070  nal map.  * "app
│ │ │ │ -000891d0: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
│ │ │ │ -000891e0: 6170 2852 6174 696f 6e61 6c4d 6170 2922  ap(RationalMap)"
│ │ │ │ -000891f0: 202d 2d20 7365 6520 2a6e 6f74 6520 6170   -- see *note ap
│ │ │ │ -00089200: 7072 6f78 696d 6174 6549 6e76 6572 7365  proximateInverse
│ │ │ │ -00089210: 4d61 703a 0a20 2020 2061 7070 726f 7869  Map:.    approxi
│ │ │ │ -00089220: 6d61 7465 496e 7665 7273 654d 6170 2c20  mateInverseMap, 
│ │ │ │ -00089230: 2d2d 2072 616e 646f 6d20 6d61 7020 7265  -- random map re
│ │ │ │ -00089240: 6c61 7465 6420 746f 2074 6865 2069 6e76  lated to the inv
│ │ │ │ -00089250: 6572 7365 206f 6620 6120 6269 7261 7469  erse of a birati
│ │ │ │ -00089260: 6f6e 616c 0a20 2020 206d 6170 0a20 202a  onal.    map.  *
│ │ │ │ -00089270: 2022 6170 7072 6f78 696d 6174 6549 6e76   "approximateInv
│ │ │ │ -00089280: 6572 7365 4d61 7028 5261 7469 6f6e 616c  erseMap(Rational
│ │ │ │ -00089290: 4d61 702c 5a5a 2922 202d 2d20 7365 6520  Map,ZZ)" -- see 
│ │ │ │ -000892a0: 2a6e 6f74 6520 6170 7072 6f78 696d 6174  *note approximat
│ │ │ │ -000892b0: 6549 6e76 6572 7365 4d61 703a 0a20 2020  eInverseMap:.   
│ │ │ │ -000892c0: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -000892d0: 7273 654d 6170 2c20 2d2d 2072 616e 646f  rseMap, -- rando
│ │ │ │ -000892e0: 6d20 6d61 7020 7265 6c61 7465 6420 746f  m map related to
│ │ │ │ -000892f0: 2074 6865 2069 6e76 6572 7365 206f 6620   the inverse of 
│ │ │ │ -00089300: 6120 6269 7261 7469 6f6e 616c 0a20 2020  a birational.   
│ │ │ │ -00089310: 206d 6170 0a20 202a 202a 6e6f 7465 2063   map.  * *note c
│ │ │ │ -00089320: 6f65 6666 6963 6965 6e74 5269 6e67 2852  oefficientRing(R
│ │ │ │ -00089330: 6174 696f 6e61 6c4d 6170 293a 2063 6f65  ationalMap): coe
│ │ │ │ -00089340: 6666 6963 6965 6e74 5269 6e67 5f6c 7052  fficientRing_lpR
│ │ │ │ -00089350: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ -00089360: 2d0a 2020 2020 636f 6566 6669 6369 656e  -.    coefficien
│ │ │ │ -00089370: 7420 7269 6e67 206f 6620 6120 7261 7469  t ring of a rati
│ │ │ │ -00089380: 6f6e 616c 206d 6170 0a20 202a 202a 6e6f  onal map.  * *no
│ │ │ │ -00089390: 7465 2063 6f65 6666 6963 6965 6e74 7328  te coefficients(
│ │ │ │ -000893a0: 5261 7469 6f6e 616c 4d61 7029 3a20 636f  RationalMap): co
│ │ │ │ -000893b0: 6566 6669 6369 656e 7473 5f6c 7052 6174  efficients_lpRat
│ │ │ │ -000893c0: 696f 6e61 6c4d 6170 5f72 702c 202d 2d0a  ionalMap_rp, --.
│ │ │ │ -000893d0: 2020 2020 636f 6566 6669 6369 656e 7420      coefficient 
│ │ │ │ -000893e0: 6d61 7472 6978 206f 6620 6120 7261 7469  matrix of a rati
│ │ │ │ -000893f0: 6f6e 616c 206d 6170 0a20 202a 202a 6e6f  onal map.  * *no
│ │ │ │ -00089400: 7465 2064 6567 7265 6528 5261 7469 6f6e  te degree(Ration
│ │ │ │ -00089410: 616c 4d61 7029 3a20 6465 6772 6565 5f6c  alMap): degree_l
│ │ │ │ -00089420: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -00089430: 202d 2d20 6465 6772 6565 206f 6620 6120   -- degree of a 
│ │ │ │ -00089440: 7261 7469 6f6e 616c 0a20 2020 206d 6170  rational.    map
│ │ │ │ -00089450: 0a20 202a 202a 6e6f 7465 2064 6567 7265  .  * *note degre
│ │ │ │ -00089460: 654d 6170 2852 6174 696f 6e61 6c4d 6170  eMap(RationalMap
│ │ │ │ -00089470: 293a 2064 6567 7265 654d 6170 5f6c 7052  ): degreeMap_lpR
│ │ │ │ -00089480: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ -00089490: 2d20 6465 6772 6565 206f 6620 610a 2020  - degree of a.  
│ │ │ │ -000894a0: 2020 7261 7469 6f6e 616c 206d 6170 0a20    rational map. 
│ │ │ │ -000894b0: 202a 202a 6e6f 7465 2064 6567 7265 6573   * *note degrees
│ │ │ │ -000894c0: 2852 6174 696f 6e61 6c4d 6170 293a 2064  (RationalMap): d
│ │ │ │ -000894d0: 6567 7265 6573 5f6c 7052 6174 696f 6e61  egrees_lpRationa
│ │ │ │ -000894e0: 6c4d 6170 5f72 702c 202d 2d20 7072 6f6a  lMap_rp, -- proj
│ │ │ │ -000894f0: 6563 7469 7665 2064 6567 7265 6573 0a20  ective degrees. 
│ │ │ │ -00089500: 2020 206f 6620 6120 7261 7469 6f6e 616c     of a rational
│ │ │ │ -00089510: 206d 6170 0a20 202a 2022 6d75 6c74 6964   map.  * "multid
│ │ │ │ -00089520: 6567 7265 6528 5261 7469 6f6e 616c 4d61  egree(RationalMa
│ │ │ │ -00089530: 7029 2220 2d2d 2073 6565 202a 6e6f 7465  p)" -- see *note
│ │ │ │ -00089540: 2064 6567 7265 6573 2852 6174 696f 6e61   degrees(Rationa
│ │ │ │ -00089550: 6c4d 6170 293a 0a20 2020 2064 6567 7265  lMap):.    degre
│ │ │ │ -00089560: 6573 5f6c 7052 6174 696f 6e61 6c4d 6170  es_lpRationalMap
│ │ │ │ -00089570: 5f72 702c 202d 2d20 7072 6f6a 6563 7469  _rp, -- projecti
│ │ │ │ -00089580: 7665 2064 6567 7265 6573 206f 6620 6120  ve degrees of a 
│ │ │ │ -00089590: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ -000895a0: 202a 6e6f 7465 2064 6573 6372 6962 6528   *note describe(
│ │ │ │ -000895b0: 5261 7469 6f6e 616c 4d61 7029 3a20 6465  RationalMap): de
│ │ │ │ -000895c0: 7363 7269 6265 5f6c 7052 6174 696f 6e61  scribe_lpRationa
│ │ │ │ -000895d0: 6c4d 6170 5f72 702c 202d 2d20 6465 7363  lMap_rp, -- desc
│ │ │ │ -000895e0: 7269 6265 2061 0a20 2020 2072 6174 696f  ribe a.    ratio
│ │ │ │ -000895f0: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74  nal map.  * *not
│ │ │ │ -00089600: 6520 656e 7472 6965 7328 5261 7469 6f6e  e entries(Ration
│ │ │ │ -00089610: 616c 4d61 7029 3a20 656e 7472 6965 735f  alMap): entries_
│ │ │ │ -00089620: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00089630: 2c20 2d2d 2074 6865 2065 6e74 7269 6573  , -- the entries
│ │ │ │ -00089640: 206f 6620 7468 650a 2020 2020 6d61 7472   of the.    matr
│ │ │ │ -00089650: 6978 2061 7373 6f63 6961 7465 6420 746f  ix associated to
│ │ │ │ -00089660: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ -00089670: 2020 2a20 2265 7863 6570 7469 6f6e 616c    * "exceptional
│ │ │ │ -00089680: 4c6f 6375 7328 5261 7469 6f6e 616c 4d61  Locus(RationalMa
│ │ │ │ -00089690: 7029 2220 2d2d 2073 6565 202a 6e6f 7465  p)" -- see *note
│ │ │ │ -000896a0: 2065 7863 6570 7469 6f6e 616c 4c6f 6375   exceptionalLocu
│ │ │ │ -000896b0: 733a 0a20 2020 2065 7863 6570 7469 6f6e  s:.    exception
│ │ │ │ -000896c0: 616c 4c6f 6375 732c 202d 2d20 6578 6365  alLocus, -- exce
│ │ │ │ -000896d0: 7074 696f 6e61 6c20 6c6f 6375 7320 6f66  ptional locus of
│ │ │ │ -000896e0: 2061 2062 6972 6174 696f 6e61 6c20 6d61   a birational ma
│ │ │ │ -000896f0: 700a 2020 2a20 2a6e 6f74 6520 666c 6174  p.  * *note flat
│ │ │ │ -00089700: 7465 6e28 5261 7469 6f6e 616c 4d61 7029  ten(RationalMap)
│ │ │ │ -00089710: 3a20 666c 6174 7465 6e5f 6c70 5261 7469  : flatten_lpRati
│ │ │ │ -00089720: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2077  onalMap_rp, -- w
│ │ │ │ -00089730: 7269 7465 2073 6f75 7263 6520 616e 640a  rite source and.
│ │ │ │ -00089740: 2020 2020 7461 7267 6574 2061 7320 6e6f      target as no
│ │ │ │ -00089750: 6e64 6567 656e 6572 6174 6520 7661 7269  ndegenerate vari
│ │ │ │ -00089760: 6574 6965 730a 2020 2a20 2266 6f72 6365  eties.  * "force
│ │ │ │ -00089770: 496d 6167 6528 5261 7469 6f6e 616c 4d61  Image(RationalMa
│ │ │ │ -00089780: 702c 4964 6561 6c29 2220 2d2d 2073 6565  p,Ideal)" -- see
│ │ │ │ -00089790: 202a 6e6f 7465 2066 6f72 6365 496d 6167   *note forceImag
│ │ │ │ -000897a0: 653a 2066 6f72 6365 496d 6167 652c 202d  e: forceImage, -
│ │ │ │ -000897b0: 2d0a 2020 2020 6465 636c 6172 6520 7768  -.    declare wh
│ │ │ │ -000897c0: 6963 6820 6973 2074 6865 2069 6d61 6765  ich is the image
│ │ │ │ -000897d0: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ -000897e0: 6170 0a20 202a 2022 666f 7263 6549 6e76  ap.  * "forceInv
│ │ │ │ -000897f0: 6572 7365 4d61 7028 5261 7469 6f6e 616c  erseMap(Rational
│ │ │ │ -00089800: 4d61 702c 5261 7469 6f6e 616c 4d61 7029  Map,RationalMap)
│ │ │ │ -00089810: 2220 2d2d 2073 6565 202a 6e6f 7465 2066  " -- see *note f
│ │ │ │ -00089820: 6f72 6365 496e 7665 7273 654d 6170 3a0a  orceInverseMap:.
│ │ │ │ -00089830: 2020 2020 666f 7263 6549 6e76 6572 7365      forceInverse
│ │ │ │ -00089840: 4d61 702c 202d 2d20 6465 636c 6172 6520  Map, -- declare 
│ │ │ │ -00089850: 7468 6174 2074 776f 2072 6174 696f 6e61  that two rationa
│ │ │ │ -00089860: 6c20 6d61 7073 2061 7265 206f 6e65 2074  l maps are one t
│ │ │ │ -00089870: 6865 2069 6e76 6572 7365 206f 660a 2020  he inverse of.  
│ │ │ │ -00089880: 2020 7468 6520 6f74 6865 720a 2020 2a20    the other.  * 
│ │ │ │ -00089890: 2267 7261 7068 2852 6174 696f 6e61 6c4d  "graph(RationalM
│ │ │ │ -000898a0: 6170 2922 202d 2d20 7365 6520 2a6e 6f74  ap)" -- see *not
│ │ │ │ -000898b0: 6520 6772 6170 683a 2067 7261 7068 2c20  e graph: graph, 
│ │ │ │ -000898c0: 2d2d 2063 6c6f 7375 7265 206f 6620 7468  -- closure of th
│ │ │ │ -000898d0: 6520 6772 6170 6820 6f66 0a20 2020 2061  e graph of.    a
│ │ │ │ -000898e0: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ -000898f0: 2a20 2a6e 6f74 6520 6964 6561 6c28 5261  * *note ideal(Ra
│ │ │ │ -00089900: 7469 6f6e 616c 4d61 7029 3a20 6964 6561  tionalMap): idea
│ │ │ │ -00089910: 6c5f 6c70 5261 7469 6f6e 616c 4d61 705f  l_lpRationalMap_
│ │ │ │ -00089920: 7270 2c20 2d2d 2062 6173 6520 6c6f 6375  rp, -- base locu
│ │ │ │ -00089930: 7320 6f66 2061 0a20 2020 2072 6174 696f  s of a.    ratio
│ │ │ │ -00089940: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74  nal map.  * *not
│ │ │ │ -00089950: 6520 696d 6167 6528 5261 7469 6f6e 616c  e image(Rational
│ │ │ │ -00089960: 4d61 702c 5374 7269 6e67 293a 2069 6d61  Map,String): ima
│ │ │ │ -00089970: 6765 5f6c 7052 6174 696f 6e61 6c4d 6170  ge_lpRationalMap
│ │ │ │ -00089980: 5f63 6d53 7472 696e 675f 7270 2c20 2d2d  _cmString_rp, --
│ │ │ │ -00089990: 0a20 2020 2063 6c6f 7375 7265 206f 6620  .    closure of 
│ │ │ │ -000899a0: 7468 6520 696d 6167 6520 6f66 2061 2072  the image of a r
│ │ │ │ -000899b0: 6174 696f 6e61 6c20 6d61 7020 7573 696e  ational map usin
│ │ │ │ -000899c0: 6720 7468 6520 4634 2061 6c67 6f72 6974  g the F4 algorit
│ │ │ │ -000899d0: 686d 0a20 2020 2028 6578 7065 7269 6d65  hm.    (experime
│ │ │ │ -000899e0: 6e74 616c 290a 2020 2a20 2269 6d61 6765  ntal).  * "image
│ │ │ │ -000899f0: 2852 6174 696f 6e61 6c4d 6170 2922 202d  (RationalMap)" -
│ │ │ │ -00089a00: 2d20 7365 6520 2a6e 6f74 6520 696d 6167  - see *note imag
│ │ │ │ -00089a10: 6528 5261 7469 6f6e 616c 4d61 702c 5a5a  e(RationalMap,ZZ
│ │ │ │ -00089a20: 293a 0a20 2020 2069 6d61 6765 5f6c 7052  ):.    image_lpR
│ │ │ │ -00089a30: 6174 696f 6e61 6c4d 6170 5f63 6d5a 5a5f  ationalMap_cmZZ_
│ │ │ │ -00089a40: 7270 2c20 2d2d 2063 6c6f 7375 7265 206f  rp, -- closure o
│ │ │ │ -00089a50: 6620 7468 6520 696d 6167 6520 6f66 2061  f the image of a
│ │ │ │ -00089a60: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ -00089a70: 2a20 2a6e 6f74 6520 696d 6167 6528 5261  * *note image(Ra
│ │ │ │ -00089a80: 7469 6f6e 616c 4d61 702c 5a5a 293a 2069  tionalMap,ZZ): i
│ │ │ │ -00089a90: 6d61 6765 5f6c 7052 6174 696f 6e61 6c4d  mage_lpRationalM
│ │ │ │ -00089aa0: 6170 5f63 6d5a 5a5f 7270 2c20 2d2d 2063  ap_cmZZ_rp, -- c
│ │ │ │ -00089ab0: 6c6f 7375 7265 206f 6620 7468 650a 2020  losure of the.  
│ │ │ │ -00089ac0: 2020 696d 6167 6520 6f66 2061 2072 6174    image of a rat
│ │ │ │ -00089ad0: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ -00089ae0: 6f74 6520 696e 7665 7273 6528 5261 7469  ote inverse(Rati
│ │ │ │ -00089af0: 6f6e 616c 4d61 7029 3a20 696e 7665 7273  onalMap): invers
│ │ │ │ -00089b00: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ -00089b10: 7270 2c20 2d2d 2069 6e76 6572 7365 206f  rp, -- inverse o
│ │ │ │ -00089b20: 6620 610a 2020 2020 6269 7261 7469 6f6e  f a.    biration
│ │ │ │ -00089b30: 616c 206d 6170 0a20 202a 2022 696e 7665  al map.  * "inve
│ │ │ │ -00089b40: 7273 6528 5261 7469 6f6e 616c 4d61 702c  rse(RationalMap,
│ │ │ │ -00089b50: 4f70 7469 6f6e 2922 202d 2d20 7365 6520  Option)" -- see 
│ │ │ │ -00089b60: 2a6e 6f74 6520 696e 7665 7273 6528 5261  *note inverse(Ra
│ │ │ │ -00089b70: 7469 6f6e 616c 4d61 7029 3a0a 2020 2020  tionalMap):.    
│ │ │ │ -00089b80: 696e 7665 7273 655f 6c70 5261 7469 6f6e  inverse_lpRation
│ │ │ │ -00089b90: 616c 4d61 705f 7270 2c20 2d2d 2069 6e76  alMap_rp, -- inv
│ │ │ │ -00089ba0: 6572 7365 206f 6620 6120 6269 7261 7469  erse of a birati
│ │ │ │ -00089bb0: 6f6e 616c 206d 6170 0a20 202a 2022 696e  onal map.  * "in
│ │ │ │ -00089bc0: 7665 7273 654d 6170 2852 6174 696f 6e61  verseMap(Rationa
│ │ │ │ -00089bd0: 6c4d 6170 2922 202d 2d20 7365 6520 2a6e  lMap)" -- see *n
│ │ │ │ -00089be0: 6f74 6520 696e 7665 7273 654d 6170 3a20  ote inverseMap: 
│ │ │ │ -00089bf0: 696e 7665 7273 654d 6170 2c20 2d2d 2069  inverseMap, -- i
│ │ │ │ -00089c00: 6e76 6572 7365 0a20 2020 206f 6620 6120  nverse.    of a 
│ │ │ │ -00089c10: 6269 7261 7469 6f6e 616c 206d 6170 0a20  birational map. 
│ │ │ │ -00089c20: 202a 2022 6973 4269 7261 7469 6f6e 616c   * "isBirational
│ │ │ │ -00089c30: 2852 6174 696f 6e61 6c4d 6170 2922 202d  (RationalMap)" -
│ │ │ │ -00089c40: 2d20 7365 6520 2a6e 6f74 6520 6973 4269  - see *note isBi
│ │ │ │ -00089c50: 7261 7469 6f6e 616c 3a20 6973 4269 7261  rational: isBira
│ │ │ │ -00089c60: 7469 6f6e 616c 2c20 2d2d 0a20 2020 2077  tional, --.    w
│ │ │ │ -00089c70: 6865 7468 6572 2061 2072 6174 696f 6e61  hether a rationa
│ │ │ │ -00089c80: 6c20 6d61 7020 6973 2062 6972 6174 696f  l map is biratio
│ │ │ │ -00089c90: 6e61 6c0a 2020 2a20 2269 7344 6f6d 696e  nal.  * "isDomin
│ │ │ │ -00089ca0: 616e 7428 5261 7469 6f6e 616c 4d61 7029  ant(RationalMap)
│ │ │ │ -00089cb0: 2220 2d2d 2073 6565 202a 6e6f 7465 2069  " -- see *note i
│ │ │ │ -00089cc0: 7344 6f6d 696e 616e 743a 2069 7344 6f6d  sDominant: isDom
│ │ │ │ -00089cd0: 696e 616e 742c 202d 2d20 7768 6574 6865  inant, -- whethe
│ │ │ │ -00089ce0: 7220 610a 2020 2020 7261 7469 6f6e 616c  r a.    rational
│ │ │ │ -00089cf0: 206d 6170 2069 7320 646f 6d69 6e61 6e74   map is dominant
│ │ │ │ -00089d00: 0a20 202a 202a 6e6f 7465 2069 7349 6e76  .  * *note isInv
│ │ │ │ -00089d10: 6572 7365 4d61 7028 5261 7469 6f6e 616c  erseMap(Rational
│ │ │ │ -00089d20: 4d61 702c 5261 7469 6f6e 616c 4d61 7029  Map,RationalMap)
│ │ │ │ -00089d30: 3a0a 2020 2020 6973 496e 7665 7273 654d  :.    isInverseM
│ │ │ │ -00089d40: 6170 5f6c 7052 6174 696f 6e61 6c4d 6170  ap_lpRationalMap
│ │ │ │ -00089d50: 5f63 6d52 6174 696f 6e61 6c4d 6170 5f72  _cmRationalMap_r
│ │ │ │ -00089d60: 702c 202d 2d20 6368 6563 6b73 2077 6865  p, -- checks whe
│ │ │ │ -00089d70: 7468 6572 2074 776f 2072 6174 696f 6e61  ther two rationa
│ │ │ │ -00089d80: 6c0a 2020 2020 6d61 7073 2061 7265 206f  l.    maps are o
│ │ │ │ -00089d90: 6e65 2074 6865 2069 6e76 6572 7365 206f  ne the inverse o
│ │ │ │ -00089da0: 6620 7468 6520 6f74 6865 720a 2020 2a20  f the other.  * 
│ │ │ │ -00089db0: 2a6e 6f74 6520 6973 4973 6f6d 6f72 7068  *note isIsomorph
│ │ │ │ -00089dc0: 6973 6d28 5261 7469 6f6e 616c 4d61 7029  ism(RationalMap)
│ │ │ │ -00089dd0: 3a20 6973 4973 6f6d 6f72 7068 6973 6d5f  : isIsomorphism_
│ │ │ │ -00089de0: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00089df0: 2c20 2d2d 0a20 2020 2077 6865 7468 6572  , --.    whether
│ │ │ │ -00089e00: 2061 2062 6972 6174 696f 6e61 6c20 6d61   a birational ma
│ │ │ │ -00089e10: 7020 6973 2061 6e20 6973 6f6d 6f72 7068  p is an isomorph
│ │ │ │ -00089e20: 6973 6d0a 2020 2a20 2269 734d 6f72 7068  ism.  * "isMorph
│ │ │ │ -00089e30: 6973 6d28 5261 7469 6f6e 616c 4d61 7029  ism(RationalMap)
│ │ │ │ -00089e40: 2220 2d2d 2073 6565 202a 6e6f 7465 2069  " -- see *note i
│ │ │ │ -00089e50: 734d 6f72 7068 6973 6d3a 2069 734d 6f72  sMorphism: isMor
│ │ │ │ -00089e60: 7068 6973 6d2c 202d 2d20 7768 6574 6865  phism, -- whethe
│ │ │ │ -00089e70: 7220 610a 2020 2020 7261 7469 6f6e 616c  r a.    rational
│ │ │ │ -00089e80: 206d 6170 2069 7320 6120 6d6f 7270 6869   map is a morphi
│ │ │ │ -00089e90: 736d 0a20 202a 202a 6e6f 7465 206d 6170  sm.  * *note map
│ │ │ │ -00089ea0: 2852 6174 696f 6e61 6c4d 6170 293a 206d  (RationalMap): m
│ │ │ │ -00089eb0: 6170 5f6c 7052 6174 696f 6e61 6c4d 6170  ap_lpRationalMap
│ │ │ │ -00089ec0: 5f72 702c 202d 2d20 6765 7420 7468 6520  _rp, -- get the 
│ │ │ │ -00089ed0: 7269 6e67 206d 6170 2064 6566 696e 696e  ring map definin
│ │ │ │ -00089ee0: 670a 2020 2020 6120 7261 7469 6f6e 616c  g.    a rational
│ │ │ │ -00089ef0: 206d 6170 0a20 202a 2022 6d61 7028 5a5a   map.  * "map(ZZ
│ │ │ │ -00089f00: 2c52 6174 696f 6e61 6c4d 6170 2922 202d  ,RationalMap)" -
│ │ │ │ -00089f10: 2d20 7365 6520 2a6e 6f74 6520 6d61 7028  - see *note map(
│ │ │ │ -00089f20: 5261 7469 6f6e 616c 4d61 7029 3a20 6d61  RationalMap): ma
│ │ │ │ -00089f30: 705f 6c70 5261 7469 6f6e 616c 4d61 705f  p_lpRationalMap_
│ │ │ │ -00089f40: 7270 2c0a 2020 2020 2d2d 2067 6574 2074  rp,.    -- get t
│ │ │ │ -00089f50: 6865 2072 696e 6720 6d61 7020 6465 6669  he ring map defi
│ │ │ │ -00089f60: 6e69 6e67 2061 2072 6174 696f 6e61 6c20  ning a rational 
│ │ │ │ -00089f70: 6d61 700a 2020 2a20 2a6e 6f74 6520 6d61  map.  * *note ma
│ │ │ │ -00089f80: 7472 6978 2852 6174 696f 6e61 6c4d 6170  trix(RationalMap
│ │ │ │ -00089f90: 293a 206d 6174 7269 785f 6c70 5261 7469  ): matrix_lpRati
│ │ │ │ -00089fa0: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2074  onalMap_rp, -- t
│ │ │ │ -00089fb0: 6865 206d 6174 7269 780a 2020 2020 6173  he matrix.    as
│ │ │ │ -00089fc0: 736f 6369 6174 6564 2074 6f20 6120 7261  sociated to a ra
│ │ │ │ -00089fd0: 7469 6f6e 616c 206d 6170 0a20 202a 2022  tional map.  * "
│ │ │ │ -00089fe0: 6d61 7472 6978 285a 5a2c 5261 7469 6f6e  matrix(ZZ,Ration
│ │ │ │ -00089ff0: 616c 4d61 7029 2220 2d2d 2073 6565 202a  alMap)" -- see *
│ │ │ │ -0008a000: 6e6f 7465 206d 6174 7269 7828 5261 7469  note matrix(Rati
│ │ │ │ -0008a010: 6f6e 616c 4d61 7029 3a0a 2020 2020 6d61  onalMap):.    ma
│ │ │ │ -0008a020: 7472 6978 5f6c 7052 6174 696f 6e61 6c4d  trix_lpRationalM
│ │ │ │ -0008a030: 6170 5f72 702c 202d 2d20 7468 6520 6d61  ap_rp, -- the ma
│ │ │ │ -0008a040: 7472 6978 2061 7373 6f63 6961 7465 6420  trix associated 
│ │ │ │ -0008a050: 746f 2061 2072 6174 696f 6e61 6c20 6d61  to a rational ma
│ │ │ │ -0008a060: 700a 2020 2a20 2a6e 6f74 6520 7072 6f6a  p.  * *note proj
│ │ │ │ -0008a070: 6563 7469 7665 4465 6772 6565 7328 5261  ectiveDegrees(Ra
│ │ │ │ -0008a080: 7469 6f6e 616c 4d61 7029 3a20 7072 6f6a  tionalMap): proj
│ │ │ │ -0008a090: 6563 7469 7665 4465 6772 6565 735f 6c70  ectiveDegrees_lp
│ │ │ │ -0008a0a0: 5261 7469 6f6e 616c 4d61 705f 7270 2c0a  RationalMap_rp,.
│ │ │ │ -0008a0b0: 2020 2020 2d2d 2070 726f 6a65 6374 6976      -- projectiv
│ │ │ │ -0008a0c0: 6520 6465 6772 6565 7320 6f66 2061 2072  e degrees of a r
│ │ │ │ -0008a0d0: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ -0008a0e0: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ -0008a0f0: 7020 213a 2052 6174 696f 6e61 6c4d 6170  p !: RationalMap
│ │ │ │ -0008a100: 2021 2c20 2d2d 2063 616c 6375 6c61 7465   !, -- calculate
│ │ │ │ -0008a110: 7320 6576 6572 7920 706f 7373 6962 6c65  s every possible
│ │ │ │ -0008a120: 2074 6869 6e67 0a20 202a 2022 636f 6d70   thing.  * "comp
│ │ │ │ -0008a130: 6f73 6528 5261 7469 6f6e 616c 4d61 702c  ose(RationalMap,
│ │ │ │ -0008a140: 5261 7469 6f6e 616c 4d61 7029 2220 2d2d  RationalMap)" --
│ │ │ │ -0008a150: 2073 6565 202a 6e6f 7465 2052 6174 696f   see *note Ratio
│ │ │ │ -0008a160: 6e61 6c4d 6170 202a 2052 6174 696f 6e61  nalMap * Rationa
│ │ │ │ -0008a170: 6c4d 6170 3a0a 2020 2020 5261 7469 6f6e  lMap:.    Ration
│ │ │ │ -0008a180: 616c 4d61 7020 5f73 7420 5261 7469 6f6e  alMap _st Ration
│ │ │ │ -0008a190: 616c 4d61 702c 202d 2d20 636f 6d70 6f73  alMap, -- compos
│ │ │ │ -0008a1a0: 6974 696f 6e20 6f66 2072 6174 696f 6e61  ition of rationa
│ │ │ │ -0008a1b0: 6c20 6d61 7073 0a20 202a 202a 6e6f 7465  l maps.  * *note
│ │ │ │ -0008a1c0: 2052 6174 696f 6e61 6c4d 6170 202a 2052   RationalMap * R
│ │ │ │ -0008a1d0: 6174 696f 6e61 6c4d 6170 3a20 5261 7469  ationalMap: Rati
│ │ │ │ -0008a1e0: 6f6e 616c 4d61 7020 5f73 7420 5261 7469  onalMap _st Rati
│ │ │ │ -0008a1f0: 6f6e 616c 4d61 702c 202d 2d0a 2020 2020  onalMap, --.    
│ │ │ │ -0008a200: 636f 6d70 6f73 6974 696f 6e20 6f66 2072  composition of r
│ │ │ │ -0008a210: 6174 696f 6e61 6c20 6d61 7073 0a20 202a  ational maps.  *
│ │ │ │ -0008a220: 202a 6e6f 7465 2052 6174 696f 6e61 6c4d   *note RationalM
│ │ │ │ -0008a230: 6170 202a 2a20 5269 6e67 3a20 5261 7469  ap ** Ring: Rati
│ │ │ │ -0008a240: 6f6e 616c 4d61 7020 5f73 745f 7374 2052  onalMap _st_st R
│ │ │ │ -0008a250: 696e 672c 202d 2d20 6368 616e 6765 2074  ing, -- change t
│ │ │ │ -0008a260: 6865 0a20 2020 2063 6f65 6666 6963 6965  he.    coefficie
│ │ │ │ -0008a270: 6e74 2072 696e 6720 6f66 2061 2072 6174  nt ring of a rat
│ │ │ │ -0008a280: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ -0008a290: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ -0008a2a0: 3d3d 2052 6174 696f 6e61 6c4d 6170 3a20  == RationalMap: 
│ │ │ │ -0008a2b0: 5261 7469 6f6e 616c 4d61 7020 3d3d 2052  RationalMap == R
│ │ │ │ -0008a2c0: 6174 696f 6e61 6c4d 6170 2c20 2d2d 2065  ationalMap, -- e
│ │ │ │ -0008a2d0: 7175 616c 6974 790a 2020 2020 6f66 2072  quality.    of r
│ │ │ │ -0008a2e0: 6174 696f 6e61 6c20 6d61 7073 0a20 202a  ational maps.  *
│ │ │ │ -0008a2f0: 2022 5261 7469 6f6e 616c 4d61 7020 3d3d   "RationalMap ==
│ │ │ │ -0008a300: 205a 5a22 202d 2d20 7365 6520 2a6e 6f74   ZZ" -- see *not
│ │ │ │ -0008a310: 6520 5261 7469 6f6e 616c 4d61 7020 3d3d  e RationalMap ==
│ │ │ │ -0008a320: 2052 6174 696f 6e61 6c4d 6170 3a20 5261   RationalMap: Ra
│ │ │ │ -0008a330: 7469 6f6e 616c 4d61 7020 3d3d 0a20 2020  tionalMap ==.   
│ │ │ │ -0008a340: 2052 6174 696f 6e61 6c4d 6170 2c20 2d2d   RationalMap, --
│ │ │ │ -0008a350: 2065 7175 616c 6974 7920 6f66 2072 6174   equality of rat
│ │ │ │ -0008a360: 696f 6e61 6c20 6d61 7073 0a20 202a 2022  ional maps.  * "
│ │ │ │ -0008a370: 5a5a 203d 3d20 5261 7469 6f6e 616c 4d61  ZZ == RationalMa
│ │ │ │ -0008a380: 7022 202d 2d20 7365 6520 2a6e 6f74 6520  p" -- see *note 
│ │ │ │ -0008a390: 5261 7469 6f6e 616c 4d61 7020 3d3d 2052  RationalMap == R
│ │ │ │ -0008a3a0: 6174 696f 6e61 6c4d 6170 3a20 5261 7469  ationalMap: Rati
│ │ │ │ -0008a3b0: 6f6e 616c 4d61 7020 3d3d 0a20 2020 2052  onalMap ==.    R
│ │ │ │ -0008a3c0: 6174 696f 6e61 6c4d 6170 2c20 2d2d 2065  ationalMap, -- e
│ │ │ │ -0008a3d0: 7175 616c 6974 7920 6f66 2072 6174 696f  quality of ratio
│ │ │ │ -0008a3e0: 6e61 6c20 6d61 7073 0a20 202a 202a 6e6f  nal maps.  * *no
│ │ │ │ -0008a3f0: 7465 2052 6174 696f 6e61 6c4d 6170 205e  te RationalMap ^
│ │ │ │ -0008a400: 205a 5a3a 2052 6174 696f 6e61 6c4d 6170   ZZ: RationalMap
│ │ │ │ -0008a410: 205e 205a 5a2c 202d 2d20 706f 7765 720a   ^ ZZ, -- power.
│ │ │ │ -0008a420: 2020 2a20 2252 6174 696f 6e61 6c4d 6170    * "RationalMap
│ │ │ │ -0008a430: 205e 2a22 202d 2d20 7365 6520 2a6e 6f74   ^*" -- see *not
│ │ │ │ -0008a440: 6520 5261 7469 6f6e 616c 4d61 7020 5e2a  e RationalMap ^*
│ │ │ │ -0008a450: 2a20 4964 6561 6c3a 2052 6174 696f 6e61  * Ideal: Rationa
│ │ │ │ -0008a460: 6c4d 6170 205e 5f73 745f 7374 0a20 2020  lMap ^_st_st.   
│ │ │ │ -0008a470: 2049 6465 616c 2c20 2d2d 2069 6e76 6572   Ideal, -- inver
│ │ │ │ -0008a480: 7365 2069 6d61 6765 2076 6961 2061 2072  se image via a r
│ │ │ │ -0008a490: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ -0008a4a0: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ -0008a4b0: 7020 5e2a 2a20 4964 6561 6c3a 2052 6174  p ^** Ideal: Rat
│ │ │ │ -0008a4c0: 696f 6e61 6c4d 6170 205e 5f73 745f 7374  ionalMap ^_st_st
│ │ │ │ -0008a4d0: 2049 6465 616c 2c20 2d2d 2069 6e76 6572   Ideal, -- inver
│ │ │ │ -0008a4e0: 7365 2069 6d61 6765 0a20 2020 2076 6961  se image.    via
│ │ │ │ -0008a4f0: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ -0008a500: 2020 2a20 2a6e 6f74 6520 5261 7469 6f6e    * *note Ration
│ │ │ │ -0008a510: 616c 4d61 7020 5f2a 3a20 5261 7469 6f6e  alMap _*: Ration
│ │ │ │ -0008a520: 616c 4d61 7020 5f75 735f 7374 2c20 2d2d  alMap _us_st, --
│ │ │ │ -0008a530: 2064 6972 6563 7420 696d 6167 6520 7669   direct image vi
│ │ │ │ -0008a540: 6120 6120 7261 7469 6f6e 616c 0a20 2020  a a rational.   
│ │ │ │ -0008a550: 206d 6170 0a20 202a 2022 5261 7469 6f6e   map.  * "Ration
│ │ │ │ -0008a560: 616c 4d61 7020 4964 6561 6c22 202d 2d20  alMap Ideal" -- 
│ │ │ │ -0008a570: 7365 6520 2a6e 6f74 6520 5261 7469 6f6e  see *note Ration
│ │ │ │ -0008a580: 616c 4d61 7020 5f2a 3a20 5261 7469 6f6e  alMap _*: Ration
│ │ │ │ -0008a590: 616c 4d61 7020 5f75 735f 7374 2c20 2d2d  alMap _us_st, --
│ │ │ │ -0008a5a0: 0a20 2020 2064 6972 6563 7420 696d 6167  .    direct imag
│ │ │ │ -0008a5b0: 6520 7669 6120 6120 7261 7469 6f6e 616c  e via a rational
│ │ │ │ -0008a5c0: 206d 6170 0a20 202a 202a 6e6f 7465 2052   map.  * *note R
│ │ │ │ -0008a5d0: 6174 696f 6e61 6c4d 6170 207c 2049 6465  ationalMap | Ide
│ │ │ │ -0008a5e0: 616c 3a20 5261 7469 6f6e 616c 4d61 7020  al: RationalMap 
│ │ │ │ -0008a5f0: 7c20 4964 6561 6c2c 202d 2d20 7265 7374  | Ideal, -- rest
│ │ │ │ -0008a600: 7269 6374 696f 6e20 6f66 2061 0a20 2020  riction of a.   
│ │ │ │ -0008a610: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ -0008a620: 2a20 2252 6174 696f 6e61 6c4d 6170 207c  * "RationalMap |
│ │ │ │ -0008a630: 2052 696e 6722 202d 2d20 7365 6520 2a6e   Ring" -- see *n
│ │ │ │ -0008a640: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ -0008a650: 7c20 4964 6561 6c3a 2052 6174 696f 6e61  | Ideal: Rationa
│ │ │ │ -0008a660: 6c4d 6170 207c 2049 6465 616c 2c0a 2020  lMap | Ideal,.  
│ │ │ │ -0008a670: 2020 2d2d 2072 6573 7472 6963 7469 6f6e    -- restriction
│ │ │ │ -0008a680: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ -0008a690: 6170 0a20 202a 2022 5261 7469 6f6e 616c  ap.  * "Rational
│ │ │ │ -0008a6a0: 4d61 7020 7c20 5269 6e67 456c 656d 656e  Map | RingElemen
│ │ │ │ -0008a6b0: 7422 202d 2d20 7365 6520 2a6e 6f74 6520  t" -- see *note 
│ │ │ │ -0008a6c0: 5261 7469 6f6e 616c 4d61 7020 7c20 4964  RationalMap | Id
│ │ │ │ -0008a6d0: 6561 6c3a 2052 6174 696f 6e61 6c4d 6170  eal: RationalMap
│ │ │ │ -0008a6e0: 207c 0a20 2020 2049 6465 616c 2c20 2d2d   |.    Ideal, --
│ │ │ │ -0008a6f0: 2072 6573 7472 6963 7469 6f6e 206f 6620   restriction of 
│ │ │ │ -0008a700: 6120 7261 7469 6f6e 616c 206d 6170 0a20  a rational map. 
│ │ │ │ -0008a710: 202a 202a 6e6f 7465 2052 6174 696f 6e61   * *note Rationa
│ │ │ │ -0008a720: 6c4d 6170 207c 7c20 4964 6561 6c3a 2052  lMap || Ideal: R
│ │ │ │ -0008a730: 6174 696f 6e61 6c4d 6170 207c 7c20 4964  ationalMap || Id
│ │ │ │ -0008a740: 6561 6c2c 202d 2d20 7265 7374 7269 6374  eal, -- restrict
│ │ │ │ -0008a750: 696f 6e20 6f66 2061 0a20 2020 2072 6174  ion of a.    rat
│ │ │ │ -0008a760: 696f 6e61 6c20 6d61 700a 2020 2a20 2252  ional map.  * "R
│ │ │ │ -0008a770: 6174 696f 6e61 6c4d 6170 207c 7c20 5269  ationalMap || Ri
│ │ │ │ -0008a780: 6e67 2220 2d2d 2073 6565 202a 6e6f 7465  ng" -- see *note
│ │ │ │ -0008a790: 2052 6174 696f 6e61 6c4d 6170 207c 7c20   RationalMap || 
│ │ │ │ -0008a7a0: 4964 6561 6c3a 2052 6174 696f 6e61 6c4d  Ideal: RationalM
│ │ │ │ -0008a7b0: 6170 207c 7c0a 2020 2020 4964 6561 6c2c  ap ||.    Ideal,
│ │ │ │ -0008a7c0: 202d 2d20 7265 7374 7269 6374 696f 6e20   -- restriction 
│ │ │ │ -0008a7d0: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ -0008a7e0: 700a 2020 2a20 2252 6174 696f 6e61 6c4d  p.  * "RationalM
│ │ │ │ -0008a7f0: 6170 207c 7c20 5269 6e67 456c 656d 656e  ap || RingElemen
│ │ │ │ -0008a800: 7422 202d 2d20 7365 6520 2a6e 6f74 6520  t" -- see *note 
│ │ │ │ -0008a810: 5261 7469 6f6e 616c 4d61 7020 7c7c 2049  RationalMap || I
│ │ │ │ -0008a820: 6465 616c 3a20 5261 7469 6f6e 616c 4d61  deal: RationalMa
│ │ │ │ -0008a830: 700a 2020 2020 7c7c 2049 6465 616c 2c20  p.    || Ideal, 
│ │ │ │ -0008a840: 2d2d 2072 6573 7472 6963 7469 6f6e 206f  -- restriction o
│ │ │ │ -0008a850: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ -0008a860: 0a20 202a 2022 7365 6772 6528 5261 7469  .  * "segre(Rati
│ │ │ │ -0008a870: 6f6e 616c 4d61 7029 2220 2d2d 2073 6565  onalMap)" -- see
│ │ │ │ -0008a880: 202a 6e6f 7465 2073 6567 7265 3a20 7365   *note segre: se
│ │ │ │ -0008a890: 6772 652c 202d 2d20 5365 6772 6520 656d  gre, -- Segre em
│ │ │ │ -0008a8a0: 6265 6464 696e 670a 2020 2a20 2253 6567  bedding.  * "Seg
│ │ │ │ -0008a8b0: 7265 436c 6173 7328 5261 7469 6f6e 616c  reClass(Rational
│ │ │ │ -0008a8c0: 4d61 7029 2220 2d2d 2073 6565 202a 6e6f  Map)" -- see *no
│ │ │ │ -0008a8d0: 7465 2053 6567 7265 436c 6173 733a 2053  te SegreClass: S
│ │ │ │ -0008a8e0: 6567 7265 436c 6173 732c 202d 2d20 5365  egreClass, -- Se
│ │ │ │ -0008a8f0: 6772 650a 2020 2020 636c 6173 7320 6f66  gre.    class of
│ │ │ │ -0008a900: 2061 2063 6c6f 7365 6420 7375 6273 6368   a closed subsch
│ │ │ │ -0008a910: 656d 6520 6f66 2061 2070 726f 6a65 6374  eme of a project
│ │ │ │ -0008a920: 6976 6520 7661 7269 6574 790a 2020 2a20  ive variety.  * 
│ │ │ │ -0008a930: 2a6e 6f74 6520 736f 7572 6365 2852 6174  *note source(Rat
│ │ │ │ -0008a940: 696f 6e61 6c4d 6170 293a 2073 6f75 7263  ionalMap): sourc
│ │ │ │ -0008a950: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ -0008a960: 7270 2c20 2d2d 2063 6f6f 7264 696e 6174  rp, -- coordinat
│ │ │ │ -0008a970: 6520 7269 6e67 206f 660a 2020 2020 7468  e ring of.    th
│ │ │ │ -0008a980: 6520 736f 7572 6365 2066 6f72 2061 2072  e source for a r
│ │ │ │ -0008a990: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ -0008a9a0: 2a6e 6f74 6520 7375 6273 7469 7475 7465  *note substitute
│ │ │ │ -0008a9b0: 2852 6174 696f 6e61 6c4d 6170 2c50 6f6c  (RationalMap,Pol
│ │ │ │ -0008a9c0: 796e 6f6d 6961 6c52 696e 672c 506f 6c79  ynomialRing,Poly
│ │ │ │ -0008a9d0: 6e6f 6d69 616c 5269 6e67 293a 0a20 2020  nomialRing):.   
│ │ │ │ -0008a9e0: 2073 7562 7374 6974 7574 655f 6c70 5261   substitute_lpRa
│ │ │ │ -0008a9f0: 7469 6f6e 616c 4d61 705f 636d 506f 6c79  tionalMap_cmPoly
│ │ │ │ -0008aa00: 6e6f 6d69 616c 5269 6e67 5f63 6d50 6f6c  nomialRing_cmPol
│ │ │ │ -0008aa10: 796e 6f6d 6961 6c52 696e 675f 7270 2c20  ynomialRing_rp, 
│ │ │ │ -0008aa20: 2d2d 0a20 2020 2073 7562 7374 6974 7574  --.    substitut
│ │ │ │ -0008aa30: 6520 7468 6520 616d 6269 656e 7420 7072  e the ambient pr
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│ │ │ │ -0008aa90: 2073 7570 6572 2852 6174 696f 6e61 6c4d   super(RationalM
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│ │ │ │ -0008ab30: 705f 7270 2c20 2d2d 2067 6574 2074 6865  p_rp, -- get the
│ │ │ │ -0008ab40: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
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│ │ │ │ -0008ad90: 6c4d 6170 2c20 4e65 7874 3a20 5261 7469  lMap, Next: Rati
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│ │ │ │ -0008adf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0008ae00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -0008ae10: 5379 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d  Synopsis.=======
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│ │ │ │ -0008af50: 2a6e 6f74 6520 446f 6d69 6e61 6e74 3a20  *note Dominant: 
│ │ │ │ -0008af60: 446f 6d69 6e61 6e74 2c20 3d3e 202e 2e2e  Dominant, => ...
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│ │ │ │ -0008afb0: 3a20 5261 7469 6f6e 616c 4d61 702c 2c20  : RationalMap,, 
│ │ │ │ -0008afc0: 7468 6520 7261 7469 6f6e 616c 206d 6170  the rational map
│ │ │ │ -0008afd0: 2072 6570 7265 7365 6e74 6564 2062 7920   represented by 
│ │ │ │ -0008afe0: 7068 690a 2020 2020 2020 2020 6f72 2062  phi.        or b
│ │ │ │ -0008aff0: 7920 7468 6520 7269 6e67 206d 6170 2064  y the ring map d
│ │ │ │ -0008b000: 6566 696e 6564 2062 7920 460a 0a44 6573  efined by F..Des
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│ │ │ │ -0008b020: 3d3d 3d3d 0a0a 5468 6973 2069 7320 7468  ====..This is th
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│ │ │ │ -0008b040: 7469 6f6e 2066 6f72 2061 202a 6e6f 7465  tion for a *note
│ │ │ │ -0008b050: 2072 6174 696f 6e61 6c20 6d61 703a 2052   rational map: R
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│ │ │ │ -0008b0a0: 2065 7863 6570 7420 7468 6174 2069 7420   except that it 
│ │ │ │ -0008b0b0: 7265 7475 726e 7320 6120 2a6e 6f74 650a  returns a *note.
│ │ │ │ -0008b0c0: 5261 7469 6f6e 616c 4d61 703a 2052 6174  RationalMap: Rat
│ │ │ │ -0008b0d0: 696f 6e61 6c4d 6170 2c20 696e 7374 6561  ionalMap, instea
│ │ │ │ -0008b0e0: 6420 6f66 2061 202a 6e6f 7465 2052 696e  d of a *note Rin
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│ │ │ │ -0008b100: 446f 6329 5269 6e67 4d61 702c 2e0a 0a2b  Doc)RingMap,...+
│ │ │ │ -0008b110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00089140: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +00089150: 6162 7374 7261 6374 5261 7469 6f6e 616c  abstractRational
│ │ │ │ +00089160: 4d61 7028 5261 7469 6f6e 616c 4d61 7029  Map(RationalMap)
│ │ │ │ +00089170: 2220 2d2d 2073 6565 202a 6e6f 7465 2061  " -- see *note a
│ │ │ │ +00089180: 6273 7472 6163 7452 6174 696f 6e61 6c4d  bstractRationalM
│ │ │ │ +00089190: 6170 3a0a 2020 2020 6162 7374 7261 6374  ap:.    abstract
│ │ │ │ +000891a0: 5261 7469 6f6e 616c 4d61 702c 202d 2d20  RationalMap, -- 
│ │ │ │ +000891b0: 6d61 6b65 2061 6e20 6162 7374 7261 6374  make an abstract
│ │ │ │ +000891c0: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ +000891d0: 2a20 2261 7070 726f 7869 6d61 7465 496e  * "approximateIn
│ │ │ │ +000891e0: 7665 7273 654d 6170 2852 6174 696f 6e61  verseMap(Rationa
│ │ │ │ +000891f0: 6c4d 6170 2922 202d 2d20 7365 6520 2a6e  lMap)" -- see *n
│ │ │ │ +00089200: 6f74 6520 6170 7072 6f78 696d 6174 6549  ote approximateI
│ │ │ │ +00089210: 6e76 6572 7365 4d61 703a 0a20 2020 2061  nverseMap:.    a
│ │ │ │ +00089220: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ +00089230: 654d 6170 2c20 2d2d 2072 616e 646f 6d20  eMap, -- random 
│ │ │ │ +00089240: 6d61 7020 7265 6c61 7465 6420 746f 2074  map related to t
│ │ │ │ +00089250: 6865 2069 6e76 6572 7365 206f 6620 6120  he inverse of a 
│ │ │ │ +00089260: 6269 7261 7469 6f6e 616c 0a20 2020 206d  birational.    m
│ │ │ │ +00089270: 6170 0a20 202a 2022 6170 7072 6f78 696d  ap.  * "approxim
│ │ │ │ +00089280: 6174 6549 6e76 6572 7365 4d61 7028 5261  ateInverseMap(Ra
│ │ │ │ +00089290: 7469 6f6e 616c 4d61 702c 5a5a 2922 202d  tionalMap,ZZ)" -
│ │ │ │ +000892a0: 2d20 7365 6520 2a6e 6f74 6520 6170 7072  - see *note appr
│ │ │ │ +000892b0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +000892c0: 703a 0a20 2020 2061 7070 726f 7869 6d61  p:.    approxima
│ │ │ │ +000892d0: 7465 496e 7665 7273 654d 6170 2c20 2d2d  teInverseMap, --
│ │ │ │ +000892e0: 2072 616e 646f 6d20 6d61 7020 7265 6c61   random map rela
│ │ │ │ +000892f0: 7465 6420 746f 2074 6865 2069 6e76 6572  ted to the inver
│ │ │ │ +00089300: 7365 206f 6620 6120 6269 7261 7469 6f6e  se of a biration
│ │ │ │ +00089310: 616c 0a20 2020 206d 6170 0a20 202a 202a  al.    map.  * *
│ │ │ │ +00089320: 6e6f 7465 2063 6f65 6666 6963 6965 6e74  note coefficient
│ │ │ │ +00089330: 5269 6e67 2852 6174 696f 6e61 6c4d 6170  Ring(RationalMap
│ │ │ │ +00089340: 293a 2063 6f65 6666 6963 6965 6e74 5269  ): coefficientRi
│ │ │ │ +00089350: 6e67 5f6c 7052 6174 696f 6e61 6c4d 6170  ng_lpRationalMap
│ │ │ │ +00089360: 5f72 702c 202d 2d0a 2020 2020 636f 6566  _rp, --.    coef
│ │ │ │ +00089370: 6669 6369 656e 7420 7269 6e67 206f 6620  ficient ring of 
│ │ │ │ +00089380: 6120 7261 7469 6f6e 616c 206d 6170 0a20  a rational map. 
│ │ │ │ +00089390: 202a 202a 6e6f 7465 2063 6f65 6666 6963   * *note coeffic
│ │ │ │ +000893a0: 6965 6e74 7328 5261 7469 6f6e 616c 4d61  ients(RationalMa
│ │ │ │ +000893b0: 7029 3a20 636f 6566 6669 6369 656e 7473  p): coefficients
│ │ │ │ +000893c0: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +000893d0: 702c 202d 2d0a 2020 2020 636f 6566 6669  p, --.    coeffi
│ │ │ │ +000893e0: 6369 656e 7420 6d61 7472 6978 206f 6620  cient matrix of 
│ │ │ │ +000893f0: 6120 7261 7469 6f6e 616c 206d 6170 0a20  a rational map. 
│ │ │ │ +00089400: 202a 202a 6e6f 7465 2064 6567 7265 6528   * *note degree(
│ │ │ │ +00089410: 5261 7469 6f6e 616c 4d61 7029 3a20 6465  RationalMap): de
│ │ │ │ +00089420: 6772 6565 5f6c 7052 6174 696f 6e61 6c4d  gree_lpRationalM
│ │ │ │ +00089430: 6170 5f72 702c 202d 2d20 6465 6772 6565  ap_rp, -- degree
│ │ │ │ +00089440: 206f 6620 6120 7261 7469 6f6e 616c 0a20   of a rational. 
│ │ │ │ +00089450: 2020 206d 6170 0a20 202a 202a 6e6f 7465     map.  * *note
│ │ │ │ +00089460: 2064 6567 7265 654d 6170 2852 6174 696f   degreeMap(Ratio
│ │ │ │ +00089470: 6e61 6c4d 6170 293a 2064 6567 7265 654d  nalMap): degreeM
│ │ │ │ +00089480: 6170 5f6c 7052 6174 696f 6e61 6c4d 6170  ap_lpRationalMap
│ │ │ │ +00089490: 5f72 702c 202d 2d20 6465 6772 6565 206f  _rp, -- degree o
│ │ │ │ +000894a0: 6620 610a 2020 2020 7261 7469 6f6e 616c  f a.    rational
│ │ │ │ +000894b0: 206d 6170 0a20 202a 202a 6e6f 7465 2064   map.  * *note d
│ │ │ │ +000894c0: 6567 7265 6573 2852 6174 696f 6e61 6c4d  egrees(RationalM
│ │ │ │ +000894d0: 6170 293a 2064 6567 7265 6573 5f6c 7052  ap): degrees_lpR
│ │ │ │ +000894e0: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +000894f0: 2d20 7072 6f6a 6563 7469 7665 2064 6567  - projective deg
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│ │ │ │ +00089510: 7469 6f6e 616c 206d 6170 0a20 202a 2022  tional map.  * "
│ │ │ │ +00089520: 6d75 6c74 6964 6567 7265 6528 5261 7469  multidegree(Rati
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│ │ │ │ +00089540: 202a 6e6f 7465 2064 6567 7265 6573 2852   *note degrees(R
│ │ │ │ +00089550: 6174 696f 6e61 6c4d 6170 293a 0a20 2020  ationalMap):.   
│ │ │ │ +00089560: 2064 6567 7265 6573 5f6c 7052 6174 696f   degrees_lpRatio
│ │ │ │ +00089570: 6e61 6c4d 6170 5f72 702c 202d 2d20 7072  nalMap_rp, -- pr
│ │ │ │ +00089580: 6f6a 6563 7469 7665 2064 6567 7265 6573  ojective degrees
│ │ │ │ +00089590: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ +000895a0: 6170 0a20 202a 202a 6e6f 7465 2064 6573  ap.  * *note des
│ │ │ │ +000895b0: 6372 6962 6528 5261 7469 6f6e 616c 4d61  cribe(RationalMa
│ │ │ │ +000895c0: 7029 3a20 6465 7363 7269 6265 5f6c 7052  p): describe_lpR
│ │ │ │ +000895d0: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +000895e0: 2d20 6465 7363 7269 6265 2061 0a20 2020  - describe a.   
│ │ │ │ +000895f0: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ +00089600: 2a20 2a6e 6f74 6520 656e 7472 6965 7328  * *note entries(
│ │ │ │ +00089610: 5261 7469 6f6e 616c 4d61 7029 3a20 656e  RationalMap): en
│ │ │ │ +00089620: 7472 6965 735f 6c70 5261 7469 6f6e 616c  tries_lpRational
│ │ │ │ +00089630: 4d61 705f 7270 2c20 2d2d 2074 6865 2065  Map_rp, -- the e
│ │ │ │ +00089640: 6e74 7269 6573 206f 6620 7468 650a 2020  ntries of the.  
│ │ │ │ +00089650: 2020 6d61 7472 6978 2061 7373 6f63 6961    matrix associa
│ │ │ │ +00089660: 7465 6420 746f 2061 2072 6174 696f 6e61  ted to a rationa
│ │ │ │ +00089670: 6c20 6d61 700a 2020 2a20 2265 7863 6570  l map.  * "excep
│ │ │ │ +00089680: 7469 6f6e 616c 4c6f 6375 7328 5261 7469  tionalLocus(Rati
│ │ │ │ +00089690: 6f6e 616c 4d61 7029 2220 2d2d 2073 6565  onalMap)" -- see
│ │ │ │ +000896a0: 202a 6e6f 7465 2065 7863 6570 7469 6f6e   *note exception
│ │ │ │ +000896b0: 616c 4c6f 6375 733a 0a20 2020 2065 7863  alLocus:.    exc
│ │ │ │ +000896c0: 6570 7469 6f6e 616c 4c6f 6375 732c 202d  eptionalLocus, -
│ │ │ │ +000896d0: 2d20 6578 6365 7074 696f 6e61 6c20 6c6f  - exceptional lo
│ │ │ │ +000896e0: 6375 7320 6f66 2061 2062 6972 6174 696f  cus of a biratio
│ │ │ │ +000896f0: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74  nal map.  * *not
│ │ │ │ +00089700: 6520 666c 6174 7465 6e28 5261 7469 6f6e  e flatten(Ration
│ │ │ │ +00089710: 616c 4d61 7029 3a20 666c 6174 7465 6e5f  alMap): flatten_
│ │ │ │ +00089720: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ +00089730: 2c20 2d2d 2077 7269 7465 2073 6f75 7263  , -- write sourc
│ │ │ │ +00089740: 6520 616e 640a 2020 2020 7461 7267 6574  e and.    target
│ │ │ │ +00089750: 2061 7320 6e6f 6e64 6567 656e 6572 6174   as nondegenerat
│ │ │ │ +00089760: 6520 7661 7269 6574 6965 730a 2020 2a20  e varieties.  * 
│ │ │ │ +00089770: 2266 6f72 6365 496d 6167 6528 5261 7469  "forceImage(Rati
│ │ │ │ +00089780: 6f6e 616c 4d61 702c 4964 6561 6c29 2220  onalMap,Ideal)" 
│ │ │ │ +00089790: 2d2d 2073 6565 202a 6e6f 7465 2066 6f72  -- see *note for
│ │ │ │ +000897a0: 6365 496d 6167 653a 2066 6f72 6365 496d  ceImage: forceIm
│ │ │ │ +000897b0: 6167 652c 202d 2d0a 2020 2020 6465 636c  age, --.    decl
│ │ │ │ +000897c0: 6172 6520 7768 6963 6820 6973 2074 6865  are which is the
│ │ │ │ +000897d0: 2069 6d61 6765 206f 6620 6120 7261 7469   image of a rati
│ │ │ │ +000897e0: 6f6e 616c 206d 6170 0a20 202a 2022 666f  onal map.  * "fo
│ │ │ │ +000897f0: 7263 6549 6e76 6572 7365 4d61 7028 5261  rceInverseMap(Ra
│ │ │ │ +00089800: 7469 6f6e 616c 4d61 702c 5261 7469 6f6e  tionalMap,Ration
│ │ │ │ +00089810: 616c 4d61 7029 2220 2d2d 2073 6565 202a  alMap)" -- see *
│ │ │ │ +00089820: 6e6f 7465 2066 6f72 6365 496e 7665 7273  note forceInvers
│ │ │ │ +00089830: 654d 6170 3a0a 2020 2020 666f 7263 6549  eMap:.    forceI
│ │ │ │ +00089840: 6e76 6572 7365 4d61 702c 202d 2d20 6465  nverseMap, -- de
│ │ │ │ +00089850: 636c 6172 6520 7468 6174 2074 776f 2072  clare that two r
│ │ │ │ +00089860: 6174 696f 6e61 6c20 6d61 7073 2061 7265  ational maps are
│ │ │ │ +00089870: 206f 6e65 2074 6865 2069 6e76 6572 7365   one the inverse
│ │ │ │ +00089880: 206f 660a 2020 2020 7468 6520 6f74 6865   of.    the othe
│ │ │ │ +00089890: 720a 2020 2a20 2267 7261 7068 2852 6174  r.  * "graph(Rat
│ │ │ │ +000898a0: 696f 6e61 6c4d 6170 2922 202d 2d20 7365  ionalMap)" -- se
│ │ │ │ +000898b0: 6520 2a6e 6f74 6520 6772 6170 683a 2067  e *note graph: g
│ │ │ │ +000898c0: 7261 7068 2c20 2d2d 2063 6c6f 7375 7265  raph, -- closure
│ │ │ │ +000898d0: 206f 6620 7468 6520 6772 6170 6820 6f66   of the graph of
│ │ │ │ +000898e0: 0a20 2020 2061 2072 6174 696f 6e61 6c20  .    a rational 
│ │ │ │ +000898f0: 6d61 700a 2020 2a20 2a6e 6f74 6520 6964  map.  * *note id
│ │ │ │ +00089900: 6561 6c28 5261 7469 6f6e 616c 4d61 7029  eal(RationalMap)
│ │ │ │ +00089910: 3a20 6964 6561 6c5f 6c70 5261 7469 6f6e  : ideal_lpRation
│ │ │ │ +00089920: 616c 4d61 705f 7270 2c20 2d2d 2062 6173  alMap_rp, -- bas
│ │ │ │ +00089930: 6520 6c6f 6375 7320 6f66 2061 0a20 2020  e locus of a.   
│ │ │ │ +00089940: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ +00089950: 2a20 2a6e 6f74 6520 696d 6167 6528 5261  * *note image(Ra
│ │ │ │ +00089960: 7469 6f6e 616c 4d61 702c 5374 7269 6e67  tionalMap,String
│ │ │ │ +00089970: 293a 2069 6d61 6765 5f6c 7052 6174 696f  ): image_lpRatio
│ │ │ │ +00089980: 6e61 6c4d 6170 5f63 6d53 7472 696e 675f  nalMap_cmString_
│ │ │ │ +00089990: 7270 2c20 2d2d 0a20 2020 2063 6c6f 7375  rp, --.    closu
│ │ │ │ +000899a0: 7265 206f 6620 7468 6520 696d 6167 6520  re of the image 
│ │ │ │ +000899b0: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ +000899c0: 7020 7573 696e 6720 7468 6520 4634 2061  p using the F4 a
│ │ │ │ +000899d0: 6c67 6f72 6974 686d 0a20 2020 2028 6578  lgorithm.    (ex
│ │ │ │ +000899e0: 7065 7269 6d65 6e74 616c 290a 2020 2a20  perimental).  * 
│ │ │ │ +000899f0: 2269 6d61 6765 2852 6174 696f 6e61 6c4d  "image(RationalM
│ │ │ │ +00089a00: 6170 2922 202d 2d20 7365 6520 2a6e 6f74  ap)" -- see *not
│ │ │ │ +00089a10: 6520 696d 6167 6528 5261 7469 6f6e 616c  e image(Rational
│ │ │ │ +00089a20: 4d61 702c 5a5a 293a 0a20 2020 2069 6d61  Map,ZZ):.    ima
│ │ │ │ +00089a30: 6765 5f6c 7052 6174 696f 6e61 6c4d 6170  ge_lpRationalMap
│ │ │ │ +00089a40: 5f63 6d5a 5a5f 7270 2c20 2d2d 2063 6c6f  _cmZZ_rp, -- clo
│ │ │ │ +00089a50: 7375 7265 206f 6620 7468 6520 696d 6167  sure of the imag
│ │ │ │ +00089a60: 6520 6f66 2061 2072 6174 696f 6e61 6c20  e of a rational 
│ │ │ │ +00089a70: 6d61 700a 2020 2a20 2a6e 6f74 6520 696d  map.  * *note im
│ │ │ │ +00089a80: 6167 6528 5261 7469 6f6e 616c 4d61 702c  age(RationalMap,
│ │ │ │ +00089a90: 5a5a 293a 2069 6d61 6765 5f6c 7052 6174  ZZ): image_lpRat
│ │ │ │ +00089aa0: 696f 6e61 6c4d 6170 5f63 6d5a 5a5f 7270  ionalMap_cmZZ_rp
│ │ │ │ +00089ab0: 2c20 2d2d 2063 6c6f 7375 7265 206f 6620  , -- closure of 
│ │ │ │ +00089ac0: 7468 650a 2020 2020 696d 6167 6520 6f66  the.    image of
│ │ │ │ +00089ad0: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +00089ae0: 2020 2a20 2a6e 6f74 6520 696e 7665 7273    * *note invers
│ │ │ │ +00089af0: 6528 5261 7469 6f6e 616c 4d61 7029 3a20  e(RationalMap): 
│ │ │ │ +00089b00: 696e 7665 7273 655f 6c70 5261 7469 6f6e  inverse_lpRation
│ │ │ │ +00089b10: 616c 4d61 705f 7270 2c20 2d2d 2069 6e76  alMap_rp, -- inv
│ │ │ │ +00089b20: 6572 7365 206f 6620 610a 2020 2020 6269  erse of a.    bi
│ │ │ │ +00089b30: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ +00089b40: 2022 696e 7665 7273 6528 5261 7469 6f6e   "inverse(Ration
│ │ │ │ +00089b50: 616c 4d61 702c 4f70 7469 6f6e 2922 202d  alMap,Option)" -
│ │ │ │ +00089b60: 2d20 7365 6520 2a6e 6f74 6520 696e 7665  - see *note inve
│ │ │ │ +00089b70: 7273 6528 5261 7469 6f6e 616c 4d61 7029  rse(RationalMap)
│ │ │ │ +00089b80: 3a0a 2020 2020 696e 7665 7273 655f 6c70  :.    inverse_lp
│ │ │ │ +00089b90: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ +00089ba0: 2d2d 2069 6e76 6572 7365 206f 6620 6120  -- inverse of a 
│ │ │ │ +00089bb0: 6269 7261 7469 6f6e 616c 206d 6170 0a20  birational map. 
│ │ │ │ +00089bc0: 202a 2022 696e 7665 7273 654d 6170 2852   * "inverseMap(R
│ │ │ │ +00089bd0: 6174 696f 6e61 6c4d 6170 2922 202d 2d20  ationalMap)" -- 
│ │ │ │ +00089be0: 7365 6520 2a6e 6f74 6520 696e 7665 7273  see *note invers
│ │ │ │ +00089bf0: 654d 6170 3a20 696e 7665 7273 654d 6170  eMap: inverseMap
│ │ │ │ +00089c00: 2c20 2d2d 2069 6e76 6572 7365 0a20 2020  , -- inverse.   
│ │ │ │ +00089c10: 206f 6620 6120 6269 7261 7469 6f6e 616c   of a birational
│ │ │ │ +00089c20: 206d 6170 0a20 202a 2022 6973 4269 7261   map.  * "isBira
│ │ │ │ +00089c30: 7469 6f6e 616c 2852 6174 696f 6e61 6c4d  tional(RationalM
│ │ │ │ +00089c40: 6170 2922 202d 2d20 7365 6520 2a6e 6f74  ap)" -- see *not
│ │ │ │ +00089c50: 6520 6973 4269 7261 7469 6f6e 616c 3a20  e isBirational: 
│ │ │ │ +00089c60: 6973 4269 7261 7469 6f6e 616c 2c20 2d2d  isBirational, --
│ │ │ │ +00089c70: 0a20 2020 2077 6865 7468 6572 2061 2072  .    whether a r
│ │ │ │ +00089c80: 6174 696f 6e61 6c20 6d61 7020 6973 2062  ational map is b
│ │ │ │ +00089c90: 6972 6174 696f 6e61 6c0a 2020 2a20 2269  irational.  * "i
│ │ │ │ +00089ca0: 7344 6f6d 696e 616e 7428 5261 7469 6f6e  sDominant(Ration
│ │ │ │ +00089cb0: 616c 4d61 7029 2220 2d2d 2073 6565 202a  alMap)" -- see *
│ │ │ │ +00089cc0: 6e6f 7465 2069 7344 6f6d 696e 616e 743a  note isDominant:
│ │ │ │ +00089cd0: 2069 7344 6f6d 696e 616e 742c 202d 2d20   isDominant, -- 
│ │ │ │ +00089ce0: 7768 6574 6865 7220 610a 2020 2020 7261  whether a.    ra
│ │ │ │ +00089cf0: 7469 6f6e 616c 206d 6170 2069 7320 646f  tional map is do
│ │ │ │ +00089d00: 6d69 6e61 6e74 0a20 202a 202a 6e6f 7465  minant.  * *note
│ │ │ │ +00089d10: 2069 7349 6e76 6572 7365 4d61 7028 5261   isInverseMap(Ra
│ │ │ │ +00089d20: 7469 6f6e 616c 4d61 702c 5261 7469 6f6e  tionalMap,Ration
│ │ │ │ +00089d30: 616c 4d61 7029 3a0a 2020 2020 6973 496e  alMap):.    isIn
│ │ │ │ +00089d40: 7665 7273 654d 6170 5f6c 7052 6174 696f  verseMap_lpRatio
│ │ │ │ +00089d50: 6e61 6c4d 6170 5f63 6d52 6174 696f 6e61  nalMap_cmRationa
│ │ │ │ +00089d60: 6c4d 6170 5f72 702c 202d 2d20 6368 6563  lMap_rp, -- chec
│ │ │ │ +00089d70: 6b73 2077 6865 7468 6572 2074 776f 2072  ks whether two r
│ │ │ │ +00089d80: 6174 696f 6e61 6c0a 2020 2020 6d61 7073  ational.    maps
│ │ │ │ +00089d90: 2061 7265 206f 6e65 2074 6865 2069 6e76   are one the inv
│ │ │ │ +00089da0: 6572 7365 206f 6620 7468 6520 6f74 6865  erse of the othe
│ │ │ │ +00089db0: 720a 2020 2a20 2a6e 6f74 6520 6973 4973  r.  * *note isIs
│ │ │ │ +00089dc0: 6f6d 6f72 7068 6973 6d28 5261 7469 6f6e  omorphism(Ration
│ │ │ │ +00089dd0: 616c 4d61 7029 3a20 6973 4973 6f6d 6f72  alMap): isIsomor
│ │ │ │ +00089de0: 7068 6973 6d5f 6c70 5261 7469 6f6e 616c  phism_lpRational
│ │ │ │ +00089df0: 4d61 705f 7270 2c20 2d2d 0a20 2020 2077  Map_rp, --.    w
│ │ │ │ +00089e00: 6865 7468 6572 2061 2062 6972 6174 696f  hether a biratio
│ │ │ │ +00089e10: 6e61 6c20 6d61 7020 6973 2061 6e20 6973  nal map is an is
│ │ │ │ +00089e20: 6f6d 6f72 7068 6973 6d0a 2020 2a20 2269  omorphism.  * "i
│ │ │ │ +00089e30: 734d 6f72 7068 6973 6d28 5261 7469 6f6e  sMorphism(Ration
│ │ │ │ +00089e40: 616c 4d61 7029 2220 2d2d 2073 6565 202a  alMap)" -- see *
│ │ │ │ +00089e50: 6e6f 7465 2069 734d 6f72 7068 6973 6d3a  note isMorphism:
│ │ │ │ +00089e60: 2069 734d 6f72 7068 6973 6d2c 202d 2d20   isMorphism, -- 
│ │ │ │ +00089e70: 7768 6574 6865 7220 610a 2020 2020 7261  whether a.    ra
│ │ │ │ +00089e80: 7469 6f6e 616c 206d 6170 2069 7320 6120  tional map is a 
│ │ │ │ +00089e90: 6d6f 7270 6869 736d 0a20 202a 202a 6e6f  morphism.  * *no
│ │ │ │ +00089ea0: 7465 206d 6170 2852 6174 696f 6e61 6c4d  te map(RationalM
│ │ │ │ +00089eb0: 6170 293a 206d 6170 5f6c 7052 6174 696f  ap): map_lpRatio
│ │ │ │ +00089ec0: 6e61 6c4d 6170 5f72 702c 202d 2d20 6765  nalMap_rp, -- ge
│ │ │ │ +00089ed0: 7420 7468 6520 7269 6e67 206d 6170 2064  t the ring map d
│ │ │ │ +00089ee0: 6566 696e 696e 670a 2020 2020 6120 7261  efining.    a ra
│ │ │ │ +00089ef0: 7469 6f6e 616c 206d 6170 0a20 202a 2022  tional map.  * "
│ │ │ │ +00089f00: 6d61 7028 5a5a 2c52 6174 696f 6e61 6c4d  map(ZZ,RationalM
│ │ │ │ +00089f10: 6170 2922 202d 2d20 7365 6520 2a6e 6f74  ap)" -- see *not
│ │ │ │ +00089f20: 6520 6d61 7028 5261 7469 6f6e 616c 4d61  e map(RationalMa
│ │ │ │ +00089f30: 7029 3a20 6d61 705f 6c70 5261 7469 6f6e  p): map_lpRation
│ │ │ │ +00089f40: 616c 4d61 705f 7270 2c0a 2020 2020 2d2d  alMap_rp,.    --
│ │ │ │ +00089f50: 2067 6574 2074 6865 2072 696e 6720 6d61   get the ring ma
│ │ │ │ +00089f60: 7020 6465 6669 6e69 6e67 2061 2072 6174  p defining a rat
│ │ │ │ +00089f70: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ +00089f80: 6f74 6520 6d61 7472 6978 2852 6174 696f  ote matrix(Ratio
│ │ │ │ +00089f90: 6e61 6c4d 6170 293a 206d 6174 7269 785f  nalMap): matrix_
│ │ │ │ +00089fa0: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ +00089fb0: 2c20 2d2d 2074 6865 206d 6174 7269 780a  , -- the matrix.
│ │ │ │ +00089fc0: 2020 2020 6173 736f 6369 6174 6564 2074      associated t
│ │ │ │ +00089fd0: 6f20 6120 7261 7469 6f6e 616c 206d 6170  o a rational map
│ │ │ │ +00089fe0: 0a20 202a 2022 6d61 7472 6978 285a 5a2c  .  * "matrix(ZZ,
│ │ │ │ +00089ff0: 5261 7469 6f6e 616c 4d61 7029 2220 2d2d  RationalMap)" --
│ │ │ │ +0008a000: 2073 6565 202a 6e6f 7465 206d 6174 7269   see *note matri
│ │ │ │ +0008a010: 7828 5261 7469 6f6e 616c 4d61 7029 3a0a  x(RationalMap):.
│ │ │ │ +0008a020: 2020 2020 6d61 7472 6978 5f6c 7052 6174      matrix_lpRat
│ │ │ │ +0008a030: 696f 6e61 6c4d 6170 5f72 702c 202d 2d20  ionalMap_rp, -- 
│ │ │ │ +0008a040: 7468 6520 6d61 7472 6978 2061 7373 6f63  the matrix assoc
│ │ │ │ +0008a050: 6961 7465 6420 746f 2061 2072 6174 696f  iated to a ratio
│ │ │ │ +0008a060: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74  nal map.  * *not
│ │ │ │ +0008a070: 6520 7072 6f6a 6563 7469 7665 4465 6772  e projectiveDegr
│ │ │ │ +0008a080: 6565 7328 5261 7469 6f6e 616c 4d61 7029  ees(RationalMap)
│ │ │ │ +0008a090: 3a20 7072 6f6a 6563 7469 7665 4465 6772  : projectiveDegr
│ │ │ │ +0008a0a0: 6565 735f 6c70 5261 7469 6f6e 616c 4d61  ees_lpRationalMa
│ │ │ │ +0008a0b0: 705f 7270 2c0a 2020 2020 2d2d 2070 726f  p_rp,.    -- pro
│ │ │ │ +0008a0c0: 6a65 6374 6976 6520 6465 6772 6565 7320  jective degrees 
│ │ │ │ +0008a0d0: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ +0008a0e0: 700a 2020 2a20 2a6e 6f74 6520 5261 7469  p.  * *note Rati
│ │ │ │ +0008a0f0: 6f6e 616c 4d61 7020 213a 2052 6174 696f  onalMap !: Ratio
│ │ │ │ +0008a100: 6e61 6c4d 6170 2021 2c20 2d2d 2063 616c  nalMap !, -- cal
│ │ │ │ +0008a110: 6375 6c61 7465 7320 6576 6572 7920 706f  culates every po
│ │ │ │ +0008a120: 7373 6962 6c65 2074 6869 6e67 0a20 202a  ssible thing.  *
│ │ │ │ +0008a130: 2022 636f 6d70 6f73 6528 5261 7469 6f6e   "compose(Ration
│ │ │ │ +0008a140: 616c 4d61 702c 5261 7469 6f6e 616c 4d61  alMap,RationalMa
│ │ │ │ +0008a150: 7029 2220 2d2d 2073 6565 202a 6e6f 7465  p)" -- see *note
│ │ │ │ +0008a160: 2052 6174 696f 6e61 6c4d 6170 202a 2052   RationalMap * R
│ │ │ │ +0008a170: 6174 696f 6e61 6c4d 6170 3a0a 2020 2020  ationalMap:.    
│ │ │ │ +0008a180: 5261 7469 6f6e 616c 4d61 7020 5f73 7420  RationalMap _st 
│ │ │ │ +0008a190: 5261 7469 6f6e 616c 4d61 702c 202d 2d20  RationalMap, -- 
│ │ │ │ +0008a1a0: 636f 6d70 6f73 6974 696f 6e20 6f66 2072  composition of r
│ │ │ │ +0008a1b0: 6174 696f 6e61 6c20 6d61 7073 0a20 202a  ational maps.  *
│ │ │ │ +0008a1c0: 202a 6e6f 7465 2052 6174 696f 6e61 6c4d   *note RationalM
│ │ │ │ +0008a1d0: 6170 202a 2052 6174 696f 6e61 6c4d 6170  ap * RationalMap
│ │ │ │ +0008a1e0: 3a20 5261 7469 6f6e 616c 4d61 7020 5f73  : RationalMap _s
│ │ │ │ +0008a1f0: 7420 5261 7469 6f6e 616c 4d61 702c 202d  t RationalMap, -
│ │ │ │ +0008a200: 2d0a 2020 2020 636f 6d70 6f73 6974 696f  -.    compositio
│ │ │ │ +0008a210: 6e20 6f66 2072 6174 696f 6e61 6c20 6d61  n of rational ma
│ │ │ │ +0008a220: 7073 0a20 202a 202a 6e6f 7465 2052 6174  ps.  * *note Rat
│ │ │ │ +0008a230: 696f 6e61 6c4d 6170 202a 2a20 5269 6e67  ionalMap ** Ring
│ │ │ │ +0008a240: 3a20 5261 7469 6f6e 616c 4d61 7020 5f73  : RationalMap _s
│ │ │ │ +0008a250: 745f 7374 2052 696e 672c 202d 2d20 6368  t_st Ring, -- ch
│ │ │ │ +0008a260: 616e 6765 2074 6865 0a20 2020 2063 6f65  ange the.    coe
│ │ │ │ +0008a270: 6666 6963 6965 6e74 2072 696e 6720 6f66  fficient ring of
│ │ │ │ +0008a280: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +0008a290: 2020 2a20 2a6e 6f74 6520 5261 7469 6f6e    * *note Ration
│ │ │ │ +0008a2a0: 616c 4d61 7020 3d3d 2052 6174 696f 6e61  alMap == Rationa
│ │ │ │ +0008a2b0: 6c4d 6170 3a20 5261 7469 6f6e 616c 4d61  lMap: RationalMa
│ │ │ │ +0008a2c0: 7020 3d3d 2052 6174 696f 6e61 6c4d 6170  p == RationalMap
│ │ │ │ +0008a2d0: 2c20 2d2d 2065 7175 616c 6974 790a 2020  , -- equality.  
│ │ │ │ +0008a2e0: 2020 6f66 2072 6174 696f 6e61 6c20 6d61    of rational ma
│ │ │ │ +0008a2f0: 7073 0a20 202a 2022 5261 7469 6f6e 616c  ps.  * "Rational
│ │ │ │ +0008a300: 4d61 7020 3d3d 205a 5a22 202d 2d20 7365  Map == ZZ" -- se
│ │ │ │ +0008a310: 6520 2a6e 6f74 6520 5261 7469 6f6e 616c  e *note Rational
│ │ │ │ +0008a320: 4d61 7020 3d3d 2052 6174 696f 6e61 6c4d  Map == RationalM
│ │ │ │ +0008a330: 6170 3a20 5261 7469 6f6e 616c 4d61 7020  ap: RationalMap 
│ │ │ │ +0008a340: 3d3d 0a20 2020 2052 6174 696f 6e61 6c4d  ==.    RationalM
│ │ │ │ +0008a350: 6170 2c20 2d2d 2065 7175 616c 6974 7920  ap, -- equality 
│ │ │ │ +0008a360: 6f66 2072 6174 696f 6e61 6c20 6d61 7073  of rational maps
│ │ │ │ +0008a370: 0a20 202a 2022 5a5a 203d 3d20 5261 7469  .  * "ZZ == Rati
│ │ │ │ +0008a380: 6f6e 616c 4d61 7022 202d 2d20 7365 6520  onalMap" -- see 
│ │ │ │ +0008a390: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ +0008a3a0: 7020 3d3d 2052 6174 696f 6e61 6c4d 6170  p == RationalMap
│ │ │ │ +0008a3b0: 3a20 5261 7469 6f6e 616c 4d61 7020 3d3d  : RationalMap ==
│ │ │ │ +0008a3c0: 0a20 2020 2052 6174 696f 6e61 6c4d 6170  .    RationalMap
│ │ │ │ +0008a3d0: 2c20 2d2d 2065 7175 616c 6974 7920 6f66  , -- equality of
│ │ │ │ +0008a3e0: 2072 6174 696f 6e61 6c20 6d61 7073 0a20   rational maps. 
│ │ │ │ +0008a3f0: 202a 202a 6e6f 7465 2052 6174 696f 6e61   * *note Rationa
│ │ │ │ +0008a400: 6c4d 6170 205e 205a 5a3a 2052 6174 696f  lMap ^ ZZ: Ratio
│ │ │ │ +0008a410: 6e61 6c4d 6170 205e 205a 5a2c 202d 2d20  nalMap ^ ZZ, -- 
│ │ │ │ +0008a420: 706f 7765 720a 2020 2a20 2252 6174 696f  power.  * "Ratio
│ │ │ │ +0008a430: 6e61 6c4d 6170 205e 2a22 202d 2d20 7365  nalMap ^*" -- se
│ │ │ │ +0008a440: 6520 2a6e 6f74 6520 5261 7469 6f6e 616c  e *note Rational
│ │ │ │ +0008a450: 4d61 7020 5e2a 2a20 4964 6561 6c3a 2052  Map ^** Ideal: R
│ │ │ │ +0008a460: 6174 696f 6e61 6c4d 6170 205e 5f73 745f  ationalMap ^_st_
│ │ │ │ +0008a470: 7374 0a20 2020 2049 6465 616c 2c20 2d2d  st.    Ideal, --
│ │ │ │ +0008a480: 2069 6e76 6572 7365 2069 6d61 6765 2076   inverse image v
│ │ │ │ +0008a490: 6961 2061 2072 6174 696f 6e61 6c20 6d61  ia a rational ma
│ │ │ │ +0008a4a0: 700a 2020 2a20 2a6e 6f74 6520 5261 7469  p.  * *note Rati
│ │ │ │ +0008a4b0: 6f6e 616c 4d61 7020 5e2a 2a20 4964 6561  onalMap ^** Idea
│ │ │ │ +0008a4c0: 6c3a 2052 6174 696f 6e61 6c4d 6170 205e  l: RationalMap ^
│ │ │ │ +0008a4d0: 5f73 745f 7374 2049 6465 616c 2c20 2d2d  _st_st Ideal, --
│ │ │ │ +0008a4e0: 2069 6e76 6572 7365 2069 6d61 6765 0a20   inverse image. 
│ │ │ │ +0008a4f0: 2020 2076 6961 2061 2072 6174 696f 6e61     via a rationa
│ │ │ │ +0008a500: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ +0008a510: 5261 7469 6f6e 616c 4d61 7020 5f2a 3a20  RationalMap _*: 
│ │ │ │ +0008a520: 5261 7469 6f6e 616c 4d61 7020 5f75 735f  RationalMap _us_
│ │ │ │ +0008a530: 7374 2c20 2d2d 2064 6972 6563 7420 696d  st, -- direct im
│ │ │ │ +0008a540: 6167 6520 7669 6120 6120 7261 7469 6f6e  age via a ration
│ │ │ │ +0008a550: 616c 0a20 2020 206d 6170 0a20 202a 2022  al.    map.  * "
│ │ │ │ +0008a560: 5261 7469 6f6e 616c 4d61 7020 4964 6561  RationalMap Idea
│ │ │ │ +0008a570: 6c22 202d 2d20 7365 6520 2a6e 6f74 6520  l" -- see *note 
│ │ │ │ +0008a580: 5261 7469 6f6e 616c 4d61 7020 5f2a 3a20  RationalMap _*: 
│ │ │ │ +0008a590: 5261 7469 6f6e 616c 4d61 7020 5f75 735f  RationalMap _us_
│ │ │ │ +0008a5a0: 7374 2c20 2d2d 0a20 2020 2064 6972 6563  st, --.    direc
│ │ │ │ +0008a5b0: 7420 696d 6167 6520 7669 6120 6120 7261  t image via a ra
│ │ │ │ +0008a5c0: 7469 6f6e 616c 206d 6170 0a20 202a 202a  tional map.  * *
│ │ │ │ +0008a5d0: 6e6f 7465 2052 6174 696f 6e61 6c4d 6170  note RationalMap
│ │ │ │ +0008a5e0: 207c 2049 6465 616c 3a20 5261 7469 6f6e   | Ideal: Ration
│ │ │ │ +0008a5f0: 616c 4d61 7020 7c20 4964 6561 6c2c 202d  alMap | Ideal, -
│ │ │ │ +0008a600: 2d20 7265 7374 7269 6374 696f 6e20 6f66  - restriction of
│ │ │ │ +0008a610: 2061 0a20 2020 2072 6174 696f 6e61 6c20   a.    rational 
│ │ │ │ +0008a620: 6d61 700a 2020 2a20 2252 6174 696f 6e61  map.  * "Rationa
│ │ │ │ +0008a630: 6c4d 6170 207c 2052 696e 6722 202d 2d20  lMap | Ring" -- 
│ │ │ │ +0008a640: 7365 6520 2a6e 6f74 6520 5261 7469 6f6e  see *note Ration
│ │ │ │ +0008a650: 616c 4d61 7020 7c20 4964 6561 6c3a 2052  alMap | Ideal: R
│ │ │ │ +0008a660: 6174 696f 6e61 6c4d 6170 207c 2049 6465  ationalMap | Ide
│ │ │ │ +0008a670: 616c 2c0a 2020 2020 2d2d 2072 6573 7472  al,.    -- restr
│ │ │ │ +0008a680: 6963 7469 6f6e 206f 6620 6120 7261 7469  iction of a rati
│ │ │ │ +0008a690: 6f6e 616c 206d 6170 0a20 202a 2022 5261  onal map.  * "Ra
│ │ │ │ +0008a6a0: 7469 6f6e 616c 4d61 7020 7c20 5269 6e67  tionalMap | Ring
│ │ │ │ +0008a6b0: 456c 656d 656e 7422 202d 2d20 7365 6520  Element" -- see 
│ │ │ │ +0008a6c0: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ +0008a6d0: 7020 7c20 4964 6561 6c3a 2052 6174 696f  p | Ideal: Ratio
│ │ │ │ +0008a6e0: 6e61 6c4d 6170 207c 0a20 2020 2049 6465  nalMap |.    Ide
│ │ │ │ +0008a6f0: 616c 2c20 2d2d 2072 6573 7472 6963 7469  al, -- restricti
│ │ │ │ +0008a700: 6f6e 206f 6620 6120 7261 7469 6f6e 616c  on of a rational
│ │ │ │ +0008a710: 206d 6170 0a20 202a 202a 6e6f 7465 2052   map.  * *note R
│ │ │ │ +0008a720: 6174 696f 6e61 6c4d 6170 207c 7c20 4964  ationalMap || Id
│ │ │ │ +0008a730: 6561 6c3a 2052 6174 696f 6e61 6c4d 6170  eal: RationalMap
│ │ │ │ +0008a740: 207c 7c20 4964 6561 6c2c 202d 2d20 7265   || Ideal, -- re
│ │ │ │ +0008a750: 7374 7269 6374 696f 6e20 6f66 2061 0a20  striction of a. 
│ │ │ │ +0008a760: 2020 2072 6174 696f 6e61 6c20 6d61 700a     rational map.
│ │ │ │ +0008a770: 2020 2a20 2252 6174 696f 6e61 6c4d 6170    * "RationalMap
│ │ │ │ +0008a780: 207c 7c20 5269 6e67 2220 2d2d 2073 6565   || Ring" -- see
│ │ │ │ +0008a790: 202a 6e6f 7465 2052 6174 696f 6e61 6c4d   *note RationalM
│ │ │ │ +0008a7a0: 6170 207c 7c20 4964 6561 6c3a 2052 6174  ap || Ideal: Rat
│ │ │ │ +0008a7b0: 696f 6e61 6c4d 6170 207c 7c0a 2020 2020  ionalMap ||.    
│ │ │ │ +0008a7c0: 4964 6561 6c2c 202d 2d20 7265 7374 7269  Ideal, -- restri
│ │ │ │ +0008a7d0: 6374 696f 6e20 6f66 2061 2072 6174 696f  ction of a ratio
│ │ │ │ +0008a7e0: 6e61 6c20 6d61 700a 2020 2a20 2252 6174  nal map.  * "Rat
│ │ │ │ +0008a7f0: 696f 6e61 6c4d 6170 207c 7c20 5269 6e67  ionalMap || Ring
│ │ │ │ +0008a800: 456c 656d 656e 7422 202d 2d20 7365 6520  Element" -- see 
│ │ │ │ +0008a810: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ +0008a820: 7020 7c7c 2049 6465 616c 3a20 5261 7469  p || Ideal: Rati
│ │ │ │ +0008a830: 6f6e 616c 4d61 700a 2020 2020 7c7c 2049  onalMap.    || I
│ │ │ │ +0008a840: 6465 616c 2c20 2d2d 2072 6573 7472 6963  deal, -- restric
│ │ │ │ +0008a850: 7469 6f6e 206f 6620 6120 7261 7469 6f6e  tion of a ration
│ │ │ │ +0008a860: 616c 206d 6170 0a20 202a 2022 7365 6772  al map.  * "segr
│ │ │ │ +0008a870: 6528 5261 7469 6f6e 616c 4d61 7029 2220  e(RationalMap)" 
│ │ │ │ +0008a880: 2d2d 2073 6565 202a 6e6f 7465 2073 6567  -- see *note seg
│ │ │ │ +0008a890: 7265 3a20 7365 6772 652c 202d 2d20 5365  re: segre, -- Se
│ │ │ │ +0008a8a0: 6772 6520 656d 6265 6464 696e 670a 2020  gre embedding.  
│ │ │ │ +0008a8b0: 2a20 2253 6567 7265 436c 6173 7328 5261  * "SegreClass(Ra
│ │ │ │ +0008a8c0: 7469 6f6e 616c 4d61 7029 2220 2d2d 2073  tionalMap)" -- s
│ │ │ │ +0008a8d0: 6565 202a 6e6f 7465 2053 6567 7265 436c  ee *note SegreCl
│ │ │ │ +0008a8e0: 6173 733a 2053 6567 7265 436c 6173 732c  ass: SegreClass,
│ │ │ │ +0008a8f0: 202d 2d20 5365 6772 650a 2020 2020 636c   -- Segre.    cl
│ │ │ │ +0008a900: 6173 7320 6f66 2061 2063 6c6f 7365 6420  ass of a closed 
│ │ │ │ +0008a910: 7375 6273 6368 656d 6520 6f66 2061 2070  subscheme of a p
│ │ │ │ +0008a920: 726f 6a65 6374 6976 6520 7661 7269 6574  rojective variet
│ │ │ │ +0008a930: 790a 2020 2a20 2a6e 6f74 6520 736f 7572  y.  * *note sour
│ │ │ │ +0008a940: 6365 2852 6174 696f 6e61 6c4d 6170 293a  ce(RationalMap):
│ │ │ │ +0008a950: 2073 6f75 7263 655f 6c70 5261 7469 6f6e   source_lpRation
│ │ │ │ +0008a960: 616c 4d61 705f 7270 2c20 2d2d 2063 6f6f  alMap_rp, -- coo
│ │ │ │ +0008a970: 7264 696e 6174 6520 7269 6e67 206f 660a  rdinate ring of.
│ │ │ │ +0008a980: 2020 2020 7468 6520 736f 7572 6365 2066      the source f
│ │ │ │ +0008a990: 6f72 2061 2072 6174 696f 6e61 6c20 6d61  or a rational ma
│ │ │ │ +0008a9a0: 700a 2020 2a20 2a6e 6f74 6520 7375 6273  p.  * *note subs
│ │ │ │ +0008a9b0: 7469 7475 7465 2852 6174 696f 6e61 6c4d  titute(RationalM
│ │ │ │ +0008a9c0: 6170 2c50 6f6c 796e 6f6d 6961 6c52 696e  ap,PolynomialRin
│ │ │ │ +0008a9d0: 672c 506f 6c79 6e6f 6d69 616c 5269 6e67  g,PolynomialRing
│ │ │ │ +0008a9e0: 293a 0a20 2020 2073 7562 7374 6974 7574  ):.    substitut
│ │ │ │ +0008a9f0: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +0008aa00: 636d 506f 6c79 6e6f 6d69 616c 5269 6e67  cmPolynomialRing
│ │ │ │ +0008aa10: 5f63 6d50 6f6c 796e 6f6d 6961 6c52 696e  _cmPolynomialRin
│ │ │ │ +0008aa20: 675f 7270 2c20 2d2d 0a20 2020 2073 7562  g_rp, --.    sub
│ │ │ │ +0008aa30: 7374 6974 7574 6520 7468 6520 616d 6269  stitute the ambi
│ │ │ │ +0008aa40: 656e 7420 7072 6f6a 6563 7469 7665 2073  ent projective s
│ │ │ │ +0008aa50: 7061 6365 7320 6f66 2073 6f75 7263 6520  paces of source 
│ │ │ │ +0008aa60: 616e 6420 7461 7267 6574 0a20 202a 2022  and target.  * "
│ │ │ │ +0008aa70: 7261 7469 6f6e 616c 4d61 7028 5261 7469  rationalMap(Rati
│ │ │ │ +0008aa80: 6f6e 616c 4d61 7029 2220 2d2d 2073 6565  onalMap)" -- see
│ │ │ │ +0008aa90: 202a 6e6f 7465 2073 7570 6572 2852 6174   *note super(Rat
│ │ │ │ +0008aaa0: 696f 6e61 6c4d 6170 293a 0a20 2020 2073  ionalMap):.    s
│ │ │ │ +0008aab0: 7570 6572 5f6c 7052 6174 696f 6e61 6c4d  uper_lpRationalM
│ │ │ │ +0008aac0: 6170 5f72 702c 202d 2d20 6765 7420 7468  ap_rp, -- get th
│ │ │ │ +0008aad0: 6520 7261 7469 6f6e 616c 206d 6170 2077  e rational map w
│ │ │ │ +0008aae0: 686f 7365 2074 6172 6765 7420 6973 2061  hose target is a
│ │ │ │ +0008aaf0: 0a20 2020 2070 726f 6a65 6374 6976 6520  .    projective 
│ │ │ │ +0008ab00: 7370 6163 650a 2020 2a20 2a6e 6f74 6520  space.  * *note 
│ │ │ │ +0008ab10: 7375 7065 7228 5261 7469 6f6e 616c 4d61  super(RationalMa
│ │ │ │ +0008ab20: 7029 3a20 7375 7065 725f 6c70 5261 7469  p): super_lpRati
│ │ │ │ +0008ab30: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2067  onalMap_rp, -- g
│ │ │ │ +0008ab40: 6574 2074 6865 2072 6174 696f 6e61 6c20  et the rational 
│ │ │ │ +0008ab50: 6d61 700a 2020 2020 7768 6f73 6520 7461  map.    whose ta
│ │ │ │ +0008ab60: 7267 6574 2069 7320 6120 7072 6f6a 6563  rget is a projec
│ │ │ │ +0008ab70: 7469 7665 2073 7061 6365 0a20 202a 202a  tive space.  * *
│ │ │ │ +0008ab80: 6e6f 7465 2074 6172 6765 7428 5261 7469  note target(Rati
│ │ │ │ +0008ab90: 6f6e 616c 4d61 7029 3a20 7461 7267 6574  onalMap): target
│ │ │ │ +0008aba0: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +0008abb0: 702c 202d 2d20 636f 6f72 6469 6e61 7465  p, -- coordinate
│ │ │ │ +0008abc0: 2072 696e 6720 6f66 0a20 2020 2074 6865   ring of.    the
│ │ │ │ +0008abd0: 2074 6172 6765 7420 666f 7220 6120 7261   target for a ra
│ │ │ │ +0008abe0: 7469 6f6e 616c 206d 6170 0a20 202a 202a  tional map.  * *
│ │ │ │ +0008abf0: 6e6f 7465 2074 6f45 7874 6572 6e61 6c53  note toExternalS
│ │ │ │ +0008ac00: 7472 696e 6728 5261 7469 6f6e 616c 4d61  tring(RationalMa
│ │ │ │ +0008ac10: 7029 3a20 746f 4578 7465 726e 616c 5374  p): toExternalSt
│ │ │ │ +0008ac20: 7269 6e67 5f6c 7052 6174 696f 6e61 6c4d  ring_lpRationalM
│ │ │ │ +0008ac30: 6170 5f72 702c 202d 2d0a 2020 2020 636f  ap_rp, --.    co
│ │ │ │ +0008ac40: 6e76 6572 7420 746f 2061 2072 6561 6461  nvert to a reada
│ │ │ │ +0008ac50: 626c 6520 7374 7269 6e67 0a0a 466f 7220  ble string..For 
│ │ │ │ +0008ac60: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +0008ac70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0008ac80: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +0008ac90: 6f74 6520 5261 7469 6f6e 616c 4d61 703a  ote RationalMap:
│ │ │ │ +0008aca0: 2052 6174 696f 6e61 6c4d 6170 2c20 6973   RationalMap, is
│ │ │ │ +0008acb0: 2061 202a 6e6f 7465 2074 7970 653a 0a28   a *note type:.(
│ │ │ │ +0008acc0: 4d61 6361 756c 6179 3244 6f63 2954 7970  Macaulay2Doc)Typ
│ │ │ │ +0008acd0: 652c 2c20 7769 7468 2061 6e63 6573 746f  e,, with ancesto
│ │ │ │ +0008ace0: 7220 636c 6173 7365 7320 2a6e 6f74 6520  r classes *note 
│ │ │ │ +0008acf0: 4d75 7461 626c 6548 6173 6854 6162 6c65  MutableHashTable
│ │ │ │ +0008ad00: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ +0008ad10: 4d75 7461 626c 6548 6173 6854 6162 6c65  MutableHashTable
│ │ │ │ +0008ad20: 2c20 3c20 2a6e 6f74 6520 4861 7368 5461  , < *note HashTa
│ │ │ │ +0008ad30: 626c 653a 2028 4d61 6361 756c 6179 3244  ble: (Macaulay2D
│ │ │ │ +0008ad40: 6f63 2948 6173 6854 6162 6c65 2c20 3c0a  oc)HashTable, <.
│ │ │ │ +0008ad50: 2a6e 6f74 6520 5468 696e 673a 2028 4d61  *note Thing: (Ma
│ │ │ │ +0008ad60: 6361 756c 6179 3244 6f63 2954 6869 6e67  caulay2Doc)Thing
│ │ │ │ +0008ad70: 2c2e 0a1f 0a46 696c 653a 2043 7265 6d6f  ,....File: Cremo
│ │ │ │ +0008ad80: 6e61 2e69 6e66 6f2c 204e 6f64 653a 2072  na.info, Node: r
│ │ │ │ +0008ad90: 6174 696f 6e61 6c4d 6170 2c20 4e65 7874  ationalMap, Next
│ │ │ │ +0008ada0: 3a20 5261 7469 6f6e 616c 4d61 7020 212c  : RationalMap !,
│ │ │ │ +0008adb0: 2050 7265 763a 2052 6174 696f 6e61 6c4d   Prev: RationalM
│ │ │ │ +0008adc0: 6170 2c20 5570 3a20 546f 700a 0a72 6174  ap, Up: Top..rat
│ │ │ │ +0008add0: 696f 6e61 6c4d 6170 202d 2d20 6d61 6b65  ionalMap -- make
│ │ │ │ +0008ade0: 7320 6120 7261 7469 6f6e 616c 206d 6170  s a rational map
│ │ │ │ +0008adf0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +0008ae00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0008ae10: 2a2a 2a2a 0a0a 5379 6e6f 7073 6973 0a3d  ****..Synopsis.=
│ │ │ │ +0008ae20: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2055 7361  =======..  * Usa
│ │ │ │ +0008ae30: 6765 3a20 0a20 2020 2020 2020 2072 6174  ge: .        rat
│ │ │ │ +0008ae40: 696f 6e61 6c4d 6170 2070 6869 200a 2020  ionalMap phi .  
│ │ │ │ +0008ae50: 2020 2020 2020 7261 7469 6f6e 616c 4d61        rationalMa
│ │ │ │ +0008ae60: 7020 460a 2020 2a20 496e 7075 7473 3a0a  p F.  * Inputs:.
│ │ │ │ +0008ae70: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ +0008ae80: 7269 6e67 206d 6170 3a20 284d 6163 6175  ring map: (Macau
│ │ │ │ +0008ae90: 6c61 7932 446f 6329 5269 6e67 4d61 702c  lay2Doc)RingMap,
│ │ │ │ +0008aea0: 2c20 6f72 2061 202a 6e6f 7465 206d 6174  , or a *note mat
│ │ │ │ +0008aeb0: 7269 783a 0a20 2020 2020 2020 2028 4d61  rix:.        (Ma
│ │ │ │ +0008aec0: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ +0008aed0: 782c 206f 7220 6120 2a6e 6f74 6520 6c69  x, or a *note li
│ │ │ │ +0008aee0: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +0008aef0: 6329 4c69 7374 2c2c 2065 7463 2e0a 2020  c)List,, etc..  
│ │ │ │ +0008af00: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c  * *note Optional
│ │ │ │ +0008af10: 2069 6e70 7574 733a 2028 4d61 6361 756c   inputs: (Macaul
│ │ │ │ +0008af20: 6179 3244 6f63 2975 7369 6e67 2066 756e  ay2Doc)using fun
│ │ │ │ +0008af30: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ +0008af40: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020  onal inputs,:.  
│ │ │ │ +0008af50: 2020 2020 2a20 2a6e 6f74 6520 446f 6d69      * *note Domi
│ │ │ │ +0008af60: 6e61 6e74 3a20 446f 6d69 6e61 6e74 2c20  nant: Dominant, 
│ │ │ │ +0008af70: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +0008af80: 7661 6c75 6520 6e75 6c6c 2c20 0a20 202a  value null, .  *
│ │ │ │ +0008af90: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +0008afa0: 2a20 6120 2a6e 6f74 6520 7261 7469 6f6e  * a *note ration
│ │ │ │ +0008afb0: 616c 206d 6170 3a20 5261 7469 6f6e 616c  al map: Rational
│ │ │ │ +0008afc0: 4d61 702c 2c20 7468 6520 7261 7469 6f6e  Map,, the ration
│ │ │ │ +0008afd0: 616c 206d 6170 2072 6570 7265 7365 6e74  al map represent
│ │ │ │ +0008afe0: 6564 2062 7920 7068 690a 2020 2020 2020  ed by phi.      
│ │ │ │ +0008aff0: 2020 6f72 2062 7920 7468 6520 7269 6e67    or by the ring
│ │ │ │ +0008b000: 206d 6170 2064 6566 696e 6564 2062 7920   map defined by 
│ │ │ │ +0008b010: 460a 0a44 6573 6372 6970 7469 6f6e 0a3d  F..Description.=
│ │ │ │ +0008b020: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ +0008b030: 2069 7320 7468 6520 6261 7369 6320 636f   is the basic co
│ │ │ │ +0008b040: 6e73 7472 7563 7469 6f6e 2066 6f72 2061  nstruction for a
│ │ │ │ +0008b050: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +0008b060: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +0008b070: 2c2e 2054 6865 0a6d 6574 686f 6420 6973  ,. The.method is
│ │ │ │ +0008b080: 2071 7569 7465 2073 696d 696c 6172 2074   quite similar t
│ │ │ │ +0008b090: 6f20 2a6e 6f74 6520 746f 4d61 703a 2074  o *note toMap: t
│ │ │ │ +0008b0a0: 6f4d 6170 2c2c 2065 7863 6570 7420 7468  oMap,, except th
│ │ │ │ +0008b0b0: 6174 2069 7420 7265 7475 726e 7320 6120  at it returns a 
│ │ │ │ +0008b0c0: 2a6e 6f74 650a 5261 7469 6f6e 616c 4d61  *note.RationalMa
│ │ │ │ +0008b0d0: 703a 2052 6174 696f 6e61 6c4d 6170 2c20  p: RationalMap, 
│ │ │ │ +0008b0e0: 696e 7374 6561 6420 6f66 2061 202a 6e6f  instead of a *no
│ │ │ │ +0008b0f0: 7465 2052 696e 674d 6170 3a20 284d 6163  te RingMap: (Mac
│ │ │ │ +0008b100: 6175 6c61 7932 446f 6329 5269 6e67 4d61  aulay2Doc)RingMa
│ │ │ │ +0008b110: 702c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  p,...+----------
│ │ │ │  0008b120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008b150: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203a  -----+.|i1 : R :
│ │ │ │ -0008b160: 3d20 5151 5b74 5f30 2e2e 745f 385d 2020  = QQ[t_0..t_8]  
│ │ │ │ -0008b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0008b160: 203a 2052 203a 3d20 5151 5b74 5f30 2e2e   : R := QQ[t_0..
│ │ │ │ +0008b170: 745f 385d 2020 2020 2020 2020 2020 2020  t_8]            
│ │ │ │  0008b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b1a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0008b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b1e0: 2020 2020 207c 0a7c 6f31 203d 2051 515b       |.|o1 = QQ[
│ │ │ │ -0008b1f0: 7420 2e2e 7420 5d20 2020 2020 2020 2020  t ..t ]         
│ │ │ │ +0008b1e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0008b1f0: 203d 2051 515b 7420 2e2e 7420 5d20 2020   = QQ[t ..t ]   
│ │ │ │  0008b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b220: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b230: 2020 2020 2020 2020 2030 2020 2038 2020           0   8  
│ │ │ │ -0008b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b230: 2020 207c 0a7c 2020 2020 2020 2020 2030     |.|         0
│ │ │ │ +0008b240: 2020 2038 2020 2020 2020 2020 2020 2020     8            
│ │ │ │  0008b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0008b270: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0008b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b2b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b2c0: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ -0008b2d0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +0008b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b2c0: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e     |.|o1 : Polyn
│ │ │ │ +0008b2d0: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │  0008b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b300: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0008b300: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0008b310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0008b350: 6932 203a 2046 203d 206d 6174 7269 787b  i2 : F = matrix{
│ │ │ │ -0008b360: 7b74 5f30 2a74 5f33 2a74 5f35 2c20 745f  {t_0*t_3*t_5, t_
│ │ │ │ -0008b370: 312a 745f 332a 745f 362c 2074 5f32 2a74  1*t_3*t_6, t_2*t
│ │ │ │ -0008b380: 5f34 2a74 5f37 2c20 745f 322a 745f 342a  _4*t_7, t_2*t_4*
│ │ │ │ -0008b390: 745f 387d 7d7c 0a7c 2020 2020 2020 2020  t_8}}|.|        
│ │ │ │ +0008b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008b350: 2d2d 2d2b 0a7c 6932 203a 2046 203d 206d  ---+.|i2 : F = m
│ │ │ │ +0008b360: 6174 7269 787b 7b74 5f30 2a74 5f33 2a74  atrix{{t_0*t_3*t
│ │ │ │ +0008b370: 5f35 2c20 745f 312a 745f 332a 745f 362c  _5, t_1*t_3*t_6,
│ │ │ │ +0008b380: 2074 5f32 2a74 5f34 2a74 5f37 2c20 745f   t_2*t_4*t_7, t_
│ │ │ │ +0008b390: 322a 745f 342a 745f 387d 7d7c 0a7c 2020  2*t_4*t_8}}|.|  
│ │ │ │  0008b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b3d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b3e0: 6f32 203d 207c 2074 5f30 745f 3374 5f35  o2 = | t_0t_3t_5
│ │ │ │ -0008b3f0: 2074 5f31 745f 3374 5f36 2074 5f32 745f   t_1t_3t_6 t_2t_
│ │ │ │ -0008b400: 3474 5f37 2074 5f32 745f 3474 5f38 207c  4t_7 t_2t_4t_8 |
│ │ │ │ -0008b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b420: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0008b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b3e0: 2020 207c 0a7c 6f32 203d 207c 2074 5f30     |.|o2 = | t_0
│ │ │ │ +0008b3f0: 745f 3374 5f35 2074 5f31 745f 3374 5f36  t_3t_5 t_1t_3t_6
│ │ │ │ +0008b400: 2074 5f32 745f 3474 5f37 2074 5f32 745f   t_2t_4t_7 t_2t_
│ │ │ │ +0008b410: 3474 5f38 207c 2020 2020 2020 2020 2020  4t_8 |          
│ │ │ │ +0008b420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0008b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b480: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -0008b490: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ -0008b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b4b0: 2020 2020 207c 0a7c 6f32 203a 204d 6174       |.|o2 : Mat
│ │ │ │ -0008b4c0: 7269 7820 2851 515b 7420 2e2e 7420 5d29  rix (QQ[t ..t ])
│ │ │ │ -0008b4d0: 2020 3c2d 2d20 2851 515b 7420 2e2e 7420    <-- (QQ[t ..t 
│ │ │ │ -0008b4e0: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ -0008b4f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b510: 2030 2020 2038 2020 2020 2020 2020 2020   0   8          
│ │ │ │ -0008b520: 2020 2030 2020 2038 2020 2020 2020 2020     0   8        
│ │ │ │ +0008b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b470: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008b480: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +0008b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b4a0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0008b4b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0008b4c0: 203a 204d 6174 7269 7820 2851 515b 7420   : Matrix (QQ[t 
│ │ │ │ +0008b4d0: 2e2e 7420 5d29 2020 3c2d 2d20 2851 515b  ..t ])  <-- (QQ[
│ │ │ │ +0008b4e0: 7420 2e2e 7420 5d29 2020 2020 2020 2020  t ..t ])        
│ │ │ │ +0008b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b500: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008b510: 2020 2020 2020 2030 2020 2038 2020 2020         0   8    
│ │ │ │ +0008b520: 2020 2020 2020 2020 2030 2020 2038 2020           0   8  
│ │ │ │  0008b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b540: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0008b540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0008b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0008b590: 6933 203a 2070 6869 203d 2074 6f4d 6170  i3 : phi = toMap
│ │ │ │ -0008b5a0: 2046 2020 2020 2020 2020 2020 2020 2020   F              
│ │ │ │ +0008b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008b590: 2d2d 2d2b 0a7c 6933 203a 2070 6869 203d  ---+.|i3 : phi =
│ │ │ │ +0008b5a0: 2074 6f4d 6170 2046 2020 2020 2020 2020   toMap F        
│ │ │ │  0008b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b5d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0008b5d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0008b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b620: 6f33 203d 206d 6170 2028 5151 5b74 202e  o3 = map (QQ[t .
│ │ │ │ -0008b630: 2e74 205d 2c20 5151 5b78 202e 2e78 205d  .t ], QQ[x ..x ]
│ │ │ │ -0008b640: 2c20 7b74 2074 2074 202c 2074 2074 2074  , {t t t , t t t
│ │ │ │ -0008b650: 202c 2074 2074 2074 202c 2074 2074 2074   , t t t , t t t
│ │ │ │ -0008b660: 207d 2920 207c 0a7c 2020 2020 2020 2020   })  |.|        
│ │ │ │ -0008b670: 2020 2020 2020 3020 2020 3820 2020 2020        0   8     
│ │ │ │ -0008b680: 2020 3020 2020 3320 2020 2020 3020 3320    0   3     0 3 
│ │ │ │ -0008b690: 3520 2020 3120 3320 3620 2020 3220 3420  5   1 3 6   2 4 
│ │ │ │ -0008b6a0: 3720 2020 3220 3420 3820 2020 207c 0a7c  7   2 4 8    |.|
│ │ │ │ -0008b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b620: 2020 207c 0a7c 6f33 203d 206d 6170 2028     |.|o3 = map (
│ │ │ │ +0008b630: 5151 5b74 202e 2e74 205d 2c20 5151 5b78  QQ[t ..t ], QQ[x
│ │ │ │ +0008b640: 202e 2e78 205d 2c20 7b74 2074 2074 202c   ..x ], {t t t ,
│ │ │ │ +0008b650: 2074 2074 2074 202c 2074 2074 2074 202c   t t t , t t t ,
│ │ │ │ +0008b660: 2074 2074 2074 207d 2920 207c 0a7c 2020   t t t })  |.|  
│ │ │ │ +0008b670: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0008b680: 3820 2020 2020 2020 3020 2020 3320 2020  8       0   3   
│ │ │ │ +0008b690: 2020 3020 3320 3520 2020 3120 3320 3620    0 3 5   1 3 6 
│ │ │ │ +0008b6a0: 2020 3220 3420 3720 2020 3220 3420 3820    2 4 7   2 4 8 
│ │ │ │ +0008b6b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0008b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b6f0: 2020 2020 207c 0a7c 6f33 203a 2052 696e       |.|o3 : Rin
│ │ │ │ -0008b700: 674d 6170 2051 515b 7420 2e2e 7420 5d20  gMap QQ[t ..t ] 
│ │ │ │ -0008b710: 3c2d 2d20 5151 5b78 202e 2e78 205d 2020  <-- QQ[x ..x ]  
│ │ │ │ -0008b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b750: 2030 2020 2038 2020 2020 2020 2020 2020   0   8          
│ │ │ │ -0008b760: 3020 2020 3320 2020 2020 2020 2020 2020  0   3           
│ │ │ │ +0008b6f0: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0008b700: 203a 2052 696e 674d 6170 2051 515b 7420   : RingMap QQ[t 
│ │ │ │ +0008b710: 2e2e 7420 5d20 3c2d 2d20 5151 5b78 202e  ..t ] <-- QQ[x .
│ │ │ │ +0008b720: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │ +0008b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b740: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008b750: 2020 2020 2020 2030 2020 2038 2020 2020         0   8    
│ │ │ │ +0008b760: 2020 2020 2020 3020 2020 3320 2020 2020        0   3     
│ │ │ │  0008b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b780: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0008b780: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0008b790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008b7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0008b7d0: 6934 203a 2072 6174 696f 6e61 6c4d 6170  i4 : rationalMap
│ │ │ │ -0008b7e0: 2070 6869 2020 2020 2020 2020 2020 2020   phi            
│ │ │ │ +0008b7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008b7d0: 2d2d 2d2b 0a7c 6934 203a 2072 6174 696f  ---+.|i4 : ratio
│ │ │ │ +0008b7e0: 6e61 6c4d 6170 2070 6869 2020 2020 2020  nalMap phi      
│ │ │ │  0008b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0008b810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0008b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b860: 6f34 203d 202d 2d20 7261 7469 6f6e 616c  o4 = -- rational
│ │ │ │ -0008b870: 206d 6170 202d 2d20 2020 2020 2020 2020   map --         
│ │ │ │ +0008b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b860: 2020 207c 0a7c 6f34 203d 202d 2d20 7261     |.|o4 = -- ra
│ │ │ │ +0008b870: 7469 6f6e 616c 206d 6170 202d 2d20 2020  tional map --   
│ │ │ │  0008b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b8a0: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -0008b8b0: 7263 653a 2050 726f 6a28 5151 5b74 202c  rce: Proj(QQ[t ,
│ │ │ │ -0008b8c0: 2074 202c 2074 202c 2074 202c 2074 202c   t , t , t , t ,
│ │ │ │ -0008b8d0: 2074 202c 2074 202c 2074 202c 2074 205d   t , t , t , t ]
│ │ │ │ -0008b8e0: 2920 2020 2020 2020 2020 2020 207c 0a7c  )            |.|
│ │ │ │ -0008b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b900: 2020 2020 2020 3020 2020 3120 2020 3220        0   1   2 
│ │ │ │ -0008b910: 2020 3320 2020 3420 2020 3520 2020 3620    3   4   5   6 
│ │ │ │ -0008b920: 2020 3720 2020 3820 2020 2020 2020 2020    7   8         
│ │ │ │ -0008b930: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -0008b940: 6765 743a 2050 726f 6a28 5151 5b78 202c  get: Proj(QQ[x ,
│ │ │ │ -0008b950: 2078 202c 2078 202c 2078 205d 2920 2020   x , x , x ])   
│ │ │ │ -0008b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b970: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b990: 2020 2020 2020 3020 2020 3120 2020 3220        0   1   2 
│ │ │ │ -0008b9a0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +0008b8a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008b8b0: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ +0008b8c0: 5151 5b74 202c 2074 202c 2074 202c 2074  QQ[t , t , t , t
│ │ │ │ +0008b8d0: 202c 2074 202c 2074 202c 2074 202c 2074   , t , t , t , t
│ │ │ │ +0008b8e0: 202c 2074 205d 2920 2020 2020 2020 2020   , t ])         
│ │ │ │ +0008b8f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008b900: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0008b910: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ +0008b920: 3520 2020 3620 2020 3720 2020 3820 2020  5   6   7   8   
│ │ │ │ +0008b930: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008b940: 2020 2074 6172 6765 743a 2050 726f 6a28     target: Proj(
│ │ │ │ +0008b950: 5151 5b78 202c 2078 202c 2078 202c 2078  QQ[x , x , x , x
│ │ │ │ +0008b960: 205d 2920 2020 2020 2020 2020 2020 2020   ])             
│ │ │ │ +0008b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008b990: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0008b9a0: 3120 2020 3220 2020 3320 2020 2020 2020  1   2   3       
│ │ │ │  0008b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b9c0: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -0008b9d0: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ -0008b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b9c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008b9d0: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
│ │ │ │ +0008b9e0: 733a 207b 2020 2020 2020 2020 2020 2020  s: {            
│ │ │ │  0008b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ba00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ba20: 2020 2020 2020 7420 7420 7420 2c20 2020        t t t ,   
│ │ │ │ -0008ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008ba10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008ba20: 2020 2020 2020 2020 2020 2020 7420 7420              t t 
│ │ │ │ +0008ba30: 7420 2c20 2020 2020 2020 2020 2020 2020  t ,             
│ │ │ │  0008ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ba50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008ba60: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -0008ba70: 2033 2035 2020 2020 2020 2020 2020 2020   3 5            
│ │ │ │ +0008ba50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008ba70: 2020 2020 2030 2033 2035 2020 2020 2020       0 3 5      
│ │ │ │  0008ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ba90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008baa0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0008bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008baf0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -0008bb00: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │ +0008bae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bb00: 2020 2020 7420 7420 7420 2c20 2020 2020      t t t ,     
│ │ │ │  0008bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bb20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bb40: 2020 2020 2020 2031 2033 2036 2020 2020         1 3 6    
│ │ │ │ -0008bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bb30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008bb40: 2020 2020 2020 2020 2020 2020 2031 2033               1 3
│ │ │ │ +0008bb50: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │  0008bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bb70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0008bb70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0008bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bbb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bbd0: 2020 2020 2020 7420 7420 7420 2c20 2020        t t t ,   
│ │ │ │ -0008bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bbc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008bbd0: 2020 2020 2020 2020 2020 2020 7420 7420              t t 
│ │ │ │ +0008bbe0: 7420 2c20 2020 2020 2020 2020 2020 2020  t ,             
│ │ │ │  0008bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bc00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008bc10: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0008bc20: 2034 2037 2020 2020 2020 2020 2020 2020   4 7            
│ │ │ │ +0008bc00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bc20: 2020 2020 2032 2034 2037 2020 2020 2020       2 4 7      
│ │ │ │  0008bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bc40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bc50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0008bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bc90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008bca0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -0008bcb0: 7420 7420 2020 2020 2020 2020 2020 2020  t t             
│ │ │ │ +0008bc90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bcb0: 2020 2020 7420 7420 7420 2020 2020 2020      t t t       
│ │ │ │  0008bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bcd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bcf0: 2020 2020 2020 2032 2034 2038 2020 2020         2 4 8    
│ │ │ │ -0008bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bce0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008bcf0: 2020 2020 2020 2020 2020 2020 2032 2034               2 4
│ │ │ │ +0008bd00: 2038 2020 2020 2020 2020 2020 2020 2020   8              
│ │ │ │  0008bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bd20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008bd30: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ -0008bd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bd20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bd40: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  0008bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bd60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bd70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0008bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bdb0: 2020 2020 207c 0a7c 6f34 203a 2052 6174       |.|o4 : Rat
│ │ │ │ -0008bdc0: 696f 6e61 6c4d 6170 2028 6375 6269 6320  ionalMap (cubic 
│ │ │ │ -0008bdd0: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ -0008bde0: 6d20 5050 5e38 2074 6f20 5050 5e33 2920  m PP^8 to PP^3) 
│ │ │ │ -0008bdf0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -0008be00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008bdb0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +0008bdc0: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +0008bdd0: 6375 6269 6320 7261 7469 6f6e 616c 206d  cubic rational m
│ │ │ │ +0008bde0: 6170 2066 726f 6d20 5050 5e38 2074 6f20  ap from PP^8 to 
│ │ │ │ +0008bdf0: 5050 5e33 2920 2020 2020 2020 2020 2020  PP^3)           
│ │ │ │ +0008be00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  0008be10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008be20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008be30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008be40: 2d2d 2d2d 2d2b 0a7c 6935 203a 2072 6174  -----+.|i5 : rat
│ │ │ │ -0008be50: 696f 6e61 6c4d 6170 2046 2020 2020 2020  ionalMap F      
│ │ │ │ +0008be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +0008be50: 203a 2072 6174 696f 6e61 6c4d 6170 2046   : rationalMap F
│ │ │ │  0008be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008be80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008be90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0008bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bed0: 2020 2020 207c 0a7c 6f35 203d 202d 2d20       |.|o5 = -- 
│ │ │ │ -0008bee0: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ -0008bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bed0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +0008bee0: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +0008bef0: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  0008bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bf10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008bf20: 2020 2020 2073 6f75 7263 653a 2050 726f       source: Pro
│ │ │ │ -0008bf30: 6a28 5151 5b74 202c 2074 202c 2074 202c  j(QQ[t , t , t ,
│ │ │ │ -0008bf40: 2074 202c 2074 202c 2074 202c 2074 202c   t , t , t , t ,
│ │ │ │ -0008bf50: 2074 202c 2074 205d 2920 2020 2020 2020   t , t ])       
│ │ │ │ -0008bf60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008bf70: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0008bf80: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ -0008bf90: 2020 3520 2020 3620 2020 3720 2020 3820    5   6   7   8 
│ │ │ │ -0008bfa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008bfb0: 2020 2020 2074 6172 6765 743a 2050 726f       target: Pro
│ │ │ │ -0008bfc0: 6a28 5151 5b78 202c 2078 202c 2078 202c  j(QQ[x , x , x ,
│ │ │ │ -0008bfd0: 2078 205d 2920 2020 2020 2020 2020 2020   x ])           
│ │ │ │ +0008bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bf20: 2020 207c 0a7c 2020 2020 2073 6f75 7263     |.|     sourc
│ │ │ │ +0008bf30: 653a 2050 726f 6a28 5151 5b74 202c 2074  e: Proj(QQ[t , t
│ │ │ │ +0008bf40: 202c 2074 202c 2074 202c 2074 202c 2074   , t , t , t , t
│ │ │ │ +0008bf50: 202c 2074 202c 2074 202c 2074 205d 2920   , t , t , t ]) 
│ │ │ │ +0008bf60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008bf80: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +0008bf90: 3320 2020 3420 2020 3520 2020 3620 2020  3   4   5   6   
│ │ │ │ +0008bfa0: 3720 2020 3820 2020 2020 2020 2020 2020  7   8           
│ │ │ │ +0008bfb0: 2020 207c 0a7c 2020 2020 2074 6172 6765     |.|     targe
│ │ │ │ +0008bfc0: 743a 2050 726f 6a28 5151 5b78 202c 2078  t: Proj(QQ[x , x
│ │ │ │ +0008bfd0: 202c 2078 202c 2078 205d 2920 2020 2020   , x , x ])     
│ │ │ │  0008bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008c000: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0008c010: 2020 3120 2020 3220 2020 3320 2020 2020    1   2   3     
│ │ │ │ -0008c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008c040: 2020 2020 2064 6566 696e 696e 6720 666f       defining fo
│ │ │ │ -0008c050: 726d 733a 207b 2020 2020 2020 2020 2020  rms: {          
│ │ │ │ +0008bff0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c010: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +0008c020: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0008c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c040: 2020 207c 0a7c 2020 2020 2064 6566 696e     |.|     defin
│ │ │ │ +0008c050: 696e 6720 666f 726d 733a 207b 2020 2020  ing forms: {    
│ │ │ │  0008c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c080: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008c090: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -0008c0a0: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │ +0008c080: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c0a0: 2020 2020 7420 7420 7420 2c20 2020 2020      t t t ,     
│ │ │ │  0008c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c0c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c0e0: 2020 2020 2020 2030 2033 2035 2020 2020         0 3 5    
│ │ │ │ -0008c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c0d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008c0e0: 2020 2020 2020 2020 2020 2020 2030 2033               0 3
│ │ │ │ +0008c0f0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │  0008c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0008c110: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0008c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c170: 2020 2020 2020 7420 7420 7420 2c20 2020        t t t ,   
│ │ │ │ -0008c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c160: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008c170: 2020 2020 2020 2020 2020 2020 7420 7420              t t 
│ │ │ │ +0008c180: 7420 2c20 2020 2020 2020 2020 2020 2020  t ,             
│ │ │ │  0008c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c1a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008c1b0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -0008c1c0: 2033 2036 2020 2020 2020 2020 2020 2020   3 6            
│ │ │ │ +0008c1a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c1c0: 2020 2020 2031 2033 2036 2020 2020 2020       1 3 6      
│ │ │ │  0008c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c1f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0008c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c230: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008c240: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -0008c250: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │ +0008c230: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c250: 2020 2020 7420 7420 7420 2c20 2020 2020      t t t ,     
│ │ │ │  0008c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c290: 2020 2020 2020 2032 2034 2037 2020 2020         2 4 7    
│ │ │ │ -0008c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c280: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008c290: 2020 2020 2020 2020 2020 2020 2032 2034               2 4
│ │ │ │ +0008c2a0: 2037 2020 2020 2020 2020 2020 2020 2020   7              
│ │ │ │  0008c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c2c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0008c2c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0008c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c320: 2020 2020 2020 7420 7420 7420 2020 2020        t t t     
│ │ │ │ -0008c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c310: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008c320: 2020 2020 2020 2020 2020 2020 7420 7420              t t 
│ │ │ │ +0008c330: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  0008c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c350: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0008c360: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0008c370: 2034 2038 2020 2020 2020 2020 2020 2020   4 8            
│ │ │ │ +0008c350: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c370: 2020 2020 2032 2034 2038 2020 2020 2020       2 4 8      
│ │ │ │  0008c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c390: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c3b0: 2020 2020 207d 2020 2020 2020 2020 2020       }          
│ │ │ │ +0008c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c3a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008c3b0: 2020 2020 2020 2020 2020 207d 2020 2020             }    
│ │ │ │  0008c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c3e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0008c3e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0008c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008c430: 6f35 203a 2052 6174 696f 6e61 6c4d 6170  o5 : RationalMap
│ │ │ │ -0008c440: 2028 6375 6269 6320 7261 7469 6f6e 616c   (cubic rational
│ │ │ │ -0008c450: 206d 6170 2066 726f 6d20 5050 5e38 2074   map from PP^8 t
│ │ │ │ -0008c460: 6f20 5050 5e33 2920 2020 2020 2020 2020  o PP^3)         
│ │ │ │ -0008c470: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0008c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c430: 2020 207c 0a7c 6f35 203a 2052 6174 696f     |.|o5 : Ratio
│ │ │ │ +0008c440: 6e61 6c4d 6170 2028 6375 6269 6320 7261  nalMap (cubic ra
│ │ │ │ +0008c450: 7469 6f6e 616c 206d 6170 2066 726f 6d20  tional map from 
│ │ │ │ +0008c460: 5050 5e38 2074 6f20 5050 5e33 2920 2020  PP^8 to PP^3)   
│ │ │ │ +0008c470: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0008c480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008c490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008c4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -0008c4c0: 4d75 6c74 6967 7261 6465 6420 7269 6e67  Multigraded ring
│ │ │ │ -0008c4d0: 7320 6172 6520 616c 736f 2070 6572 6d69  s are also permi
│ │ │ │ -0008c4e0: 7474 6564 2062 7574 2069 6e20 7468 6973  tted but in this
│ │ │ │ -0008c4f0: 2063 6173 6520 7468 6520 6d65 7468 6f64   case the method
│ │ │ │ -0008c500: 2072 6574 7572 6e73 2061 6e0a 6f62 6a65   returns an.obje
│ │ │ │ -0008c510: 6374 206f 6620 7468 6520 636c 6173 7320  ct of the class 
│ │ │ │ -0008c520: 4d75 6c74 6968 6f6d 6f67 656e 656f 7573  Multihomogeneous
│ │ │ │ -0008c530: 5261 7469 6f6e 616c 4d61 702c 2077 6869  RationalMap, whi
│ │ │ │ -0008c540: 6368 2063 616e 2062 6520 636f 6e73 6964  ch can be consid
│ │ │ │ -0008c550: 6572 6564 2061 7320 616e 0a65 7874 656e  ered as an.exten
│ │ │ │ -0008c560: 7369 6f6e 206f 6620 7468 6520 636c 6173  sion of the clas
│ │ │ │ -0008c570: 7320 5261 7469 6f6e 616c 4d61 702e 0a0a  s RationalMap...
│ │ │ │ -0008c580: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0008c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008c4c0: 2d2d 2d2b 0a0a 4d75 6c74 6967 7261 6465  ---+..Multigrade
│ │ │ │ +0008c4d0: 6420 7269 6e67 7320 6172 6520 616c 736f  d rings are also
│ │ │ │ +0008c4e0: 2070 6572 6d69 7474 6564 2062 7574 2069   permitted but i
│ │ │ │ +0008c4f0: 6e20 7468 6973 2063 6173 6520 7468 6520  n this case the 
│ │ │ │ +0008c500: 6d65 7468 6f64 2072 6574 7572 6e73 2061  method returns a
│ │ │ │ +0008c510: 6e0a 6f62 6a65 6374 206f 6620 7468 6520  n.object of the 
│ │ │ │ +0008c520: 636c 6173 7320 4d75 6c74 6968 6f6d 6f67  class Multihomog
│ │ │ │ +0008c530: 656e 656f 7573 5261 7469 6f6e 616c 4d61  eneousRationalMa
│ │ │ │ +0008c540: 702c 2077 6869 6368 2063 616e 2062 6520  p, which can be 
│ │ │ │ +0008c550: 636f 6e73 6964 6572 6564 2061 7320 616e  considered as an
│ │ │ │ +0008c560: 0a65 7874 656e 7369 6f6e 206f 6620 7468  .extension of th
│ │ │ │ +0008c570: 6520 636c 6173 7320 5261 7469 6f6e 616c  e class Rational
│ │ │ │ +0008c580: 4d61 702e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  Map...+---------
│ │ │ │  0008c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008c5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0008c5d0: 7c69 3620 3a20 5227 203a 3d20 6e65 7752  |i6 : R' := newR
│ │ │ │ -0008c5e0: 696e 6728 522c 4465 6772 6565 733d 3e7b  ing(R,Degrees=>{
│ │ │ │ -0008c5f0: 333a 7b31 2c30 2c30 7d2c 323a 7b30 2c31  3:{1,0,0},2:{0,1
│ │ │ │ -0008c600: 2c30 7d2c 343a 7b30 2c30 2c31 7d7d 2920  ,0},4:{0,0,1}}) 
│ │ │ │ -0008c610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c620: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008c5d0: 2d2d 2d2d 2b0a 7c69 3620 3a20 5227 203a  ----+.|i6 : R' :
│ │ │ │ +0008c5e0: 3d20 6e65 7752 696e 6728 522c 4465 6772  = newRing(R,Degr
│ │ │ │ +0008c5f0: 6565 733d 3e7b 333a 7b31 2c30 2c30 7d2c  ees=>{3:{1,0,0},
│ │ │ │ +0008c600: 323a 7b30 2c31 2c30 7d2c 343a 7b30 2c30  2:{0,1,0},4:{0,0
│ │ │ │ +0008c610: 2c31 7d7d 2920 2020 2020 2020 2020 2020  ,1}})           
│ │ │ │ +0008c620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c670: 7c6f 3620 3d20 5151 5b74 202e 2e74 205d  |o6 = QQ[t ..t ]
│ │ │ │ -0008c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c670: 2020 2020 7c0a 7c6f 3620 3d20 5151 5b74      |.|o6 = QQ[t
│ │ │ │ +0008c680: 202e 2e74 205d 2020 2020 2020 2020 2020   ..t ]          
│ │ │ │  0008c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c6b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c6c0: 7c20 2020 2020 2020 2020 3020 2020 3820  |         0   8 
│ │ │ │ -0008c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c6c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008c6d0: 3020 2020 3820 2020 2020 2020 2020 2020  0   8           
│ │ │ │  0008c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c710: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c710: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c760: 7c6f 3620 3a20 506f 6c79 6e6f 6d69 616c  |o6 : Polynomial
│ │ │ │ -0008c770: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0008c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c760: 2020 2020 7c0a 7c6f 3620 3a20 506f 6c79      |.|o6 : Poly
│ │ │ │ +0008c770: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │  0008c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c7a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c7b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0008c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c7b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  0008c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008c7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008c7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0008c800: 7c69 3720 3a20 4627 203d 2073 7562 2846  |i7 : F' = sub(F
│ │ │ │ -0008c810: 2c52 2729 2020 2020 2020 2020 2020 2020  ,R')            
│ │ │ │ +0008c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008c800: 2d2d 2d2d 2b0a 7c69 3720 3a20 4627 203d  ----+.|i7 : F' =
│ │ │ │ +0008c810: 2073 7562 2846 2c52 2729 2020 2020 2020   sub(F,R')      
│ │ │ │  0008c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c850: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c8a0: 7c6f 3720 3d20 7c20 745f 3074 5f33 745f  |o7 = | t_0t_3t_
│ │ │ │ -0008c8b0: 3520 745f 3174 5f33 745f 3620 745f 3274  5 t_1t_3t_6 t_2t
│ │ │ │ -0008c8c0: 5f34 745f 3720 745f 3274 5f34 745f 3820  _4t_7 t_2t_4t_8 
│ │ │ │ -0008c8d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0008c8e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c8f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c8a0: 2020 2020 7c0a 7c6f 3720 3d20 7c20 745f      |.|o7 = | t_
│ │ │ │ +0008c8b0: 3074 5f33 745f 3520 745f 3174 5f33 745f  0t_3t_5 t_1t_3t_
│ │ │ │ +0008c8c0: 3620 745f 3274 5f34 745f 3720 745f 3274  6 t_2t_4t_7 t_2t
│ │ │ │ +0008c8d0: 5f34 745f 3820 7c20 2020 2020 2020 2020  _4t_8 |         
│ │ │ │ +0008c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c8f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c940: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0008c950: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -0008c960: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ -0008c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c990: 7c6f 3720 3a20 4d61 7472 6978 2028 5151  |o7 : Matrix (QQ
│ │ │ │ -0008c9a0: 5b74 202e 2e74 205d 2920 203c 2d2d 2028  [t ..t ])  <-- (
│ │ │ │ -0008c9b0: 5151 5b74 202e 2e74 205d 2920 2020 2020  QQ[t ..t ])     
│ │ │ │ -0008c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c9d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c9e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0008c9f0: 2020 3020 2020 3820 2020 2020 2020 2020    0   8         
│ │ │ │ -0008ca00: 2020 2020 3020 2020 3820 2020 2020 2020      0   8       
│ │ │ │ +0008c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c940: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008c950: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0008c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c970: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0008c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c990: 2020 2020 7c0a 7c6f 3720 3a20 4d61 7472      |.|o7 : Matr
│ │ │ │ +0008c9a0: 6978 2028 5151 5b74 202e 2e74 205d 2920  ix (QQ[t ..t ]) 
│ │ │ │ +0008c9b0: 203c 2d2d 2028 5151 5b74 202e 2e74 205d   <-- (QQ[t ..t ]
│ │ │ │ +0008c9c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0008c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c9e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008c9f0: 2020 2020 2020 2020 3020 2020 3820 2020          0   8   
│ │ │ │ +0008ca00: 2020 2020 2020 2020 2020 3020 2020 3820            0   8 
│ │ │ │  0008ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ca20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008ca30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0008ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008ca30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  0008ca40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008ca50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0008ca80: 7c69 3820 3a20 7068 6927 203d 2074 6f4d  |i8 : phi' = toM
│ │ │ │ -0008ca90: 6170 2046 2720 2020 2020 2020 2020 2020  ap F'           
│ │ │ │ +0008ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008ca80: 2d2d 2d2d 2b0a 7c69 3820 3a20 7068 6927  ----+.|i8 : phi'
│ │ │ │ +0008ca90: 203d 2074 6f4d 6170 2046 2720 2020 2020   = toMap F'     
│ │ │ │  0008caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008cac0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008cad0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008cad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008cb10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008cb20: 7c6f 3820 3d20 6d61 7020 2851 515b 7420  |o8 = map (QQ[t 
│ │ │ │ -0008cb30: 2e2e 7420 5d2c 2051 515b 7820 2e2e 7820  ..t ], QQ[x ..x 
│ │ │ │ -0008cb40: 5d2c 207b 7420 7420 7420 2c20 7420 7420  ], {t t t , t t 
│ │ │ │ -0008cb50: 7420 2c20 7420 7420 7420 2c20 7420 7420  t , t t t , t t 
│ │ │ │ -0008cb60: 7420 7d29 2020 2020 2020 2020 2020 7c0a  t })          |.
│ │ │ │ -0008cb70: 7c20 2020 2020 2020 2020 2020 2020 2030  |              0
│ │ │ │ -0008cb80: 2020 2038 2020 2020 2020 2030 2020 2033     8       0   3
│ │ │ │ -0008cb90: 2020 2020 2030 2033 2035 2020 2031 2033       0 3 5   1 3
│ │ │ │ -0008cba0: 2036 2020 2032 2034 2037 2020 2032 2034   6   2 4 7   2 4
│ │ │ │ -0008cbb0: 2038 2020 2020 2020 2020 2020 2020 7c0a   8            |.
│ │ │ │ -0008cbc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008cb20: 2020 2020 7c0a 7c6f 3820 3d20 6d61 7020      |.|o8 = map 
│ │ │ │ +0008cb30: 2851 515b 7420 2e2e 7420 5d2c 2051 515b  (QQ[t ..t ], QQ[
│ │ │ │ +0008cb40: 7820 2e2e 7820 5d2c 207b 7420 7420 7420  x ..x ], {t t t 
│ │ │ │ +0008cb50: 2c20 7420 7420 7420 2c20 7420 7420 7420  , t t t , t t t 
│ │ │ │ +0008cb60: 2c20 7420 7420 7420 7d29 2020 2020 2020  , t t t })      
│ │ │ │ +0008cb70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008cb80: 2020 2020 2030 2020 2038 2020 2020 2020       0   8      
│ │ │ │ +0008cb90: 2030 2020 2033 2020 2020 2030 2033 2035   0   3     0 3 5
│ │ │ │ +0008cba0: 2020 2031 2033 2036 2020 2032 2034 2037     1 3 6   2 4 7
│ │ │ │ +0008cbb0: 2020 2032 2034 2038 2020 2020 2020 2020     2 4 8        
│ │ │ │ +0008cbc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008cc00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008cc10: 7c6f 3820 3a20 5269 6e67 4d61 7020 5151  |o8 : RingMap QQ
│ │ │ │ -0008cc20: 5b74 202e 2e74 205d 203c 2d2d 2051 515b  [t ..t ] <-- QQ[
│ │ │ │ -0008cc30: 7820 2e2e 7820 5d20 2020 2020 2020 2020  x ..x ]         
│ │ │ │ +0008cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008cc10: 2020 2020 7c0a 7c6f 3820 3a20 5269 6e67      |.|o8 : Ring
│ │ │ │ +0008cc20: 4d61 7020 5151 5b74 202e 2e74 205d 203c  Map QQ[t ..t ] <
│ │ │ │ +0008cc30: 2d2d 2051 515b 7820 2e2e 7820 5d20 2020  -- QQ[x ..x ]   
│ │ │ │  0008cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008cc50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008cc60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0008cc70: 2020 3020 2020 3820 2020 2020 2020 2020    0   8         
│ │ │ │ -0008cc80: 2030 2020 2033 2020 2020 2020 2020 2020   0   3          
│ │ │ │ +0008cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008cc60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008cc70: 2020 2020 2020 2020 3020 2020 3820 2020          0   8   
│ │ │ │ +0008cc80: 2020 2020 2020 2030 2020 2033 2020 2020         0   3    
│ │ │ │  0008cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008cca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008ccb0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0008cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008ccb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  0008ccc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008ccd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0008cd00: 7c69 3920 3a20 7261 7469 6f6e 616c 4d61  |i9 : rationalMa
│ │ │ │ -0008cd10: 7020 7068 6927 2020 2020 2020 2020 2020  p phi'          
│ │ │ │ +0008ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008cd00: 2d2d 2d2d 2b0a 7c69 3920 3a20 7261 7469  ----+.|i9 : rati
│ │ │ │ +0008cd10: 6f6e 616c 4d61 7020 7068 6927 2020 2020  onalMap phi'    
│ │ │ │  0008cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008cd40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008cd50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008cd50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008cd90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008cda0: 7c6f 3920 3d20 2d2d 2072 6174 696f 6e61  |o9 = -- rationa
│ │ │ │ -0008cdb0: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │ +0008cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008cda0: 2020 2020 7c0a 7c6f 3920 3d20 2d2d 2072      |.|o9 = -- r
│ │ │ │ +0008cdb0: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │  0008cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008cde0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008cdf0: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ -0008ce00: 6f6a 2851 515b 7420 2c20 7420 2c20 7420  oj(QQ[t , t , t 
│ │ │ │ -0008ce10: 5d29 2078 2050 726f 6a28 5151 5b74 202c  ]) x Proj(QQ[t ,
│ │ │ │ -0008ce20: 2074 205d 2920 7820 5072 6f6a 2851 515b   t ]) x Proj(QQ[
│ │ │ │ -0008ce30: 7420 2c20 7420 2c20 2020 2020 2020 7c0a  t , t ,       |.
│ │ │ │ -0008ce40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0008ce50: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ -0008ce60: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -0008ce70: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ -0008ce80: 2035 2020 2036 2020 2020 2020 2020 7c0a   5   6        |.
│ │ │ │ -0008ce90: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ -0008cea0: 6f6a 2851 515b 7820 2c20 7820 2c20 7820  oj(QQ[x , x , x 
│ │ │ │ -0008ceb0: 2c20 7820 5d29 2020 2020 2020 2020 2020  , x ])          
│ │ │ │ +0008cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008cdf0: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ +0008ce00: 6365 3a20 5072 6f6a 2851 515b 7420 2c20  ce: Proj(QQ[t , 
│ │ │ │ +0008ce10: 7420 2c20 7420 5d29 2078 2050 726f 6a28  t , t ]) x Proj(
│ │ │ │ +0008ce20: 5151 5b74 202c 2074 205d 2920 7820 5072  QQ[t , t ]) x Pr
│ │ │ │ +0008ce30: 6f6a 2851 515b 7420 2c20 7420 2c20 2020  oj(QQ[t , t ,   
│ │ │ │ +0008ce40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008ce50: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +0008ce60: 2031 2020 2032 2020 2020 2020 2020 2020   1   2          
│ │ │ │ +0008ce70: 2020 2020 3320 2020 3420 2020 2020 2020      3   4       
│ │ │ │ +0008ce80: 2020 2020 2020 2035 2020 2036 2020 2020         5   6    
│ │ │ │ +0008ce90: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ +0008cea0: 6574 3a20 5072 6f6a 2851 515b 7820 2c20  et: Proj(QQ[x , 
│ │ │ │ +0008ceb0: 7820 2c20 7820 2c20 7820 5d29 2020 2020  x , x , x ])    
│ │ │ │  0008cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ced0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008cee0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0008cef0: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ -0008cf00: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +0008ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008cee0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008cef0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +0008cf00: 2031 2020 2032 2020 2033 2020 2020 2020   1   2   3      
│ │ │ │  0008cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008cf20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008cf30: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ -0008cf40: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │ +0008cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008cf30: 2020 2020 7c0a 7c20 2020 2020 6465 6669      |.|     defi
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│ │ │ │ -0008d150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  0008d320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d330: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d340: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d380: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d390: 7c6f 3920 3a20 4d75 6c74 6968 6f6d 6f67  |o9 : Multihomog
│ │ │ │ -0008d3a0: 656e 656f 7573 5261 7469 6f6e 616c 4d61  eneousRationalMa
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│ │ │ │ +0008d380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d390: 2020 2020 7c0a 7c6f 3920 3a20 4d75 6c74      |.|o9 : Mult
│ │ │ │ +0008d3a0: 6968 6f6d 6f67 656e 656f 7573 5261 7469  ihomogeneousRati
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│ │ │ │  0008d410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008d420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0008d430: 7c74 202c 2074 205d 2920 2020 2020 2020  |t , t ])       
│ │ │ │ +0008d420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0008d440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d480: 7c20 3720 2020 3820 2020 2020 2020 2020  | 7   8         
│ │ │ │ +0008d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d480: 2020 2020 7c0a 7c20 3720 2020 3820 2020      |.| 7   8   
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│ │ │ │  0008d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d4c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d4d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d4d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d520: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d520: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d570: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +0008d570: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d5b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d5c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0008d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0008d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d610: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d650: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d660: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d660: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0008d6a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d6b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d6b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d6f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d700: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d750: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  0008d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d7a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +0008d7a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d7e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d7f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +0008d7f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d830: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d840: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d840: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d890: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d8d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d8e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d8e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d930: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d930: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d980: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d980: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008d9c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008d9d0: 7c74 6f20 5050 5e33 2920 2020 2020 2020  |to PP^3)       
│ │ │ │ +0008d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008d9d0: 2020 2020 7c0a 7c74 6f20 5050 5e33 2920      |.|to PP^3) 
│ │ │ │  0008d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008da10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008da20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0008da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008da20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  0008da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0008da70: 7c69 3130 203a 2072 6174 696f 6e61 6c4d  |i10 : rationalM
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│ │ │ │  0008daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0008dac0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0008dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008db00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008db10: 7c6f 3130 203d 202d 2d20 7261 7469 6f6e  |o10 = -- ration
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│ │ │ │ -0008db60: 7c20 2020 2020 2073 6f75 7263 653a 2050  |      source: P
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│ │ │ │ -0008df60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  0008e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008e050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -0008e0f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008e100: 7c6f 3130 203a 204d 756c 7469 686f 6d6f  |o10 : Multihomo
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│ │ │ │ +0008e100: 2020 2020 7c0a 7c6f 3130 203a 204d 756c      |.|o10 : Mul
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│ │ │ │  0008e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008e230: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008e240: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008e240: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  0008e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008e280: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008e290: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │  0008e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0008e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008e3c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008e3d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0008e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0008e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008e410: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008e420: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008e420: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008e460: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008e470: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │  0008e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008e4b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008e4c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ -0008e500: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  0008e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008e550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  0008e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008e640: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008e650: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ -0008e6a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0008e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0008e730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008e740: 7c74 6f20 5050 5e33 2920 2020 2020 2020  |to PP^3)       
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│ │ │ │ +0008e740: 2020 2020 7c0a 7c74 6f20 5050 5e33 2920      |.|to PP^3) 
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│ │ │ │ -0008e790: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
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│ │ │ │ +0008e790: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
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│ │ │ │ -0008e7e0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -0008e7f0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 746f  ==..  * *note to
│ │ │ │ -0008e800: 4d61 703a 2074 6f4d 6170 2c20 2d2d 2072  Map: toMap, -- r
│ │ │ │ -0008e810: 6174 696f 6e61 6c20 6d61 7020 6465 6669  ational map defi
│ │ │ │ -0008e820: 6e65 6420 6279 2061 206c 696e 6561 7220  ned by a linear 
│ │ │ │ -0008e830: 7379 7374 656d 0a20 202a 202a 6e6f 7465  system.  * *note
│ │ │ │ -0008e840: 2072 6174 696f 6e61 6c4d 6170 2849 6465   rationalMap(Ide
│ │ │ │ -0008e850: 616c 293a 2072 6174 696f 6e61 6c4d 6170  al): rationalMap
│ │ │ │ -0008e860: 5f6c 7049 6465 616c 5f63 6d5a 5a5f 636d  _lpIdeal_cmZZ_cm
│ │ │ │ -0008e870: 5a5a 5f72 702c 202d 2d20 6d61 6b65 7320  ZZ_rp, -- makes 
│ │ │ │ -0008e880: 610a 2020 2020 7261 7469 6f6e 616c 206d  a.    rational m
│ │ │ │ -0008e890: 6170 2066 726f 6d20 616e 2069 6465 616c  ap from an ideal
│ │ │ │ -0008e8a0: 0a20 202a 202a 6e6f 7465 2072 6174 696f  .  * *note ratio
│ │ │ │ -0008e8b0: 6e61 6c4d 6170 2854 616c 6c79 293a 2072  nalMap(Tally): r
│ │ │ │ -0008e8c0: 6174 696f 6e61 6c4d 6170 5f6c 7052 696e  ationalMap_lpRin
│ │ │ │ -0008e8d0: 675f 636d 5461 6c6c 795f 7270 2c20 2d2d  g_cmTally_rp, --
│ │ │ │ -0008e8e0: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ -0008e8f0: 2020 6465 6669 6e65 6420 6279 2061 6e20    defined by an 
│ │ │ │ -0008e900: 6566 6665 6374 6976 6520 6469 7669 736f  effective diviso
│ │ │ │ -0008e910: 720a 0a57 6179 7320 746f 2075 7365 2072  r..Ways to use r
│ │ │ │ -0008e920: 6174 696f 6e61 6c4d 6170 3a0a 3d3d 3d3d  ationalMap:.====
│ │ │ │ -0008e930: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0008e940: 3d3d 3d3d 0a0a 2020 2a20 2272 6174 696f  ====..  * "ratio
│ │ │ │ -0008e950: 6e61 6c4d 6170 284c 6973 7429 220a 2020  nalMap(List)".  
│ │ │ │ -0008e960: 2a20 2272 6174 696f 6e61 6c4d 6170 284d  * "rationalMap(M
│ │ │ │ -0008e970: 6174 7269 7829 220a 2020 2a20 2272 6174  atrix)".  * "rat
│ │ │ │ -0008e980: 696f 6e61 6c4d 6170 2852 696e 6729 220a  ionalMap(Ring)".
│ │ │ │ -0008e990: 2020 2a20 2272 6174 696f 6e61 6c4d 6170    * "rationalMap
│ │ │ │ -0008e9a0: 2852 696e 672c 5269 6e67 2922 0a20 202a  (Ring,Ring)".  *
│ │ │ │ -0008e9b0: 2022 7261 7469 6f6e 616c 4d61 7028 5269   "rationalMap(Ri
│ │ │ │ -0008e9c0: 6e67 2c52 696e 672c 4c69 7374 2922 0a20  ng,Ring,List)". 
│ │ │ │ -0008e9d0: 202a 2022 7261 7469 6f6e 616c 4d61 7028   * "rationalMap(
│ │ │ │ -0008e9e0: 5269 6e67 2c52 696e 672c 4d61 7472 6978  Ring,Ring,Matrix
│ │ │ │ -0008e9f0: 2922 0a20 202a 2022 7261 7469 6f6e 616c  )".  * "rational
│ │ │ │ -0008ea00: 4d61 7028 5269 6e67 4d61 7029 220a 2020  Map(RingMap)".  
│ │ │ │ -0008ea10: 2a20 2272 6174 696f 6e61 6c4d 6170 2849  * "rationalMap(I
│ │ │ │ -0008ea20: 6465 616c 2922 202d 2d20 7365 6520 2a6e  deal)" -- see *n
│ │ │ │ -0008ea30: 6f74 6520 7261 7469 6f6e 616c 4d61 7028  ote rationalMap(
│ │ │ │ -0008ea40: 4964 6561 6c2c 5a5a 2c5a 5a29 3a0a 2020  Ideal,ZZ,ZZ):.  
│ │ │ │ -0008ea50: 2020 7261 7469 6f6e 616c 4d61 705f 6c70    rationalMap_lp
│ │ │ │ -0008ea60: 4964 6561 6c5f 636d 5a5a 5f63 6d5a 5a5f  Ideal_cmZZ_cmZZ_
│ │ │ │ -0008ea70: 7270 2c20 2d2d 206d 616b 6573 2061 2072  rp, -- makes a r
│ │ │ │ -0008ea80: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ -0008ea90: 2061 6e20 6964 6561 6c0a 2020 2a20 2272   an ideal.  * "r
│ │ │ │ -0008eaa0: 6174 696f 6e61 6c4d 6170 2849 6465 616c  ationalMap(Ideal
│ │ │ │ -0008eab0: 2c4c 6973 7429 2220 2d2d 2073 6565 202a  ,List)" -- see *
│ │ │ │ -0008eac0: 6e6f 7465 2072 6174 696f 6e61 6c4d 6170  note rationalMap
│ │ │ │ -0008ead0: 2849 6465 616c 2c5a 5a2c 5a5a 293a 0a20  (Ideal,ZZ,ZZ):. 
│ │ │ │ -0008eae0: 2020 2072 6174 696f 6e61 6c4d 6170 5f6c     rationalMap_l
│ │ │ │ -0008eaf0: 7049 6465 616c 5f63 6d5a 5a5f 636d 5a5a  pIdeal_cmZZ_cmZZ
│ │ │ │ -0008eb00: 5f72 702c 202d 2d20 6d61 6b65 7320 6120  _rp, -- makes a 
│ │ │ │ -0008eb10: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ -0008eb20: 6d20 616e 2069 6465 616c 0a20 202a 2022  m an ideal.  * "
│ │ │ │ -0008eb30: 7261 7469 6f6e 616c 4d61 7028 4964 6561  rationalMap(Idea
│ │ │ │ -0008eb40: 6c2c 5a5a 2922 202d 2d20 7365 6520 2a6e  l,ZZ)" -- see *n
│ │ │ │ -0008eb50: 6f74 6520 7261 7469 6f6e 616c 4d61 7028  ote rationalMap(
│ │ │ │ -0008eb60: 4964 6561 6c2c 5a5a 2c5a 5a29 3a0a 2020  Ideal,ZZ,ZZ):.  
│ │ │ │ -0008eb70: 2020 7261 7469 6f6e 616c 4d61 705f 6c70    rationalMap_lp
│ │ │ │ -0008eb80: 4964 6561 6c5f 636d 5a5a 5f63 6d5a 5a5f  Ideal_cmZZ_cmZZ_
│ │ │ │ -0008eb90: 7270 2c20 2d2d 206d 616b 6573 2061 2072  rp, -- makes a r
│ │ │ │ -0008eba0: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ -0008ebb0: 2061 6e20 6964 6561 6c0a 2020 2a20 2a6e   an ideal.  * *n
│ │ │ │ -0008ebc0: 6f74 6520 7261 7469 6f6e 616c 4d61 7028  ote rationalMap(
│ │ │ │ -0008ebd0: 4964 6561 6c2c 5a5a 2c5a 5a29 3a20 7261  Ideal,ZZ,ZZ): ra
│ │ │ │ -0008ebe0: 7469 6f6e 616c 4d61 705f 6c70 4964 6561  tionalMap_lpIdea
│ │ │ │ -0008ebf0: 6c5f 636d 5a5a 5f63 6d5a 5a5f 7270 2c20  l_cmZZ_cmZZ_rp, 
│ │ │ │ -0008ec00: 2d2d 206d 616b 6573 0a20 2020 2061 2072  -- makes.    a r
│ │ │ │ -0008ec10: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ -0008ec20: 2061 6e20 6964 6561 6c0a 2020 2a20 2a6e   an ideal.  * *n
│ │ │ │ -0008ec30: 6f74 6520 7261 7469 6f6e 616c 4d61 7028  ote rationalMap(
│ │ │ │ -0008ec40: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c4c  PolynomialRing,L
│ │ │ │ -0008ec50: 6973 7429 3a0a 2020 2020 7261 7469 6f6e  ist):.    ration
│ │ │ │ -0008ec60: 616c 4d61 705f 6c70 506f 6c79 6e6f 6d69  alMap_lpPolynomi
│ │ │ │ -0008ec70: 616c 5269 6e67 5f63 6d4c 6973 745f 7270  alRing_cmList_rp
│ │ │ │ -0008ec80: 2c20 2d2d 2072 6174 696f 6e61 6c20 6d61  , -- rational ma
│ │ │ │ -0008ec90: 7020 6465 6669 6e65 6420 6279 2074 6865  p defined by the
│ │ │ │ -0008eca0: 0a20 2020 206c 696e 6561 7220 7379 7374  .    linear syst
│ │ │ │ -0008ecb0: 656d 206f 6620 6879 7065 7273 7572 6661  em of hypersurfa
│ │ │ │ -0008ecc0: 6365 7320 7061 7373 696e 6720 7468 726f  ces passing thro
│ │ │ │ -0008ecd0: 7567 6820 7261 6e64 6f6d 2070 6f69 6e74  ugh random point
│ │ │ │ -0008ece0: 7320 7769 7468 0a20 2020 206d 756c 7469  s with.    multi
│ │ │ │ -0008ecf0: 706c 6963 6974 790a 2020 2a20 2a6e 6f74  plicity.  * *not
│ │ │ │ -0008ed00: 6520 7261 7469 6f6e 616c 4d61 7028 5269  e rationalMap(Ri
│ │ │ │ -0008ed10: 6e67 2c54 616c 6c79 293a 2072 6174 696f  ng,Tally): ratio
│ │ │ │ -0008ed20: 6e61 6c4d 6170 5f6c 7052 696e 675f 636d  nalMap_lpRing_cm
│ │ │ │ -0008ed30: 5461 6c6c 795f 7270 2c20 2d2d 2072 6174  Tally_rp, -- rat
│ │ │ │ -0008ed40: 696f 6e61 6c0a 2020 2020 6d61 7020 6465  ional.    map de
│ │ │ │ -0008ed50: 6669 6e65 6420 6279 2061 6e20 6566 6665  fined by an effe
│ │ │ │ -0008ed60: 6374 6976 6520 6469 7669 736f 720a 2020  ctive divisor.  
│ │ │ │ -0008ed70: 2a20 2272 6174 696f 6e61 6c4d 6170 2854  * "rationalMap(T
│ │ │ │ -0008ed80: 616c 6c79 2922 202d 2d20 7365 6520 2a6e  ally)" -- see *n
│ │ │ │ -0008ed90: 6f74 6520 7261 7469 6f6e 616c 4d61 7028  ote rationalMap(
│ │ │ │ -0008eda0: 5269 6e67 2c54 616c 6c79 293a 0a20 2020  Ring,Tally):.   
│ │ │ │ -0008edb0: 2072 6174 696f 6e61 6c4d 6170 5f6c 7052   rationalMap_lpR
│ │ │ │ -0008edc0: 696e 675f 636d 5461 6c6c 795f 7270 2c20  ing_cmTally_rp, 
│ │ │ │ -0008edd0: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -0008ede0: 6465 6669 6e65 6420 6279 2061 6e20 6566  defined by an ef
│ │ │ │ -0008edf0: 6665 6374 6976 650a 2020 2020 6469 7669  fective.    divi
│ │ │ │ -0008ee00: 736f 720a 2020 2a20 2272 6174 696f 6e61  sor.  * "rationa
│ │ │ │ -0008ee10: 6c4d 6170 2852 6174 696f 6e61 6c4d 6170  lMap(RationalMap
│ │ │ │ -0008ee20: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -0008ee30: 7375 7065 7228 5261 7469 6f6e 616c 4d61  super(RationalMa
│ │ │ │ -0008ee40: 7029 3a0a 2020 2020 7375 7065 725f 6c70  p):.    super_lp
│ │ │ │ -0008ee50: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -0008ee60: 2d2d 2067 6574 2074 6865 2072 6174 696f  -- get the ratio
│ │ │ │ -0008ee70: 6e61 6c20 6d61 7020 7768 6f73 6520 7461  nal map whose ta
│ │ │ │ -0008ee80: 7267 6574 2069 7320 610a 2020 2020 7072  rget is a.    pr
│ │ │ │ -0008ee90: 6f6a 6563 7469 7665 2073 7061 6365 0a0a  ojective space..
│ │ │ │ -0008eea0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -0008eeb0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -0008eec0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -0008eed0: 7420 2a6e 6f74 6520 7261 7469 6f6e 616c  t *note rational
│ │ │ │ -0008eee0: 4d61 703a 2072 6174 696f 6e61 6c4d 6170  Map: rationalMap
│ │ │ │ -0008eef0: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -0008ef00: 686f 6420 6675 6e63 7469 6f6e 2077 6974  hod function wit
│ │ │ │ -0008ef10: 680a 6f70 7469 6f6e 733a 2028 4d61 6361  h.options: (Maca
│ │ │ │ -0008ef20: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ -0008ef30: 756e 6374 696f 6e57 6974 684f 7074 696f  unctionWithOptio
│ │ │ │ -0008ef40: 6e73 2c2e 0a1f 0a46 696c 653a 2043 7265  ns,....File: Cre
│ │ │ │ -0008ef50: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
│ │ │ │ -0008ef60: 2052 6174 696f 6e61 6c4d 6170 2021 2c20   RationalMap !, 
│ │ │ │ -0008ef70: 4e65 7874 3a20 5261 7469 6f6e 616c 4d61  Next: RationalMa
│ │ │ │ -0008ef80: 7020 5f73 7420 5261 7469 6f6e 616c 4d61  p _st RationalMa
│ │ │ │ -0008ef90: 702c 2050 7265 763a 2072 6174 696f 6e61  p, Prev: rationa
│ │ │ │ -0008efa0: 6c4d 6170 2c20 5570 3a20 546f 700a 0a52  lMap, Up: Top..R
│ │ │ │ -0008efb0: 6174 696f 6e61 6c4d 6170 2021 202d 2d20  ationalMap ! -- 
│ │ │ │ -0008efc0: 6361 6c63 756c 6174 6573 2065 7665 7279  calculates every
│ │ │ │ -0008efd0: 2070 6f73 7369 626c 6520 7468 696e 670a   possible thing.
│ │ │ │ -0008efe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0008e7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0008e7e0: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ +0008e7f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +0008e800: 6f74 6520 746f 4d61 703a 2074 6f4d 6170  ote toMap: toMap
│ │ │ │ +0008e810: 2c20 2d2d 2072 6174 696f 6e61 6c20 6d61  , -- rational ma
│ │ │ │ +0008e820: 7020 6465 6669 6e65 6420 6279 2061 206c  p defined by a l
│ │ │ │ +0008e830: 696e 6561 7220 7379 7374 656d 0a20 202a  inear system.  *
│ │ │ │ +0008e840: 202a 6e6f 7465 2072 6174 696f 6e61 6c4d   *note rationalM
│ │ │ │ +0008e850: 6170 2849 6465 616c 293a 2072 6174 696f  ap(Ideal): ratio
│ │ │ │ +0008e860: 6e61 6c4d 6170 5f6c 7049 6465 616c 5f63  nalMap_lpIdeal_c
│ │ │ │ +0008e870: 6d5a 5a5f 636d 5a5a 5f72 702c 202d 2d20  mZZ_cmZZ_rp, -- 
│ │ │ │ +0008e880: 6d61 6b65 7320 610a 2020 2020 7261 7469  makes a.    rati
│ │ │ │ +0008e890: 6f6e 616c 206d 6170 2066 726f 6d20 616e  onal map from an
│ │ │ │ +0008e8a0: 2069 6465 616c 0a20 202a 202a 6e6f 7465   ideal.  * *note
│ │ │ │ +0008e8b0: 2072 6174 696f 6e61 6c4d 6170 2854 616c   rationalMap(Tal
│ │ │ │ +0008e8c0: 6c79 293a 2072 6174 696f 6e61 6c4d 6170  ly): rationalMap
│ │ │ │ +0008e8d0: 5f6c 7052 696e 675f 636d 5461 6c6c 795f  _lpRing_cmTally_
│ │ │ │ +0008e8e0: 7270 2c20 2d2d 2072 6174 696f 6e61 6c20  rp, -- rational 
│ │ │ │ +0008e8f0: 6d61 700a 2020 2020 6465 6669 6e65 6420  map.    defined 
│ │ │ │ +0008e900: 6279 2061 6e20 6566 6665 6374 6976 6520  by an effective 
│ │ │ │ +0008e910: 6469 7669 736f 720a 0a57 6179 7320 746f  divisor..Ways to
│ │ │ │ +0008e920: 2075 7365 2072 6174 696f 6e61 6c4d 6170   use rationalMap
│ │ │ │ +0008e930: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +0008e940: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0008e950: 2272 6174 696f 6e61 6c4d 6170 284c 6973  "rationalMap(Lis
│ │ │ │ +0008e960: 7429 220a 2020 2a20 2272 6174 696f 6e61  t)".  * "rationa
│ │ │ │ +0008e970: 6c4d 6170 284d 6174 7269 7829 220a 2020  lMap(Matrix)".  
│ │ │ │ +0008e980: 2a20 2272 6174 696f 6e61 6c4d 6170 2852  * "rationalMap(R
│ │ │ │ +0008e990: 696e 6729 220a 2020 2a20 2272 6174 696f  ing)".  * "ratio
│ │ │ │ +0008e9a0: 6e61 6c4d 6170 2852 696e 672c 5269 6e67  nalMap(Ring,Ring
│ │ │ │ +0008e9b0: 2922 0a20 202a 2022 7261 7469 6f6e 616c  )".  * "rational
│ │ │ │ +0008e9c0: 4d61 7028 5269 6e67 2c52 696e 672c 4c69  Map(Ring,Ring,Li
│ │ │ │ +0008e9d0: 7374 2922 0a20 202a 2022 7261 7469 6f6e  st)".  * "ration
│ │ │ │ +0008e9e0: 616c 4d61 7028 5269 6e67 2c52 696e 672c  alMap(Ring,Ring,
│ │ │ │ +0008e9f0: 4d61 7472 6978 2922 0a20 202a 2022 7261  Matrix)".  * "ra
│ │ │ │ +0008ea00: 7469 6f6e 616c 4d61 7028 5269 6e67 4d61  tionalMap(RingMa
│ │ │ │ +0008ea10: 7029 220a 2020 2a20 2272 6174 696f 6e61  p)".  * "rationa
│ │ │ │ +0008ea20: 6c4d 6170 2849 6465 616c 2922 202d 2d20  lMap(Ideal)" -- 
│ │ │ │ +0008ea30: 7365 6520 2a6e 6f74 6520 7261 7469 6f6e  see *note ration
│ │ │ │ +0008ea40: 616c 4d61 7028 4964 6561 6c2c 5a5a 2c5a  alMap(Ideal,ZZ,Z
│ │ │ │ +0008ea50: 5a29 3a0a 2020 2020 7261 7469 6f6e 616c  Z):.    rational
│ │ │ │ +0008ea60: 4d61 705f 6c70 4964 6561 6c5f 636d 5a5a  Map_lpIdeal_cmZZ
│ │ │ │ +0008ea70: 5f63 6d5a 5a5f 7270 2c20 2d2d 206d 616b  _cmZZ_rp, -- mak
│ │ │ │ +0008ea80: 6573 2061 2072 6174 696f 6e61 6c20 6d61  es a rational ma
│ │ │ │ +0008ea90: 7020 6672 6f6d 2061 6e20 6964 6561 6c0a  p from an ideal.
│ │ │ │ +0008eaa0: 2020 2a20 2272 6174 696f 6e61 6c4d 6170    * "rationalMap
│ │ │ │ +0008eab0: 2849 6465 616c 2c4c 6973 7429 2220 2d2d  (Ideal,List)" --
│ │ │ │ +0008eac0: 2073 6565 202a 6e6f 7465 2072 6174 696f   see *note ratio
│ │ │ │ +0008ead0: 6e61 6c4d 6170 2849 6465 616c 2c5a 5a2c  nalMap(Ideal,ZZ,
│ │ │ │ +0008eae0: 5a5a 293a 0a20 2020 2072 6174 696f 6e61  ZZ):.    rationa
│ │ │ │ +0008eaf0: 6c4d 6170 5f6c 7049 6465 616c 5f63 6d5a  lMap_lpIdeal_cmZ
│ │ │ │ +0008eb00: 5a5f 636d 5a5a 5f72 702c 202d 2d20 6d61  Z_cmZZ_rp, -- ma
│ │ │ │ +0008eb10: 6b65 7320 6120 7261 7469 6f6e 616c 206d  kes a rational m
│ │ │ │ +0008eb20: 6170 2066 726f 6d20 616e 2069 6465 616c  ap from an ideal
│ │ │ │ +0008eb30: 0a20 202a 2022 7261 7469 6f6e 616c 4d61  .  * "rationalMa
│ │ │ │ +0008eb40: 7028 4964 6561 6c2c 5a5a 2922 202d 2d20  p(Ideal,ZZ)" -- 
│ │ │ │ +0008eb50: 7365 6520 2a6e 6f74 6520 7261 7469 6f6e  see *note ration
│ │ │ │ +0008eb60: 616c 4d61 7028 4964 6561 6c2c 5a5a 2c5a  alMap(Ideal,ZZ,Z
│ │ │ │ +0008eb70: 5a29 3a0a 2020 2020 7261 7469 6f6e 616c  Z):.    rational
│ │ │ │ +0008eb80: 4d61 705f 6c70 4964 6561 6c5f 636d 5a5a  Map_lpIdeal_cmZZ
│ │ │ │ +0008eb90: 5f63 6d5a 5a5f 7270 2c20 2d2d 206d 616b  _cmZZ_rp, -- mak
│ │ │ │ +0008eba0: 6573 2061 2072 6174 696f 6e61 6c20 6d61  es a rational ma
│ │ │ │ +0008ebb0: 7020 6672 6f6d 2061 6e20 6964 6561 6c0a  p from an ideal.
│ │ │ │ +0008ebc0: 2020 2a20 2a6e 6f74 6520 7261 7469 6f6e    * *note ration
│ │ │ │ +0008ebd0: 616c 4d61 7028 4964 6561 6c2c 5a5a 2c5a  alMap(Ideal,ZZ,Z
│ │ │ │ +0008ebe0: 5a29 3a20 7261 7469 6f6e 616c 4d61 705f  Z): rationalMap_
│ │ │ │ +0008ebf0: 6c70 4964 6561 6c5f 636d 5a5a 5f63 6d5a  lpIdeal_cmZZ_cmZ
│ │ │ │ +0008ec00: 5a5f 7270 2c20 2d2d 206d 616b 6573 0a20  Z_rp, -- makes. 
│ │ │ │ +0008ec10: 2020 2061 2072 6174 696f 6e61 6c20 6d61     a rational ma
│ │ │ │ +0008ec20: 7020 6672 6f6d 2061 6e20 6964 6561 6c0a  p from an ideal.
│ │ │ │ +0008ec30: 2020 2a20 2a6e 6f74 6520 7261 7469 6f6e    * *note ration
│ │ │ │ +0008ec40: 616c 4d61 7028 506f 6c79 6e6f 6d69 616c  alMap(Polynomial
│ │ │ │ +0008ec50: 5269 6e67 2c4c 6973 7429 3a0a 2020 2020  Ring,List):.    
│ │ │ │ +0008ec60: 7261 7469 6f6e 616c 4d61 705f 6c70 506f  rationalMap_lpPo
│ │ │ │ +0008ec70: 6c79 6e6f 6d69 616c 5269 6e67 5f63 6d4c  lynomialRing_cmL
│ │ │ │ +0008ec80: 6973 745f 7270 2c20 2d2d 2072 6174 696f  ist_rp, -- ratio
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│ │ │ │ +0008ecb0: 7220 7379 7374 656d 206f 6620 6879 7065  r system of hype
│ │ │ │ +0008ecc0: 7273 7572 6661 6365 7320 7061 7373 696e  rsurfaces passin
│ │ │ │ +0008ecd0: 6720 7468 726f 7567 6820 7261 6e64 6f6d  g through random
│ │ │ │ +0008ece0: 2070 6f69 6e74 7320 7769 7468 0a20 2020   points with.   
│ │ │ │ +0008ecf0: 206d 756c 7469 706c 6963 6974 790a 2020   multiplicity.  
│ │ │ │ +0008ed00: 2a20 2a6e 6f74 6520 7261 7469 6f6e 616c  * *note rational
│ │ │ │ +0008ed10: 4d61 7028 5269 6e67 2c54 616c 6c79 293a  Map(Ring,Tally):
│ │ │ │ +0008ed20: 2072 6174 696f 6e61 6c4d 6170 5f6c 7052   rationalMap_lpR
│ │ │ │ +0008ed30: 696e 675f 636d 5461 6c6c 795f 7270 2c20  ing_cmTally_rp, 
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│ │ │ │ +0008ed50: 6d61 7020 6465 6669 6e65 6420 6279 2061  map defined by a
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│ │ │ │ +0008ed80: 6c4d 6170 2854 616c 6c79 2922 202d 2d20  lMap(Tally)" -- 
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│ │ │ │ +0008eda0: 616c 4d61 7028 5269 6e67 2c54 616c 6c79  alMap(Ring,Tally
│ │ │ │ +0008edb0: 293a 0a20 2020 2072 6174 696f 6e61 6c4d  ):.    rationalM
│ │ │ │ +0008edc0: 6170 5f6c 7052 696e 675f 636d 5461 6c6c  ap_lpRing_cmTall
│ │ │ │ +0008edd0: 795f 7270 2c20 2d2d 2072 6174 696f 6e61  y_rp, -- rationa
│ │ │ │ +0008ede0: 6c20 6d61 7020 6465 6669 6e65 6420 6279  l map defined by
│ │ │ │ +0008edf0: 2061 6e20 6566 6665 6374 6976 650a 2020   an effective.  
│ │ │ │ +0008ee00: 2020 6469 7669 736f 720a 2020 2a20 2272    divisor.  * "r
│ │ │ │ +0008ee10: 6174 696f 6e61 6c4d 6170 2852 6174 696f  ationalMap(Ratio
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│ │ │ │ +0008ee30: 2a6e 6f74 6520 7375 7065 7228 5261 7469  *note super(Rati
│ │ │ │ +0008ee40: 6f6e 616c 4d61 7029 3a0a 2020 2020 7375  onalMap):.    su
│ │ │ │ +0008ee50: 7065 725f 6c70 5261 7469 6f6e 616c 4d61  per_lpRationalMa
│ │ │ │ +0008ee60: 705f 7270 2c20 2d2d 2067 6574 2074 6865  p_rp, -- get the
│ │ │ │ +0008ee70: 2072 6174 696f 6e61 6c20 6d61 7020 7768   rational map wh
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│ │ │ │ +0008ee90: 2020 2020 7072 6f6a 6563 7469 7665 2073      projective s
│ │ │ │ +0008eea0: 7061 6365 0a0a 466f 7220 7468 6520 7072  pace..For the pr
│ │ │ │ +0008eeb0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +0008eec0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
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│ │ │ │ +0008eee0: 7469 6f6e 616c 4d61 703a 2072 6174 696f  tionalMap: ratio
│ │ │ │ +0008eef0: 6e61 6c4d 6170 2c20 6973 2061 202a 6e6f  nalMap, is a *no
│ │ │ │ +0008ef00: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
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│ │ │ │ +0008ef20: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +0008ef30: 6574 686f 6446 756e 6374 696f 6e57 6974  ethodFunctionWit
│ │ │ │ +0008ef40: 684f 7074 696f 6e73 2c2e 0a1f 0a46 696c  hOptions,....Fil
│ │ │ │ +0008ef50: 653a 2043 7265 6d6f 6e61 2e69 6e66 6f2c  e: Cremona.info,
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│ │ │ │ +0008ef80: 6f6e 616c 4d61 7020 5f73 7420 5261 7469  onalMap _st Rati
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│ │ │ │ +0008efc0: 2021 202d 2d20 6361 6c63 756c 6174 6573   ! -- calculates
│ │ │ │ +0008efd0: 2065 7665 7279 2070 6f73 7369 626c 6520   every possible 
│ │ │ │ +0008efe0: 7468 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a  thing.**********
│ │ │ │  0008eff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0008f000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0008f010: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ -0008f020: 3d3d 3d0a 0a20 202a 204f 7065 7261 746f  ===..  * Operato
│ │ │ │ -0008f030: 723a 202a 6e6f 7465 2021 3a20 284d 6163  r: *note !: (Mac
│ │ │ │ -0008f040: 6175 6c61 7932 446f 6329 212c 0a20 202a  aulay2Doc)!,.  *
│ │ │ │ -0008f050: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -0008f060: 2070 6869 210a 2020 2a20 496e 7075 7473   phi!.  * Inputs
│ │ │ │ -0008f070: 3a0a 2020 2020 2020 2a20 7068 692c 2061  :.      * phi, a
│ │ │ │ -0008f080: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ -0008f090: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ -0008f0a0: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ -0008f0b0: 2020 2020 202a 2061 202a 6e6f 7465 2072       * a *note r
│ │ │ │ -0008f0c0: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ -0008f0d0: 696f 6e61 6c4d 6170 2c2c 2074 6865 2073  ionalMap,, the s
│ │ │ │ -0008f0e0: 616d 6520 7261 7469 6f6e 616c 206d 6170  ame rational map
│ │ │ │ -0008f0f0: 2070 6869 0a0a 4465 7363 7269 7074 696f   phi..Descriptio
│ │ │ │ -0008f100: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
│ │ │ │ -0008f110: 6869 7320 6d65 7468 6f64 2028 6d61 696e  his method (main
│ │ │ │ -0008f120: 6c79 2075 7365 6420 666f 7220 7465 7374  ly used for test
│ │ │ │ -0008f130: 7329 2061 7070 6c69 6573 2061 6c6d 6f73  s) applies almos
│ │ │ │ -0008f140: 7420 616c 6c20 7468 6520 6465 7465 726d  t all the determ
│ │ │ │ -0008f150: 696e 6973 7469 630a 6d65 7468 6f64 7320  inistic.methods 
│ │ │ │ -0008f160: 7468 6174 2061 7265 2061 7661 696c 6162  that are availab
│ │ │ │ -0008f170: 6c65 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  le...+----------
│ │ │ │ +0008f010: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +0008f020: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 204f  .========..  * O
│ │ │ │ +0008f030: 7065 7261 746f 723a 202a 6e6f 7465 2021  perator: *note !
│ │ │ │ +0008f040: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0008f050: 212c 0a20 202a 2055 7361 6765 3a20 0a20  !,.  * Usage: . 
│ │ │ │ +0008f060: 2020 2020 2020 2070 6869 210a 2020 2a20         phi!.  * 
│ │ │ │ +0008f070: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +0008f080: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ +0008f090: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +0008f0a0: 6e61 6c4d 6170 2c0a 2020 2a20 4f75 7470  nalMap,.  * Outp
│ │ │ │ +0008f0b0: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +0008f0c0: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ +0008f0d0: 703a 2052 6174 696f 6e61 6c4d 6170 2c2c  p: RationalMap,,
│ │ │ │ +0008f0e0: 2074 6865 2073 616d 6520 7261 7469 6f6e   the same ration
│ │ │ │ +0008f0f0: 616c 206d 6170 2070 6869 0a0a 4465 7363  al map phi..Desc
│ │ │ │ +0008f100: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0008f110: 3d3d 3d0a 0a54 6869 7320 6d65 7468 6f64  ===..This method
│ │ │ │ +0008f120: 2028 6d61 696e 6c79 2075 7365 6420 666f   (mainly used fo
│ │ │ │ +0008f130: 7220 7465 7374 7329 2061 7070 6c69 6573  r tests) applies
│ │ │ │ +0008f140: 2061 6c6d 6f73 7420 616c 6c20 7468 6520   almost all the 
│ │ │ │ +0008f150: 6465 7465 726d 696e 6973 7469 630a 6d65  deterministic.me
│ │ │ │ +0008f160: 7468 6f64 7320 7468 6174 2061 7265 2061  thods that are a
│ │ │ │ +0008f170: 7661 696c 6162 6c65 2e0a 0a2b 2d2d 2d2d  vailable...+----
│ │ │ │  0008f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f1c0: 2d2d 2d2b 0a7c 6931 203a 2051 515b 785f  ---+.|i1 : QQ[x_
│ │ │ │ -0008f1d0: 302e 2e78 5f35 5d3b 2070 6869 203d 2072  0..x_5]; phi = r
│ │ │ │ -0008f1e0: 6174 696f 6e61 6c4d 6170 2020 2020 2020  ationalMap      
│ │ │ │ +0008f1c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +0008f1d0: 2051 515b 785f 302e 2e78 5f35 5d3b 2070   QQ[x_0..x_5]; p
│ │ │ │ +0008f1e0: 6869 203d 2072 6174 696f 6e61 6c4d 6170  hi = rationalMap
│ │ │ │  0008f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f210: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008f210: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0008f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f260: 2020 207c 0a7c 6f32 203a 2052 6174 696f     |.|o2 : Ratio
│ │ │ │ -0008f270: 6e61 6c4d 6170 2028 7175 6164 7261 7469  nalMap (quadrati
│ │ │ │ -0008f280: 6320 7261 7469 6f6e 616c 2020 2020 2020  c rational      
│ │ │ │ +0008f260: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ +0008f270: 2052 6174 696f 6e61 6c4d 6170 2028 7175   RationalMap (qu
│ │ │ │ +0008f280: 6164 7261 7469 6320 7261 7469 6f6e 616c  adratic rational
│ │ │ │  0008f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f2b0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +0008f2b0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │  0008f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f300: 2d2d 2d7c 0a7c 7b78 5f34 5e32 2d78 5f33  ---|.|{x_4^2-x_3
│ │ │ │ -0008f310: 2a78 5f35 2c78 5f32 2a78 5f34 2d78 5f31  *x_5,x_2*x_4-x_1
│ │ │ │ -0008f320: 2a78 5f35 2c78 5f32 2a78 5f33 2d78 5f31  *x_5,x_2*x_3-x_1
│ │ │ │ -0008f330: 2a78 5f34 2c78 5f32 5e32 2d78 5f30 2a78  *x_4,x_2^2-x_0*x
│ │ │ │ -0008f340: 5f35 2c78 5f31 2a78 5f32 2d78 5f30 2a78  _5,x_1*x_2-x_0*x
│ │ │ │ -0008f350: 5f34 2c7c 0a7c 2020 2020 2020 2020 2020  _4,|.|          
│ │ │ │ +0008f300: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7b78 5f34  ---------|.|{x_4
│ │ │ │ +0008f310: 5e32 2d78 5f33 2a78 5f35 2c78 5f32 2a78  ^2-x_3*x_5,x_2*x
│ │ │ │ +0008f320: 5f34 2d78 5f31 2a78 5f35 2c78 5f32 2a78  _4-x_1*x_5,x_2*x
│ │ │ │ +0008f330: 5f33 2d78 5f31 2a78 5f34 2c78 5f32 5e32  _3-x_1*x_4,x_2^2
│ │ │ │ +0008f340: 2d78 5f30 2a78 5f35 2c78 5f31 2a78 5f32  -x_0*x_5,x_1*x_2
│ │ │ │ +0008f350: 2d78 5f30 2a78 5f34 2c7c 0a7c 2020 2020  -x_0*x_4,|.|    
│ │ │ │  0008f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f3a0: 2020 207c 0a7c 6d61 7020 6672 6f6d 2050     |.|map from P
│ │ │ │ -0008f3b0: 505e 3520 746f 2050 505e 3529 2020 2020  P^5 to PP^5)    
│ │ │ │ -0008f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008f3a0: 2020 2020 2020 2020 207c 0a7c 6d61 7020           |.|map 
│ │ │ │ +0008f3b0: 6672 6f6d 2050 505e 3520 746f 2050 505e  from PP^5 to PP^
│ │ │ │ +0008f3c0: 3529 2020 2020 2020 2020 2020 2020 2020  5)              
│ │ │ │  0008f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f3f0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +0008f3f0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │  0008f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f440: 2d2d 2d7c 0a7c 785f 315e 322d 785f 302a  ---|.|x_1^2-x_0*
│ │ │ │ -0008f450: 785f 337d 3b20 2020 2020 2020 2020 2020  x_3};           
│ │ │ │ +0008f440: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 785f 315e  ---------|.|x_1^
│ │ │ │ +0008f450: 322d 785f 302a 785f 337d 3b20 2020 2020  2-x_0*x_3};     
│ │ │ │  0008f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f490: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0008f490: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0008f4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f4e0: 2d2d 2d2b 0a7c 6933 203a 2064 6573 6372  ---+.|i3 : descr
│ │ │ │ -0008f4f0: 6962 6520 7068 6920 2020 2020 2020 2020  ibe phi         
│ │ │ │ +0008f4e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +0008f4f0: 2064 6573 6372 6962 6520 7068 6920 2020   describe phi   
│ │ │ │  0008f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f530: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008f530: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0008f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f580: 2020 207c 0a7c 6f33 203d 2072 6174 696f     |.|o3 = ratio
│ │ │ │ -0008f590: 6e61 6c20 6d61 7020 6465 6669 6e65 6420  nal map defined 
│ │ │ │ -0008f5a0: 6279 2066 6f72 6d73 206f 6620 6465 6772  by forms of degr
│ │ │ │ -0008f5b0: 6565 2032 2020 2020 2020 2020 2020 2020  ee 2            
│ │ │ │ +0008f580: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +0008f590: 2072 6174 696f 6e61 6c20 6d61 7020 6465   rational map de
│ │ │ │ +0008f5a0: 6669 6e65 6420 6279 2066 6f72 6d73 206f  fined by forms o
│ │ │ │ +0008f5b0: 6620 6465 6772 6565 2032 2020 2020 2020  f degree 2      
│ │ │ │  0008f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f5d0: 2020 207c 0a7c 2020 2020 2073 6f75 7263     |.|     sourc
│ │ │ │ -0008f5e0: 6520 7661 7269 6574 793a 2050 505e 3520  e variety: PP^5 
│ │ │ │ -0008f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008f5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008f5e0: 2073 6f75 7263 6520 7661 7269 6574 793a   source variety:
│ │ │ │ +0008f5f0: 2050 505e 3520 2020 2020 2020 2020 2020   PP^5           
│ │ │ │  0008f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f620: 2020 207c 0a7c 2020 2020 2074 6172 6765     |.|     targe
│ │ │ │ -0008f630: 7420 7661 7269 6574 793a 2050 505e 3520  t variety: PP^5 
│ │ │ │ -0008f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008f620: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008f630: 2074 6172 6765 7420 7661 7269 6574 793a   target variety:
│ │ │ │ +0008f640: 2050 505e 3520 2020 2020 2020 2020 2020   PP^5           
│ │ │ │  0008f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f670: 2020 207c 0a7c 2020 2020 2063 6f65 6666     |.|     coeff
│ │ │ │ -0008f680: 6963 6965 6e74 2072 696e 673a 2051 5120  icient ring: QQ 
│ │ │ │ -0008f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008f670: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008f680: 2063 6f65 6666 6963 6965 6e74 2072 696e   coefficient rin
│ │ │ │ +0008f690: 673a 2051 5120 2020 2020 2020 2020 2020  g: QQ           
│ │ │ │  0008f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f6c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0008f6c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0008f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f710: 2d2d 2d2b 0a7c 6934 203a 2074 696d 6520  ---+.|i4 : time 
│ │ │ │ -0008f720: 7068 6921 203b 2020 2020 2020 2020 2020  phi! ;          
│ │ │ │ +0008f710: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +0008f720: 2074 696d 6520 7068 6921 203b 2020 2020   time phi! ;    
│ │ │ │  0008f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f760: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -0008f770: 2e31 3335 3139 3173 2028 6370 7529 3b20  .135191s (cpu); 
│ │ │ │ -0008f780: 302e 3037 3632 3231 3273 2028 7468 7265  0.0762212s (thre
│ │ │ │ -0008f790: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ -0008f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f7b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008f760: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +0008f770: 7573 6564 2030 2e32 3032 3832 3373 2028  used 0.202823s (
│ │ │ │ +0008f780: 6370 7529 3b20 302e 3038 3636 3230 3873  cpu); 0.0866208s
│ │ │ │ +0008f790: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +0008f7a0: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +0008f7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0008f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f800: 2020 207c 0a7c 6f34 203a 2052 6174 696f     |.|o4 : Ratio
│ │ │ │ -0008f810: 6e61 6c4d 6170 2028 4372 656d 6f6e 6120  nalMap (Cremona 
│ │ │ │ -0008f820: 7472 616e 7366 6f72 6d61 7469 6f6e 206f  transformation o
│ │ │ │ -0008f830: 6620 5050 5e35 206f 6620 7479 7065 2028  f PP^5 of type (
│ │ │ │ -0008f840: 322c 3229 2920 2020 2020 2020 2020 2020  2,2))           
│ │ │ │ -0008f850: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0008f800: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ +0008f810: 2052 6174 696f 6e61 6c4d 6170 2028 4372   RationalMap (Cr
│ │ │ │ +0008f820: 656d 6f6e 6120 7472 616e 7366 6f72 6d61  emona transforma
│ │ │ │ +0008f830: 7469 6f6e 206f 6620 5050 5e35 206f 6620  tion of PP^5 of 
│ │ │ │ +0008f840: 7479 7065 2028 322c 3229 2920 2020 2020  type (2,2))     
│ │ │ │ +0008f850: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0008f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f8a0: 2d2d 2d2b 0a7c 6935 203a 2064 6573 6372  ---+.|i5 : descr
│ │ │ │ -0008f8b0: 6962 6520 7068 6920 2020 2020 2020 2020  ibe phi         
│ │ │ │ +0008f8a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +0008f8b0: 2064 6573 6372 6962 6520 7068 6920 2020   describe phi   
│ │ │ │  0008f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f8f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008f8f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0008f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f940: 2020 207c 0a7c 6f35 203d 2072 6174 696f     |.|o5 = ratio
│ │ │ │ -0008f950: 6e61 6c20 6d61 7020 6465 6669 6e65 6420  nal map defined 
│ │ │ │ -0008f960: 6279 2066 6f72 6d73 206f 6620 6465 6772  by forms of degr
│ │ │ │ -0008f970: 6565 2032 2020 2020 2020 2020 2020 2020  ee 2            
│ │ │ │ +0008f940: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +0008f950: 2072 6174 696f 6e61 6c20 6d61 7020 6465   rational map de
│ │ │ │ +0008f960: 6669 6e65 6420 6279 2066 6f72 6d73 206f  fined by forms o
│ │ │ │ +0008f970: 6620 6465 6772 6565 2032 2020 2020 2020  f degree 2      
│ │ │ │  0008f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f990: 2020 207c 0a7c 2020 2020 2073 6f75 7263     |.|     sourc
│ │ │ │ -0008f9a0: 6520 7661 7269 6574 793a 2050 505e 3520  e variety: PP^5 
│ │ │ │ -0008f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008f990: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008f9a0: 2073 6f75 7263 6520 7661 7269 6574 793a   source variety:
│ │ │ │ +0008f9b0: 2050 505e 3520 2020 2020 2020 2020 2020   PP^5           
│ │ │ │  0008f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f9e0: 2020 207c 0a7c 2020 2020 2074 6172 6765     |.|     targe
│ │ │ │ -0008f9f0: 7420 7661 7269 6574 793a 2050 505e 3520  t variety: PP^5 
│ │ │ │ -0008fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008f9e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008f9f0: 2074 6172 6765 7420 7661 7269 6574 793a   target variety:
│ │ │ │ +0008fa00: 2050 505e 3520 2020 2020 2020 2020 2020   PP^5           
│ │ │ │  0008fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fa30: 2020 207c 0a7c 2020 2020 2064 6f6d 696e     |.|     domin
│ │ │ │ -0008fa40: 616e 6365 3a20 7472 7565 2020 2020 2020  ance: true      
│ │ │ │ +0008fa30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008fa40: 2064 6f6d 696e 616e 6365 3a20 7472 7565   dominance: true
│ │ │ │  0008fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fa80: 2020 207c 0a7c 2020 2020 2062 6972 6174     |.|     birat
│ │ │ │ -0008fa90: 696f 6e61 6c69 7479 3a20 7472 7565 2028  ionality: true (
│ │ │ │ -0008faa0: 7468 6520 696e 7665 7273 6520 6d61 7020  the inverse map 
│ │ │ │ -0008fab0: 6973 2061 6c72 6561 6479 2063 616c 6375  is already calcu
│ │ │ │ -0008fac0: 6c61 7465 6429 2020 2020 2020 2020 2020  lated)          
│ │ │ │ -0008fad0: 2020 207c 0a7c 2020 2020 2070 726f 6a65     |.|     proje
│ │ │ │ -0008fae0: 6374 6976 6520 6465 6772 6565 733a 207b  ctive degrees: {
│ │ │ │ -0008faf0: 312c 2032 2c20 342c 2034 2c20 322c 2031  1, 2, 4, 4, 2, 1
│ │ │ │ -0008fb00: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0008fa80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008fa90: 2062 6972 6174 696f 6e61 6c69 7479 3a20   birationality: 
│ │ │ │ +0008faa0: 7472 7565 2028 7468 6520 696e 7665 7273  true (the invers
│ │ │ │ +0008fab0: 6520 6d61 7020 6973 2061 6c72 6561 6479  e map is already
│ │ │ │ +0008fac0: 2063 616c 6375 6c61 7465 6429 2020 2020   calculated)    
│ │ │ │ +0008fad0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008fae0: 2070 726f 6a65 6374 6976 6520 6465 6772   projective degr
│ │ │ │ +0008faf0: 6565 733a 207b 312c 2032 2c20 342c 2034  ees: {1, 2, 4, 4
│ │ │ │ +0008fb00: 2c20 322c 2031 7d20 2020 2020 2020 2020  , 2, 1}         
│ │ │ │  0008fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fb20: 2020 207c 0a7c 2020 2020 206e 756d 6265     |.|     numbe
│ │ │ │ -0008fb30: 7220 6f66 206d 696e 696d 616c 2072 6570  r of minimal rep
│ │ │ │ -0008fb40: 7265 7365 6e74 6174 6976 6573 3a20 3120  resentatives: 1 
│ │ │ │ -0008fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008fb20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008fb30: 206e 756d 6265 7220 6f66 206d 696e 696d   number of minim
│ │ │ │ +0008fb40: 616c 2072 6570 7265 7365 6e74 6174 6976  al representativ
│ │ │ │ +0008fb50: 6573 3a20 3120 2020 2020 2020 2020 2020  es: 1           
│ │ │ │  0008fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fb70: 2020 207c 0a7c 2020 2020 2064 696d 656e     |.|     dimen
│ │ │ │ -0008fb80: 7369 6f6e 2062 6173 6520 6c6f 6375 733a  sion base locus:
│ │ │ │ -0008fb90: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0008fb70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008fb80: 2064 696d 656e 7369 6f6e 2062 6173 6520   dimension base 
│ │ │ │ +0008fb90: 6c6f 6375 733a 2032 2020 2020 2020 2020  locus: 2        
│ │ │ │  0008fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fbc0: 2020 207c 0a7c 2020 2020 2064 6567 7265     |.|     degre
│ │ │ │ -0008fbd0: 6520 6261 7365 206c 6f63 7573 3a20 3420  e base locus: 4 
│ │ │ │ -0008fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008fbc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008fbd0: 2064 6567 7265 6520 6261 7365 206c 6f63   degree base loc
│ │ │ │ +0008fbe0: 7573 3a20 3420 2020 2020 2020 2020 2020  us: 4           
│ │ │ │  0008fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fc10: 2020 207c 0a7c 2020 2020 2063 6f65 6666     |.|     coeff
│ │ │ │ -0008fc20: 6963 6965 6e74 2072 696e 673a 2051 5120  icient ring: QQ 
│ │ │ │ -0008fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008fc10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0008fc20: 2063 6f65 6666 6963 6965 6e74 2072 696e   coefficient rin
│ │ │ │ +0008fc30: 673a 2051 5120 2020 2020 2020 2020 2020  g: QQ           
│ │ │ │  0008fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fc60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0008fc60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0008fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008fc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008fca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008fcb0: 2d2d 2d2b 0a7c 6936 203a 2051 515b 785f  ---+.|i6 : QQ[x_
│ │ │ │ -0008fcc0: 302e 2e78 5f34 5d3b 2070 6869 203d 2072  0..x_4]; phi = r
│ │ │ │ -0008fcd0: 6174 696f 6e61 6c4d 6170 2020 2020 2020  ationalMap      
│ │ │ │ +0008fcb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +0008fcc0: 2051 515b 785f 302e 2e78 5f34 5d3b 2070   QQ[x_0..x_4]; p
│ │ │ │ +0008fcd0: 6869 203d 2072 6174 696f 6e61 6c4d 6170  hi = rationalMap
│ │ │ │  0008fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fd00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008fd00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0008fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fd50: 2020 207c 0a7c 6f37 203a 2052 6174 696f     |.|o7 : Ratio
│ │ │ │ -0008fd60: 6e61 6c4d 6170 2028 7175 6164 7261 7469  nalMap (quadrati
│ │ │ │ -0008fd70: 6320 7261 7469 6f6e 616c 2020 2020 2020  c rational      
│ │ │ │ +0008fd50: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ +0008fd60: 2052 6174 696f 6e61 6c4d 6170 2028 7175   RationalMap (qu
│ │ │ │ +0008fd70: 6164 7261 7469 6320 7261 7469 6f6e 616c  adratic rational
│ │ │ │  0008fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fda0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +0008fda0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │  0008fdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008fdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008fdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008fdf0: 2d2d 2d7c 0a7c 7b2d 785f 315e 322b 785f  ---|.|{-x_1^2+x_
│ │ │ │ -0008fe00: 302a 785f 322c 2d78 5f31 2a78 5f32 2b78  0*x_2,-x_1*x_2+x
│ │ │ │ -0008fe10: 5f30 2a78 5f33 2c2d 785f 325e 322b 785f  _0*x_3,-x_2^2+x_
│ │ │ │ -0008fe20: 312a 785f 332c 2d78 5f31 2a78 5f33 2b78  1*x_3,-x_1*x_3+x
│ │ │ │ -0008fe30: 5f30 2a78 5f34 2c2d 785f 322a 785f 332b  _0*x_4,-x_2*x_3+
│ │ │ │ -0008fe40: 785f 317c 0a7c 2020 2020 2020 2020 2020  x_1|.|          
│ │ │ │ +0008fdf0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7b2d 785f  ---------|.|{-x_
│ │ │ │ +0008fe00: 315e 322b 785f 302a 785f 322c 2d78 5f31  1^2+x_0*x_2,-x_1
│ │ │ │ +0008fe10: 2a78 5f32 2b78 5f30 2a78 5f33 2c2d 785f  *x_2+x_0*x_3,-x_
│ │ │ │ +0008fe20: 325e 322b 785f 312a 785f 332c 2d78 5f31  2^2+x_1*x_3,-x_1
│ │ │ │ +0008fe30: 2a78 5f33 2b78 5f30 2a78 5f34 2c2d 785f  *x_3+x_0*x_4,-x_
│ │ │ │ +0008fe40: 322a 785f 332b 785f 317c 0a7c 2020 2020  2*x_3+x_1|.|    
│ │ │ │  0008fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fe90: 2020 207c 0a7c 6d61 7020 6672 6f6d 2050     |.|map from P
│ │ │ │ -0008fea0: 505e 3420 746f 2050 505e 3529 2020 2020  P^4 to PP^5)    
│ │ │ │ -0008feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008fe90: 2020 2020 2020 2020 207c 0a7c 6d61 7020           |.|map 
│ │ │ │ +0008fea0: 6672 6f6d 2050 505e 3420 746f 2050 505e  from PP^4 to PP^
│ │ │ │ +0008feb0: 3529 2020 2020 2020 2020 2020 2020 2020  5)              
│ │ │ │  0008fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fee0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +0008fee0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │  0008fef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008ff00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008ff10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008ff20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008ff30: 2d2d 2d7c 0a7c 2a78 5f34 2c2d 785f 335e  ---|.|*x_4,-x_3^
│ │ │ │ -0008ff40: 322b 785f 322a 785f 347d 3b20 2020 2020  2+x_2*x_4};     
│ │ │ │ -0008ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008ff30: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f34  ---------|.|*x_4
│ │ │ │ +0008ff40: 2c2d 785f 335e 322b 785f 322a 785f 347d  ,-x_3^2+x_2*x_4}
│ │ │ │ +0008ff50: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  0008ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ff80: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0008ff80: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0008ff90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008ffa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008ffc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008ffd0: 2d2d 2d2b 0a7c 6938 203a 2064 6573 6372  ---+.|i8 : descr
│ │ │ │ -0008ffe0: 6962 6520 7068 6920 2020 2020 2020 2020  ibe phi         
│ │ │ │ +0008ffd0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ +0008ffe0: 2064 6573 6372 6962 6520 7068 6920 2020   describe phi   
│ │ │ │  0008fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00090020: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00090030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090070: 2020 207c 0a7c 6f38 203d 2072 6174 696f     |.|o8 = ratio
│ │ │ │ -00090080: 6e61 6c20 6d61 7020 6465 6669 6e65 6420  nal map defined 
│ │ │ │ -00090090: 6279 2066 6f72 6d73 206f 6620 6465 6772  by forms of degr
│ │ │ │ -000900a0: 6565 2032 2020 2020 2020 2020 2020 2020  ee 2            
│ │ │ │ +00090070: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ +00090080: 2072 6174 696f 6e61 6c20 6d61 7020 6465   rational map de
│ │ │ │ +00090090: 6669 6e65 6420 6279 2066 6f72 6d73 206f  fined by forms o
│ │ │ │ +000900a0: 6620 6465 6772 6565 2032 2020 2020 2020  f degree 2      
│ │ │ │  000900b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000900c0: 2020 207c 0a7c 2020 2020 2073 6f75 7263     |.|     sourc
│ │ │ │ -000900d0: 6520 7661 7269 6574 793a 2050 505e 3420  e variety: PP^4 
│ │ │ │ -000900e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000900c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000900d0: 2073 6f75 7263 6520 7661 7269 6574 793a   source variety:
│ │ │ │ +000900e0: 2050 505e 3420 2020 2020 2020 2020 2020   PP^4           
│ │ │ │  000900f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090110: 2020 207c 0a7c 2020 2020 2074 6172 6765     |.|     targe
│ │ │ │ -00090120: 7420 7661 7269 6574 793a 2050 505e 3520  t variety: PP^5 
│ │ │ │ -00090130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090110: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00090120: 2074 6172 6765 7420 7661 7269 6574 793a   target variety:
│ │ │ │ +00090130: 2050 505e 3520 2020 2020 2020 2020 2020   PP^5           
│ │ │ │  00090140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090160: 2020 207c 0a7c 2020 2020 2063 6f65 6666     |.|     coeff
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│ │ │ │ -00090180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090160: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00090170: 2063 6f65 6666 6963 6965 6e74 2072 696e   coefficient rin
│ │ │ │ +00090180: 673a 2051 5120 2020 2020 2020 2020 2020  g: QQ           
│ │ │ │  00090190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000901a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000901b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000901b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  000901c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000901d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000901e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000901f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00090210: 7068 6921 203b 2020 2020 2020 2020 2020  phi! ;          
│ │ │ │ +00090200: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a  ---------+.|i9 :
│ │ │ │ +00090210: 2074 696d 6520 7068 6921 203b 2020 2020   time phi! ;    
│ │ │ │  00090220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090250: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
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│ │ │ │ -00090290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000902a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00090250: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
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│ │ │ │ +00090270: 2863 7075 293b 2030 2e30 3434 3535 3038  (cpu); 0.0445508
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│ │ │ │ +00090290: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +000902a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000902b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000902c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000902d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000902e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00090320: 726f 6d20 5050 5e34 2074 6f20 5050 5e35  rom PP^4 to PP^5
│ │ │ │ -00090330: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00090340: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000902f0: 2020 2020 2020 2020 207c 0a7c 6f39 203a           |.|o9 :
│ │ │ │ +00090300: 2052 6174 696f 6e61 6c4d 6170 2028 7175   RationalMap (qu
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│ │ │ │ +00090320: 206d 6170 2066 726f 6d20 5050 5e34 2074   map from PP^4 t
│ │ │ │ +00090330: 6f20 5050 5e35 2920 2020 2020 2020 2020  o PP^5)         
│ │ │ │ +00090340: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  00090350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090390: 2d2d 2d2b 0a7c 6931 3020 3a20 6465 7363  ---+.|i10 : desc
│ │ │ │ -000903a0: 7269 6265 2070 6869 2020 2020 2020 2020  ribe phi        
│ │ │ │ +00090390: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │ +000903a0: 3a20 6465 7363 7269 6265 2070 6869 2020  : describe phi  
│ │ │ │  000903b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000903c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000903d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000903e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000903e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000903f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090430: 2020 207c 0a7c 6f31 3020 3d20 7261 7469     |.|o10 = rati
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│ │ │ │ +00090430: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
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│ │ │ │ +00090460: 6f66 2064 6567 7265 6520 3220 2020 2020  of degree 2     
│ │ │ │  00090470: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000904a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090480: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00090490: 2020 736f 7572 6365 2076 6172 6965 7479    source variety
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│ │ │ │  000904b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000904c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000904d0: 2020 207c 0a7c 2020 2020 2020 7461 7267     |.|      targ
│ │ │ │ -000904e0: 6574 2076 6172 6965 7479 3a20 5050 5e35  et variety: PP^5
│ │ │ │ -000904f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000904d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000904e0: 2020 7461 7267 6574 2076 6172 6965 7479    target variety
│ │ │ │ +000904f0: 3a20 5050 5e35 2020 2020 2020 2020 2020  : PP^5          
│ │ │ │  00090500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090520: 2020 207c 0a7c 2020 2020 2020 696d 6167     |.|      imag
│ │ │ │ -00090530: 653a 2073 6d6f 6f74 6820 7175 6164 7269  e: smooth quadri
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│ │ │ │ -00090550: 6e20 5050 5e35 2020 2020 2020 2020 2020  n PP^5          
│ │ │ │ +00090520: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00090530: 2020 696d 6167 653a 2073 6d6f 6f74 6820    image: smooth 
│ │ │ │ +00090540: 7175 6164 7269 6320 6879 7065 7273 7572  quadric hypersur
│ │ │ │ +00090550: 6661 6365 2069 6e20 5050 5e35 2020 2020  face in PP^5    
│ │ │ │  00090560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090570: 2020 207c 0a7c 2020 2020 2020 646f 6d69     |.|      domi
│ │ │ │ -00090580: 6e61 6e63 653a 2066 616c 7365 2020 2020  nance: false    
│ │ │ │ -00090590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090570: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00090580: 2020 646f 6d69 6e61 6e63 653a 2066 616c    dominance: fal
│ │ │ │ +00090590: 7365 2020 2020 2020 2020 2020 2020 2020  se              
│ │ │ │  000905a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000905b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000905c0: 2020 207c 0a7c 2020 2020 2020 6269 7261     |.|      bira
│ │ │ │ -000905d0: 7469 6f6e 616c 6974 793a 2066 616c 7365  tionality: false
│ │ │ │ -000905e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000905c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000905d0: 2020 6269 7261 7469 6f6e 616c 6974 793a    birationality:
│ │ │ │ +000905e0: 2066 616c 7365 2020 2020 2020 2020 2020   false          
│ │ │ │  000905f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090610: 2020 207c 0a7c 2020 2020 2020 6465 6772     |.|      degr
│ │ │ │ -00090620: 6565 206f 6620 6d61 703a 2031 2020 2020  ee of map: 1    
│ │ │ │ -00090630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090610: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00090620: 2020 6465 6772 6565 206f 6620 6d61 703a    degree of map:
│ │ │ │ +00090630: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  00090640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090660: 2020 207c 0a7c 2020 2020 2020 7072 6f6a     |.|      proj
│ │ │ │ -00090670: 6563 7469 7665 2064 6567 7265 6573 3a20  ective degrees: 
│ │ │ │ -00090680: 7b31 2c20 322c 2034 2c20 342c 2032 7d20  {1, 2, 4, 4, 2} 
│ │ │ │ -00090690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090660: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00090670: 2020 7072 6f6a 6563 7469 7665 2064 6567    projective deg
│ │ │ │ +00090680: 7265 6573 3a20 7b31 2c20 322c 2034 2c20  rees: {1, 2, 4, 
│ │ │ │ +00090690: 342c 2032 7d20 2020 2020 2020 2020 2020  4, 2}           
│ │ │ │  000906a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000906b0: 2020 207c 0a7c 2020 2020 2020 6e75 6d62     |.|      numb
│ │ │ │ -000906c0: 6572 206f 6620 6d69 6e69 6d61 6c20 7265  er of minimal re
│ │ │ │ -000906d0: 7072 6573 656e 7461 7469 7665 733a 2031  presentatives: 1
│ │ │ │ -000906e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000906b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000906c0: 2020 6e75 6d62 6572 206f 6620 6d69 6e69    number of mini
│ │ │ │ +000906d0: 6d61 6c20 7265 7072 6573 656e 7461 7469  mal representati
│ │ │ │ +000906e0: 7665 733a 2031 2020 2020 2020 2020 2020  ves: 1          
│ │ │ │  000906f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090700: 2020 207c 0a7c 2020 2020 2020 6469 6d65     |.|      dime
│ │ │ │ -00090710: 6e73 696f 6e20 6261 7365 206c 6f63 7573  nsion base locus
│ │ │ │ -00090720: 3a20 3120 2020 2020 2020 2020 2020 2020  : 1             
│ │ │ │ +00090700: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00090710: 2020 6469 6d65 6e73 696f 6e20 6261 7365    dimension base
│ │ │ │ +00090720: 206c 6f63 7573 3a20 3120 2020 2020 2020   locus: 1       
│ │ │ │  00090730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090750: 2020 207c 0a7c 2020 2020 2020 6465 6772     |.|      degr
│ │ │ │ -00090760: 6565 2062 6173 6520 6c6f 6375 733a 2034  ee base locus: 4
│ │ │ │ -00090770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090750: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00090760: 2020 6465 6772 6565 2062 6173 6520 6c6f    degree base lo
│ │ │ │ +00090770: 6375 733a 2034 2020 2020 2020 2020 2020  cus: 4          
│ │ │ │  00090780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000907a0: 2020 207c 0a7c 2020 2020 2020 636f 6566     |.|      coef
│ │ │ │ -000907b0: 6669 6369 656e 7420 7269 6e67 3a20 5151  ficient ring: QQ
│ │ │ │ -000907c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000907a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000907b0: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ +000907c0: 6e67 3a20 5151 2020 2020 2020 2020 2020  ng: QQ          
│ │ │ │  000907d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000907e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000907f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000907f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  00090800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090840: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d  ---+..See also.=
│ │ │ │ -00090850: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ -00090860: 7465 2064 6573 6372 6962 6528 5261 7469  te describe(Rati
│ │ │ │ -00090870: 6f6e 616c 4d61 7029 3a20 6465 7363 7269  onalMap): descri
│ │ │ │ -00090880: 6265 5f6c 7052 6174 696f 6e61 6c4d 6170  be_lpRationalMap
│ │ │ │ -00090890: 5f72 702c 202d 2d20 6465 7363 7269 6265  _rp, -- describe
│ │ │ │ -000908a0: 2061 0a20 2020 2072 6174 696f 6e61 6c20   a.    rational 
│ │ │ │ -000908b0: 6d61 700a 0a57 6179 7320 746f 2075 7365  map..Ways to use
│ │ │ │ -000908c0: 2074 6869 7320 6d65 7468 6f64 3a0a 3d3d   this method:.==
│ │ │ │ -000908d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000908e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -000908f0: 6520 5261 7469 6f6e 616c 4d61 7020 213a  e RationalMap !:
│ │ │ │ -00090900: 2052 6174 696f 6e61 6c4d 6170 2021 2c20   RationalMap !, 
│ │ │ │ -00090910: 2d2d 2063 616c 6375 6c61 7465 7320 6576  -- calculates ev
│ │ │ │ -00090920: 6572 7920 706f 7373 6962 6c65 2074 6869  ery possible thi
│ │ │ │ -00090930: 6e67 0a1f 0a46 696c 653a 2043 7265 6d6f  ng...File: Cremo
│ │ │ │ -00090940: 6e61 2e69 6e66 6f2c 204e 6f64 653a 2052  na.info, Node: R
│ │ │ │ -00090950: 6174 696f 6e61 6c4d 6170 205f 7374 2052  ationalMap _st R
│ │ │ │ -00090960: 6174 696f 6e61 6c4d 6170 2c20 4e65 7874  ationalMap, Next
│ │ │ │ -00090970: 3a20 5261 7469 6f6e 616c 4d61 7020 5f73  : RationalMap _s
│ │ │ │ -00090980: 745f 7374 2052 696e 672c 2050 7265 763a  t_st Ring, Prev:
│ │ │ │ -00090990: 2052 6174 696f 6e61 6c4d 6170 2021 2c20   RationalMap !, 
│ │ │ │ -000909a0: 5570 3a20 546f 700a 0a52 6174 696f 6e61  Up: Top..Rationa
│ │ │ │ -000909b0: 6c4d 6170 202a 2052 6174 696f 6e61 6c4d  lMap * RationalM
│ │ │ │ -000909c0: 6170 202d 2d20 636f 6d70 6f73 6974 696f  ap -- compositio
│ │ │ │ -000909d0: 6e20 6f66 2072 6174 696f 6e61 6c20 6d61  n of rational ma
│ │ │ │ -000909e0: 7073 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ps.*************
│ │ │ │ +00090840: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ +00090850: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ +00090860: 202a 202a 6e6f 7465 2064 6573 6372 6962   * *note describ
│ │ │ │ +00090870: 6528 5261 7469 6f6e 616c 4d61 7029 3a20  e(RationalMap): 
│ │ │ │ +00090880: 6465 7363 7269 6265 5f6c 7052 6174 696f  describe_lpRatio
│ │ │ │ +00090890: 6e61 6c4d 6170 5f72 702c 202d 2d20 6465  nalMap_rp, -- de
│ │ │ │ +000908a0: 7363 7269 6265 2061 0a20 2020 2072 6174  scribe a.    rat
│ │ │ │ +000908b0: 696f 6e61 6c20 6d61 700a 0a57 6179 7320  ional map..Ways 
│ │ │ │ +000908c0: 746f 2075 7365 2074 6869 7320 6d65 7468  to use this meth
│ │ │ │ +000908d0: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  od:.============
│ │ │ │ +000908e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +000908f0: 2a20 2a6e 6f74 6520 5261 7469 6f6e 616c  * *note Rational
│ │ │ │ +00090900: 4d61 7020 213a 2052 6174 696f 6e61 6c4d  Map !: RationalM
│ │ │ │ +00090910: 6170 2021 2c20 2d2d 2063 616c 6375 6c61  ap !, -- calcula
│ │ │ │ +00090920: 7465 7320 6576 6572 7920 706f 7373 6962  tes every possib
│ │ │ │ +00090930: 6c65 2074 6869 6e67 0a1f 0a46 696c 653a  le thing...File:
│ │ │ │ +00090940: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ +00090950: 6f64 653a 2052 6174 696f 6e61 6c4d 6170  ode: RationalMap
│ │ │ │ +00090960: 205f 7374 2052 6174 696f 6e61 6c4d 6170   _st RationalMap
│ │ │ │ +00090970: 2c20 4e65 7874 3a20 5261 7469 6f6e 616c  , Next: Rational
│ │ │ │ +00090980: 4d61 7020 5f73 745f 7374 2052 696e 672c  Map _st_st Ring,
│ │ │ │ +00090990: 2050 7265 763a 2052 6174 696f 6e61 6c4d   Prev: RationalM
│ │ │ │ +000909a0: 6170 2021 2c20 5570 3a20 546f 700a 0a52  ap !, Up: Top..R
│ │ │ │ +000909b0: 6174 696f 6e61 6c4d 6170 202a 2052 6174  ationalMap * Rat
│ │ │ │ +000909c0: 696f 6e61 6c4d 6170 202d 2d20 636f 6d70  ionalMap -- comp
│ │ │ │ +000909d0: 6f73 6974 696f 6e20 6f66 2072 6174 696f  osition of ratio
│ │ │ │ +000909e0: 6e61 6c20 6d61 7073 0a2a 2a2a 2a2a 2a2a  nal maps.*******
│ │ │ │  000909f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00090a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00090a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -00090a20: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -00090a30: 0a20 202a 204f 7065 7261 746f 723a 202a  .  * Operator: *
│ │ │ │ -00090a40: 6e6f 7465 202a 3a20 284d 6163 6175 6c61  note *: (Macaula
│ │ │ │ -00090a50: 7932 446f 6329 5f73 742c 0a20 202a 2055  y2Doc)_st,.  * U
│ │ │ │ -00090a60: 7361 6765 3a20 0a20 2020 2020 2020 2070  sage: .        p
│ │ │ │ -00090a70: 6869 202a 2070 7369 200a 2020 2020 2020  hi * psi .      
│ │ │ │ -00090a80: 2020 636f 6d70 6f73 6528 7068 692c 7073    compose(phi,ps
│ │ │ │ -00090a90: 6929 0a20 202a 2049 6e70 7574 733a 0a20  i).  * Inputs:. 
│ │ │ │ -00090aa0: 2020 2020 202a 2070 6869 2c20 6120 2a6e       * phi, a *n
│ │ │ │ -00090ab0: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ -00090ac0: 3a20 5261 7469 6f6e 616c 4d61 702c 2c20  : RationalMap,, 
│ │ │ │ -00090ad0: 2458 205c 6461 7368 7269 6768 7461 7272  $X \dashrightarr
│ │ │ │ -00090ae0: 6f77 2059 240a 2020 2020 2020 2a20 7073  ow Y$.      * ps
│ │ │ │ -00090af0: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ -00090b00: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ -00090b10: 6c4d 6170 2c2c 2024 5920 5c64 6173 6872  lMap,, $Y \dashr
│ │ │ │ -00090b20: 6967 6874 6172 726f 7720 5a24 0a20 202a  ightarrow Z$.  *
│ │ │ │ -00090b30: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -00090b40: 2a20 6120 2a6e 6f74 6520 7261 7469 6f6e  * a *note ration
│ │ │ │ -00090b50: 616c 206d 6170 3a20 5261 7469 6f6e 616c  al map: Rational
│ │ │ │ -00090b60: 4d61 702c 2c20 2458 205c 6461 7368 7269  Map,, $X \dashri
│ │ │ │ -00090b70: 6768 7461 7272 6f77 205a 242c 2074 6865  ghtarrow Z$, the
│ │ │ │ -00090b80: 0a20 2020 2020 2020 2063 6f6d 706f 7369  .        composi
│ │ │ │ -00090b90: 7469 6f6e 206f 6620 7068 6920 616e 6420  tion of phi and 
│ │ │ │ -00090ba0: 7073 690a 0a44 6573 6372 6970 7469 6f6e  psi..Description
│ │ │ │ -00090bb0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ -00090bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00090a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00090a20: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +00090a30: 3d3d 3d3d 3d0a 0a20 202a 204f 7065 7261  =====..  * Opera
│ │ │ │ +00090a40: 746f 723a 202a 6e6f 7465 202a 3a20 284d  tor: *note *: (M
│ │ │ │ +00090a50: 6163 6175 6c61 7932 446f 6329 5f73 742c  acaulay2Doc)_st,
│ │ │ │ +00090a60: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00090a70: 2020 2020 2070 6869 202a 2070 7369 200a       phi * psi .
│ │ │ │ +00090a80: 2020 2020 2020 2020 636f 6d70 6f73 6528          compose(
│ │ │ │ +00090a90: 7068 692c 7073 6929 0a20 202a 2049 6e70  phi,psi).  * Inp
│ │ │ │ +00090aa0: 7574 733a 0a20 2020 2020 202a 2070 6869  uts:.      * phi
│ │ │ │ +00090ab0: 2c20 6120 2a6e 6f74 6520 7261 7469 6f6e  , a *note ration
│ │ │ │ +00090ac0: 616c 206d 6170 3a20 5261 7469 6f6e 616c  al map: Rational
│ │ │ │ +00090ad0: 4d61 702c 2c20 2458 205c 6461 7368 7269  Map,, $X \dashri
│ │ │ │ +00090ae0: 6768 7461 7272 6f77 2059 240a 2020 2020  ghtarrow Y$.    
│ │ │ │ +00090af0: 2020 2a20 7073 692c 2061 202a 6e6f 7465    * psi, a *note
│ │ │ │ +00090b00: 2072 6174 696f 6e61 6c20 6d61 703a 2052   rational map: R
│ │ │ │ +00090b10: 6174 696f 6e61 6c4d 6170 2c2c 2024 5920  ationalMap,, $Y 
│ │ │ │ +00090b20: 5c64 6173 6872 6967 6874 6172 726f 7720  \dashrightarrow 
│ │ │ │ +00090b30: 5a24 0a20 202a 204f 7574 7075 7473 3a0a  Z$.  * Outputs:.
│ │ │ │ +00090b40: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ +00090b50: 7261 7469 6f6e 616c 206d 6170 3a20 5261  rational map: Ra
│ │ │ │ +00090b60: 7469 6f6e 616c 4d61 702c 2c20 2458 205c  tionalMap,, $X \
│ │ │ │ +00090b70: 6461 7368 7269 6768 7461 7272 6f77 205a  dashrightarrow Z
│ │ │ │ +00090b80: 242c 2074 6865 0a20 2020 2020 2020 2063  $, the.        c
│ │ │ │ +00090b90: 6f6d 706f 7369 7469 6f6e 206f 6620 7068  omposition of ph
│ │ │ │ +00090ba0: 6920 616e 6420 7073 690a 0a44 6573 6372  i and psi..Descr
│ │ │ │ +00090bb0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00090bc0: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ==..+-----------
│ │ │ │  00090bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090c00: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5220  ------+.|i1 : R 
│ │ │ │ -00090c10: 3d20 5151 5b78 5f30 2e2e 785f 335d 3b20  = QQ[x_0..x_3]; 
│ │ │ │ -00090c20: 5320 3d20 5151 5b79 5f30 2e2e 795f 345d  S = QQ[y_0..y_4]
│ │ │ │ -00090c30: 3b20 5420 3d20 5151 5b7a 5f30 2e2e 7a5f  ; T = QQ[z_0..z_
│ │ │ │ -00090c40: 345d 3b20 2020 2020 2020 2020 2020 2020  4];             
│ │ │ │ -00090c50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00090c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00090c10: 3120 3a20 5220 3d20 5151 5b78 5f30 2e2e  1 : R = QQ[x_0..
│ │ │ │ +00090c20: 785f 335d 3b20 5320 3d20 5151 5b79 5f30  x_3]; S = QQ[y_0
│ │ │ │ +00090c30: 2e2e 795f 345d 3b20 5420 3d20 5151 5b7a  ..y_4]; T = QQ[z
│ │ │ │ +00090c40: 5f30 2e2e 7a5f 345d 3b20 2020 2020 2020  _0..z_4];       
│ │ │ │ +00090c50: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00090c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -00090ca0: 3a20 7068 6920 3d20 7261 7469 6f6e 616c  : phi = rational
│ │ │ │ -00090cb0: 4d61 7028 522c 532c 7b78 5f30 2a78 5f32  Map(R,S,{x_0*x_2
│ │ │ │ -00090cc0: 2c78 5f30 2a78 5f33 2c78 5f31 2a78 5f32  ,x_0*x_3,x_1*x_2
│ │ │ │ -00090cd0: 2c78 5f31 2a78 5f33 2c78 5f32 2a78 5f33  ,x_1*x_3,x_2*x_3
│ │ │ │ -00090ce0: 7d29 2020 7c0a 7c20 2020 2020 2020 2020  })  |.|         
│ │ │ │ +00090c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00090ca0: 2b0a 7c69 3420 3a20 7068 6920 3d20 7261  +.|i4 : phi = ra
│ │ │ │ +00090cb0: 7469 6f6e 616c 4d61 7028 522c 532c 7b78  tionalMap(R,S,{x
│ │ │ │ +00090cc0: 5f30 2a78 5f32 2c78 5f30 2a78 5f33 2c78  _0*x_2,x_0*x_3,x
│ │ │ │ +00090cd0: 5f31 2a78 5f32 2c78 5f31 2a78 5f33 2c78  _1*x_2,x_1*x_3,x
│ │ │ │ +00090ce0: 5f32 2a78 5f33 7d29 2020 7c0a 7c20 2020  _2*x_3})  |.|   
│ │ │ │  00090cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090d20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00090d30: 7c6f 3420 3d20 2d2d 2072 6174 696f 6e61  |o4 = -- rationa
│ │ │ │ -00090d40: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │ +00090d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090d30: 2020 2020 7c0a 7c6f 3420 3d20 2d2d 2072      |.|o4 = -- r
│ │ │ │ +00090d40: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │  00090d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090d70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00090d80: 736f 7572 6365 3a20 5072 6f6a 2851 515b  source: Proj(QQ[
│ │ │ │ -00090d90: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ -00090da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090d70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090d80: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ +00090d90: 6f6a 2851 515b 7820 2c20 7820 2c20 7820  oj(QQ[x , x , x 
│ │ │ │ +00090da0: 2c20 7820 5d29 2020 2020 2020 2020 2020  , x ])          
│ │ │ │  00090db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090dc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00090dd0: 2020 2020 2020 2020 2020 2030 2020 2031             0   1
│ │ │ │ -00090de0: 2020 2032 2020 2033 2020 2020 2020 2020     2   3        
│ │ │ │ +00090dc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00090dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090de0: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │  00090df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00090e10: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -00090e20: 2851 515b 7920 2c20 7920 2c20 7920 2c20  (QQ[y , y , y , 
│ │ │ │ -00090e30: 7920 2c20 7920 5d29 2020 2020 2020 2020  y , y ])        
│ │ │ │ +00090e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090e10: 2020 7c0a 7c20 2020 2020 7461 7267 6574    |.|     target
│ │ │ │ +00090e20: 3a20 5072 6f6a 2851 515b 7920 2c20 7920  : Proj(QQ[y , y 
│ │ │ │ +00090e30: 2c20 7920 2c20 7920 2c20 7920 5d29 2020  , y , y , y ])  
│ │ │ │  00090e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090e50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00090e60: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00090e70: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ -00090e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090e50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00090e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090e70: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ +00090e80: 2033 2020 2034 2020 2020 2020 2020 2020   3   4          
│ │ │ │  00090e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090ea0: 7c0a 7c20 2020 2020 6465 6669 6e69 6e67  |.|     defining
│ │ │ │ -00090eb0: 2066 6f72 6d73 3a20 7b20 2020 2020 2020   forms: {       
│ │ │ │ +00090ea0: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +00090eb0: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │  00090ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090ee0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00090ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090f00: 2020 2078 2078 202c 2020 2020 2020 2020     x x ,        
│ │ │ │ +00090ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090ef0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00090f00: 2020 2020 2020 2020 2078 2078 202c 2020           x x ,  
│ │ │ │  00090f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090f30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00090f40: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00090f50: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00090f30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00090f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090f50: 2020 2020 3020 3220 2020 2020 2020 2020      0 2         
│ │ │ │  00090f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00090f80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00090f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090f80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00090f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090fc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00090fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090fe0: 2078 2078 202c 2020 2020 2020 2020 2020   x x ,          
│ │ │ │ +00090fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090fd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00090fe0: 2020 2020 2020 2078 2078 202c 2020 2020         x x ,    
│ │ │ │  00090ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091010: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00091020: 2020 2020 2020 2020 2020 2020 3020 3320              0 3 
│ │ │ │ -00091030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091010: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00091020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091030: 2020 3020 3320 2020 2020 2020 2020 2020    0 3           
│ │ │ │  00091040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091050: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00091060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091060: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00091070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000910a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000910b0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -000910c0: 2078 202c 2020 2020 2020 2020 2020 2020   x ,            
│ │ │ │ +000910a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000910b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000910c0: 2020 2020 2078 2078 202c 2020 2020 2020       x x ,      
│ │ │ │  000910d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000910e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000910f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00091100: 2020 2020 2020 2020 2020 3120 3220 2020            1 2   
│ │ │ │ -00091110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000910f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00091100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091110: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
│ │ │ │  00091120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091130: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00091140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091140: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00091150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091180: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00091190: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ -000911a0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +00091180: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00091190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000911a0: 2020 2078 2078 202c 2020 2020 2020 2020     x x ,        
│ │ │ │  000911b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000911c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000911d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000911e0: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ -000911f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000911c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000911d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000911e0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +000911f0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  00091200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091210: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00091220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00091220: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00091230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091260: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00091270: 2020 2020 2020 2020 2020 2078 2078 2020             x x  
│ │ │ │ -00091280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00091270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091280: 2078 2078 2020 2020 2020 2020 2020 2020   x x            
│ │ │ │  00091290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000912a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000912b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000912c0: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ +000912a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000912b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000912c0: 2020 2020 2020 2020 2020 2020 3220 3320              2 3 
│ │ │ │  000912d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000912e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000912f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00091300: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ -00091310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000912f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00091300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091310: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │  00091320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091340: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00091340: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00091350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091380: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -00091390: 3a20 5261 7469 6f6e 616c 4d61 7020 2871  : RationalMap (q
│ │ │ │ -000913a0: 7561 6472 6174 6963 2072 6174 696f 6e61  uadratic rationa
│ │ │ │ -000913b0: 6c20 6d61 7020 6672 6f6d 2050 505e 3320  l map from PP^3 
│ │ │ │ -000913c0: 746f 2050 505e 3429 2020 2020 2020 2020  to PP^4)        
│ │ │ │ -000913d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00091380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091390: 7c0a 7c6f 3420 3a20 5261 7469 6f6e 616c  |.|o4 : Rational
│ │ │ │ +000913a0: 4d61 7020 2871 7561 6472 6174 6963 2072  Map (quadratic r
│ │ │ │ +000913b0: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ +000913c0: 2050 505e 3320 746f 2050 505e 3429 2020   PP^3 to PP^4)  
│ │ │ │ +000913d0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  000913e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000913f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00091400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00091410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00091420: 7c69 3520 3a20 7073 6920 3d20 7261 7469  |i5 : psi = rati
│ │ │ │ -00091430: 6f6e 616c 4d61 7028 532c 542c 7b79 5f30  onalMap(S,T,{y_0
│ │ │ │ -00091440: 2a79 5f33 2c2d 795f 322a 795f 332c 795f  *y_3,-y_2*y_3,y_
│ │ │ │ -00091450: 312a 795f 322c 795f 322a 795f 342c 2d79  1*y_2,y_2*y_4,-y
│ │ │ │ -00091460: 5f33 2a79 5f34 7d29 7c0a 7c20 2020 2020  _3*y_4})|.|     
│ │ │ │ -00091470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00091420: 2d2d 2d2d 2b0a 7c69 3520 3a20 7073 6920  ----+.|i5 : psi 
│ │ │ │ +00091430: 3d20 7261 7469 6f6e 616c 4d61 7028 532c  = rationalMap(S,
│ │ │ │ +00091440: 542c 7b79 5f30 2a79 5f33 2c2d 795f 322a  T,{y_0*y_3,-y_2*
│ │ │ │ +00091450: 795f 332c 795f 312a 795f 322c 795f 322a  y_3,y_1*y_2,y_2*
│ │ │ │ +00091460: 795f 342c 2d79 5f33 2a79 5f34 7d29 7c0a  y_4,-y_3*y_4})|.
│ │ │ │ +00091470: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00091480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000914a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000914b0: 2020 7c0a 7c6f 3520 3d20 2d2d 2072 6174    |.|o5 = -- rat
│ │ │ │ -000914c0: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │ -000914d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000914b0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ +000914c0: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ +000914d0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  000914e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000914f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00091500: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -00091510: 2851 515b 7920 2c20 7920 2c20 7920 2c20  (QQ[y , y , y , 
│ │ │ │ -00091520: 7920 2c20 7920 5d29 2020 2020 2020 2020  y , y ])        
│ │ │ │ +000914f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091500: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +00091510: 3a20 5072 6f6a 2851 515b 7920 2c20 7920  : Proj(QQ[y , y 
│ │ │ │ +00091520: 2c20 7920 2c20 7920 2c20 7920 5d29 2020  , y , y , y ])  
│ │ │ │  00091530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091540: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00091550: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00091560: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ -00091570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00091550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091560: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ +00091570: 2033 2020 2034 2020 2020 2020 2020 2020   3   4          
│ │ │ │  00091580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091590: 7c0a 7c20 2020 2020 7461 7267 6574 3a20  |.|     target: 
│ │ │ │ -000915a0: 5072 6f6a 2851 515b 7a20 2c20 7a20 2c20  Proj(QQ[z , z , 
│ │ │ │ -000915b0: 7a20 2c20 7a20 2c20 7a20 5d29 2020 2020  z , z , z ])    
│ │ │ │ -000915c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000915d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000915e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000915f0: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
│ │ │ │ -00091600: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00091590: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +000915a0: 7267 6574 3a20 5072 6f6a 2851 515b 7a20  rget: Proj(QQ[z 
│ │ │ │ +000915b0: 2c20 7a20 2c20 7a20 2c20 7a20 2c20 7a20  , z , z , z , z 
│ │ │ │ +000915c0: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +000915d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000915e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000915f0: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +00091600: 2032 2020 2033 2020 2034 2020 2020 2020   2   3   4      
│ │ │ │  00091610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091620: 2020 2020 7c0a 7c20 2020 2020 6465 6669      |.|     defi
│ │ │ │ -00091630: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │ -00091640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091620: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00091630: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ +00091640: 3a20 7b20 2020 2020 2020 2020 2020 2020  : {             
│ │ │ │  00091650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00091670: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00091680: 2020 2020 2020 2079 2079 202c 2020 2020         y y ,    
│ │ │ │ -00091690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091670: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00091680: 2020 2020 2020 2020 2020 2020 2079 2079               y y
│ │ │ │ +00091690: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  000916a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000916b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000916c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000916d0: 2020 3020 3320 2020 2020 2020 2020 2020    0 3           
│ │ │ │ +000916b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000916c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000916d0: 2020 2020 2020 2020 3020 3320 2020 2020          0 3     
│ │ │ │  000916e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000916f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00091700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00091710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091740: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00091750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091760: 2020 2020 202d 7920 7920 2c20 2020 2020       -y y ,     
│ │ │ │ -00091770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091750: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00091760: 2020 2020 2020 2020 2020 202d 7920 7920             -y y 
│ │ │ │ +00091770: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  00091780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091790: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00091790: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000917a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000917b0: 2032 2033 2020 2020 2020 2020 2020 2020   2 3            
│ │ │ │ +000917b0: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │  000917c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000917d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000917e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000917e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000917f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091820: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00091830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091840: 2020 2079 2079 202c 2020 2020 2020 2020     y y ,        
│ │ │ │ +00091820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091830: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00091840: 2020 2020 2020 2020 2079 2079 202c 2020           y y ,  
│ │ │ │  00091850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091870: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00091880: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00091890: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00091870: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00091880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091890: 2020 2020 3120 3220 2020 2020 2020 2020      1 2         
│ │ │ │  000918a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000918b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000918c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000918b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000918c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000918d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000918e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000918f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091900: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00091910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091920: 2079 2079 202c 2020 2020 2020 2020 2020   y y ,          
│ │ │ │ +00091900: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00091910: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00091920: 2020 2020 2020 2079 2079 202c 2020 2020         y y ,    
│ │ │ │  00091930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091950: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00091960: 2020 2020 2020 2020 2020 2020 3220 3420              2 4 
│ │ │ │ -00091970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091950: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00091960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091970: 2020 3220 3420 2020 2020 2020 2020 2020    2 4           
│ │ │ │  00091980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091990: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000919a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000919a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000919b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000919c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000919d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000919e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000919f0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -00091a00: 7920 7920 2020 2020 2020 2020 2020 2020  y y             
│ │ │ │ +000919e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000919f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091a00: 2020 2020 202d 7920 7920 2020 2020 2020       -y y       
│ │ │ │  00091a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091a30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00091a40: 2020 2020 2020 2020 2020 2033 2034 2020             3 4  
│ │ │ │ -00091a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091a30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00091a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091a50: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │  00091a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091a70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00091a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091a90: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │ +00091a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091a80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00091a90: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │  00091aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091ac0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00091ac0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00091ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091b00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00091b10: 7c6f 3520 3a20 5261 7469 6f6e 616c 4d61  |o5 : RationalMa
│ │ │ │ -00091b20: 7020 2871 7561 6472 6174 6963 2072 6174  p (quadratic rat
│ │ │ │ -00091b30: 696f 6e61 6c20 6d61 7020 6672 6f6d 2050  ional map from P
│ │ │ │ -00091b40: 505e 3420 746f 2050 505e 3429 2020 2020  P^4 to PP^4)    
│ │ │ │ -00091b50: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00091b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00091b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091b10: 2020 2020 7c0a 7c6f 3520 3a20 5261 7469      |.|o5 : Rati
│ │ │ │ +00091b20: 6f6e 616c 4d61 7020 2871 7561 6472 6174  onalMap (quadrat
│ │ │ │ +00091b30: 6963 2072 6174 696f 6e61 6c20 6d61 7020  ic rational map 
│ │ │ │ +00091b40: 6672 6f6d 2050 505e 3420 746f 2050 505e  from PP^4 to PP^
│ │ │ │ +00091b50: 3429 2020 2020 2020 2020 2020 2020 7c0a  4)            |.
│ │ │ │ +00091b60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00091b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00091b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00091b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00091ba0: 2d2d 2b0a 7c69 3620 3a20 7068 6920 2a20  --+.|i6 : phi * 
│ │ │ │ -00091bb0: 7073 6920 2020 2020 2020 2020 2020 2020  psi             
│ │ │ │ +00091ba0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +00091bb0: 7068 6920 2a20 7073 6920 2020 2020 2020  phi * psi       
│ │ │ │  00091bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091be0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00091bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091bf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00091c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091c30: 2020 2020 2020 7c0a 7c6f 3620 3d20 2d2d        |.|o6 = --
│ │ │ │ -00091c40: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ -00091c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091c30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00091c40: 3620 3d20 2d2d 2072 6174 696f 6e61 6c20  6 = -- rational 
│ │ │ │ +00091c50: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │  00091c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091c80: 7c0a 7c20 2020 2020 736f 7572 6365 3a20  |.|     source: 
│ │ │ │ -00091c90: 5072 6f6a 2851 515b 7820 2c20 7820 2c20  Proj(QQ[x , x , 
│ │ │ │ -00091ca0: 7820 2c20 7820 5d29 2020 2020 2020 2020  x , x ])        
│ │ │ │ +00091c80: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +00091c90: 7572 6365 3a20 5072 6f6a 2851 515b 7820  urce: Proj(QQ[x 
│ │ │ │ +00091ca0: 2c20 7820 2c20 7820 2c20 7820 5d29 2020  , x , x , x ])  
│ │ │ │  00091cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091cc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00091cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091ce0: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
│ │ │ │ -00091cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091cd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00091ce0: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +00091cf0: 2032 2020 2033 2020 2020 2020 2020 2020   2   3          
│ │ │ │  00091d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091d10: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ -00091d20: 6574 3a20 5072 6f6a 2851 515b 7a20 2c20  et: Proj(QQ[z , 
│ │ │ │ -00091d30: 7a20 2c20 7a20 2c20 7a20 2c20 7a20 5d29  z , z , z , z ])
│ │ │ │ -00091d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091d50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00091d60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00091d70: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ -00091d80: 2020 2033 2020 2034 2020 2020 2020 2020     3   4        
│ │ │ │ +00091d10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00091d20: 2020 7461 7267 6574 3a20 5072 6f6a 2851    target: Proj(Q
│ │ │ │ +00091d30: 515b 7a20 2c20 7a20 2c20 7a20 2c20 7a20  Q[z , z , z , z 
│ │ │ │ +00091d40: 2c20 7a20 5d29 2020 2020 2020 2020 2020  , z ])          
│ │ │ │ +00091d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00091d70: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +00091d80: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │  00091d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091da0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00091db0: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -00091dc0: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +00091da0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00091db0: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ +00091dc0: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │  00091dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091df0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00091e00: 2020 2020 2020 2020 2020 2078 202c 2020             x ,  
│ │ │ │ -00091e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091df0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00091e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091e10: 2078 202c 2020 2020 2020 2020 2020 2020   x ,            
│ │ │ │  00091e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091e30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00091e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091e50: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00091e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091e40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00091e50: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │  00091e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091e80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00091e80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00091e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091ed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00091ee0: 2020 2020 2020 2020 202d 7820 2c20 2020           -x ,   
│ │ │ │ -00091ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091ed0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00091ee0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +00091ef0: 7820 2c20 2020 2020 2020 2020 2020 2020  x ,             
│ │ │ │  00091f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091f10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00091f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091f30: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00091f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091f20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00091f30: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │  00091f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091f60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00091f60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00091f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091fa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00091fb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00091fc0: 2020 2020 2020 2078 202c 2020 2020 2020         x ,      
│ │ │ │ +00091fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00091fb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00091fc0: 2020 2020 2020 2020 2020 2020 2078 202c               x ,
│ │ │ │  00091fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091ff0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00092000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092010: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00091ff0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00092000: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00092010: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │  00092020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092040: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00092040: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00092050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092080: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00092090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000920a0: 2020 2020 2078 202c 2020 2020 2020 2020       x ,        
│ │ │ │ +00092080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092090: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000920a0: 2020 2020 2020 2020 2020 2078 202c 2020             x ,  
│ │ │ │  000920b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000920c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000920d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000920d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000920e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000920f0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000920f0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00092100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092120: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00092120: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00092130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092160: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00092170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092180: 2020 202d 7820 2020 2020 2020 2020 2020     -x           
│ │ │ │ +00092160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092170: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00092180: 2020 2020 2020 2020 202d 7820 2020 2020           -x     
│ │ │ │  00092190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000921a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000921b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000921c0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -000921d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000921b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000921c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000921d0: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │  000921e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000921f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00092200: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00092210: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +000921f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092200: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00092210: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │  00092220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092240: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00092250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092240: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00092250: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00092260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092290: 2020 7c0a 7c6f 3620 3a20 5261 7469 6f6e    |.|o6 : Ration
│ │ │ │ -000922a0: 616c 4d61 7020 286c 696e 6561 7220 7261  alMap (linear ra
│ │ │ │ -000922b0: 7469 6f6e 616c 206d 6170 2066 726f 6d20  tional map from 
│ │ │ │ -000922c0: 5050 5e33 2074 6f20 5050 5e34 2920 2020  PP^3 to PP^4)   
│ │ │ │ -000922d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -000922e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00092290: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ +000922a0: 5261 7469 6f6e 616c 4d61 7020 286c 696e  RationalMap (lin
│ │ │ │ +000922b0: 6561 7220 7261 7469 6f6e 616c 206d 6170  ear rational map
│ │ │ │ +000922c0: 2066 726f 6d20 5050 5e33 2074 6f20 5050   from PP^3 to PP
│ │ │ │ +000922d0: 5e34 2920 2020 2020 2020 2020 2020 2020  ^4)             
│ │ │ │ +000922e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000922f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00092320: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 286d  ------+.|i7 : (m
│ │ │ │ -00092330: 6170 2070 6869 2920 2a20 286d 6170 2070  ap phi) * (map p
│ │ │ │ -00092340: 7369 2920 2020 2020 2020 2020 2020 2020  si)             
│ │ │ │ +00092320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00092330: 3720 3a20 286d 6170 2070 6869 2920 2a20  7 : (map phi) * 
│ │ │ │ +00092340: 286d 6170 2070 7369 2920 2020 2020 2020  (map psi)       
│ │ │ │  00092350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092370: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00092370: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00092380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000923a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000923b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000923c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000923d0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000923e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000923f0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00092400: 2020 2020 7c0a 7c6f 3720 3d20 6d61 7020      |.|o7 = map 
│ │ │ │ -00092410: 2852 2c20 542c 207b 7820 7820 7820 7820  (R, T, {x x x x 
│ │ │ │ -00092420: 2c20 2d78 2078 2078 202c 2078 2078 2078  , -x x x , x x x
│ │ │ │ -00092430: 2078 202c 2078 2078 2078 202c 202d 7820   x , x x x , -x 
│ │ │ │ -00092440: 7820 7820 7d29 2020 2020 2020 2020 7c0a  x x })        |.
│ │ │ │ -00092450: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00092460: 2020 2030 2031 2032 2033 2020 2020 3120     0 1 2 3    1 
│ │ │ │ -00092470: 3220 3320 2020 3020 3120 3220 3320 2020  2 3   0 1 2 3   
│ │ │ │ -00092480: 3120 3220 3320 2020 2031 2032 2033 2020  1 2 3    1 2 3  
│ │ │ │ -00092490: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000924a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000923b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000923c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000923d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000923e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000923f0: 2020 2020 3220 2020 2020 2020 2020 2032      2          2
│ │ │ │ +00092400: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +00092410: 3d20 6d61 7020 2852 2c20 542c 207b 7820  = map (R, T, {x 
│ │ │ │ +00092420: 7820 7820 7820 2c20 2d78 2078 2078 202c  x x x , -x x x ,
│ │ │ │ +00092430: 2078 2078 2078 2078 202c 2078 2078 2078   x x x x , x x x
│ │ │ │ +00092440: 202c 202d 7820 7820 7820 7d29 2020 2020   , -x x x })    
│ │ │ │ +00092450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00092460: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ +00092470: 2020 2020 3120 3220 3320 2020 3020 3120      1 2 3   0 1 
│ │ │ │ +00092480: 3220 3320 2020 3120 3220 3320 2020 2031  2 3   1 2 3    1
│ │ │ │ +00092490: 2032 2033 2020 2020 2020 2020 2020 7c0a   2 3          |.
│ │ │ │ +000924a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000924b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000924c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000924d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000924e0: 2020 7c0a 7c6f 3720 3a20 5269 6e67 4d61    |.|o7 : RingMa
│ │ │ │ -000924f0: 7020 5220 3c2d 2d20 5420 2020 2020 2020  p R <-- T       
│ │ │ │ +000924e0: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ +000924f0: 5269 6e67 4d61 7020 5220 3c2d 2d20 5420  RingMap R <-- T 
│ │ │ │  00092500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092520: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00092530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00092520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092530: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00092540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00092570: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ -00092580: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ -00092590: 2a6e 6f74 6520 5269 6e67 4d61 7020 2a20  *note RingMap * 
│ │ │ │ -000925a0: 5269 6e67 4d61 703a 2028 4d61 6361 756c  RingMap: (Macaul
│ │ │ │ -000925b0: 6179 3244 6f63 295f 7374 2c20 2d2d 2061  ay2Doc)_st, -- a
│ │ │ │ -000925c0: 2062 696e 6172 7920 6f70 6572 6174 6f72   binary operator
│ │ │ │ -000925d0: 2c20 7573 7561 6c6c 790a 2020 2020 7573  , usually.    us
│ │ │ │ -000925e0: 6564 2066 6f72 206d 756c 7469 706c 6963  ed for multiplic
│ │ │ │ -000925f0: 6174 696f 6e0a 0a57 6179 7320 746f 2075  ation..Ways to u
│ │ │ │ -00092600: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ -00092610: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00092620: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -00092630: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ -00092640: 2a20 5261 7469 6f6e 616c 4d61 703a 2052  * RationalMap: R
│ │ │ │ -00092650: 6174 696f 6e61 6c4d 6170 205f 7374 2052  ationalMap _st R
│ │ │ │ -00092660: 6174 696f 6e61 6c4d 6170 2c20 2d2d 0a20  ationalMap, --. 
│ │ │ │ -00092670: 2020 2063 6f6d 706f 7369 7469 6f6e 206f     composition o
│ │ │ │ -00092680: 6620 7261 7469 6f6e 616c 206d 6170 730a  f rational maps.
│ │ │ │ -00092690: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -000926a0: 696e 666f 2c20 4e6f 6465 3a20 5261 7469  info, Node: Rati
│ │ │ │ -000926b0: 6f6e 616c 4d61 7020 5f73 745f 7374 2052  onalMap _st_st R
│ │ │ │ -000926c0: 696e 672c 204e 6578 743a 2052 6174 696f  ing, Next: Ratio
│ │ │ │ -000926d0: 6e61 6c4d 6170 203d 3d20 5261 7469 6f6e  nalMap == Ration
│ │ │ │ -000926e0: 616c 4d61 702c 2050 7265 763a 2052 6174  alMap, Prev: Rat
│ │ │ │ -000926f0: 696f 6e61 6c4d 6170 205f 7374 2052 6174  ionalMap _st Rat
│ │ │ │ -00092700: 696f 6e61 6c4d 6170 2c20 5570 3a20 546f  ionalMap, Up: To
│ │ │ │ -00092710: 700a 0a52 6174 696f 6e61 6c4d 6170 202a  p..RationalMap *
│ │ │ │ -00092720: 2a20 5269 6e67 202d 2d20 6368 616e 6765  * Ring -- change
│ │ │ │ -00092730: 2074 6865 2063 6f65 6666 6963 6965 6e74   the coefficient
│ │ │ │ -00092740: 2072 696e 6720 6f66 2061 2072 6174 696f   ring of a ratio
│ │ │ │ -00092750: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a  nal map.********
│ │ │ │ +00092570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +00092580: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +00092590: 0a0a 2020 2a20 2a6e 6f74 6520 5269 6e67  ..  * *note Ring
│ │ │ │ +000925a0: 4d61 7020 2a20 5269 6e67 4d61 703a 2028  Map * RingMap: (
│ │ │ │ +000925b0: 4d61 6361 756c 6179 3244 6f63 295f 7374  Macaulay2Doc)_st
│ │ │ │ +000925c0: 2c20 2d2d 2061 2062 696e 6172 7920 6f70  , -- a binary op
│ │ │ │ +000925d0: 6572 6174 6f72 2c20 7573 7561 6c6c 790a  erator, usually.
│ │ │ │ +000925e0: 2020 2020 7573 6564 2066 6f72 206d 756c      used for mul
│ │ │ │ +000925f0: 7469 706c 6963 6174 696f 6e0a 0a57 6179  tiplication..Way
│ │ │ │ +00092600: 7320 746f 2075 7365 2074 6869 7320 6d65  s to use this me
│ │ │ │ +00092610: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  thod:.==========
│ │ │ │ +00092620: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00092630: 2020 2a20 2a6e 6f74 6520 5261 7469 6f6e    * *note Ration
│ │ │ │ +00092640: 616c 4d61 7020 2a20 5261 7469 6f6e 616c  alMap * Rational
│ │ │ │ +00092650: 4d61 703a 2052 6174 696f 6e61 6c4d 6170  Map: RationalMap
│ │ │ │ +00092660: 205f 7374 2052 6174 696f 6e61 6c4d 6170   _st RationalMap
│ │ │ │ +00092670: 2c20 2d2d 0a20 2020 2063 6f6d 706f 7369  , --.    composi
│ │ │ │ +00092680: 7469 6f6e 206f 6620 7261 7469 6f6e 616c  tion of rational
│ │ │ │ +00092690: 206d 6170 730a 1f0a 4669 6c65 3a20 4372   maps...File: Cr
│ │ │ │ +000926a0: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ +000926b0: 3a20 5261 7469 6f6e 616c 4d61 7020 5f73  : RationalMap _s
│ │ │ │ +000926c0: 745f 7374 2052 696e 672c 204e 6578 743a  t_st Ring, Next:
│ │ │ │ +000926d0: 2052 6174 696f 6e61 6c4d 6170 203d 3d20   RationalMap == 
│ │ │ │ +000926e0: 5261 7469 6f6e 616c 4d61 702c 2050 7265  RationalMap, Pre
│ │ │ │ +000926f0: 763a 2052 6174 696f 6e61 6c4d 6170 205f  v: RationalMap _
│ │ │ │ +00092700: 7374 2052 6174 696f 6e61 6c4d 6170 2c20  st RationalMap, 
│ │ │ │ +00092710: 5570 3a20 546f 700a 0a52 6174 696f 6e61  Up: Top..Rationa
│ │ │ │ +00092720: 6c4d 6170 202a 2a20 5269 6e67 202d 2d20  lMap ** Ring -- 
│ │ │ │ +00092730: 6368 616e 6765 2074 6865 2063 6f65 6666  change the coeff
│ │ │ │ +00092740: 6963 6965 6e74 2072 696e 6720 6f66 2061  icient ring of a
│ │ │ │ +00092750: 2072 6174 696f 6e61 6c20 6d61 700a 2a2a   rational map.**
│ │ │ │  00092760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00092770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00092780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00092790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -000927a0: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -000927b0: 0a20 202a 204f 7065 7261 746f 723a 202a  .  * Operator: *
│ │ │ │ -000927c0: 6e6f 7465 202a 2a3a 2028 4d61 6361 756c  note **: (Macaul
│ │ │ │ -000927d0: 6179 3244 6f63 295f 7374 5f73 742c 0a20  ay2Doc)_st_st,. 
│ │ │ │ -000927e0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -000927f0: 2020 2070 6869 202a 2a20 4b0a 2020 2a20     phi ** K.  * 
│ │ │ │ -00092800: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00092810: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ -00092820: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00092830: 6e61 6c4d 6170 2c2c 2064 6566 696e 6564  nalMap,, defined
│ │ │ │ -00092840: 206f 7665 7220 6120 636f 6566 6669 6369   over a coeffici
│ │ │ │ -00092850: 656e 740a 2020 2020 2020 2020 7269 6e67  ent.        ring
│ │ │ │ -00092860: 2046 0a20 2020 2020 202a 204b 2c20 6120   F.      * K, a 
│ │ │ │ -00092870: 2a6e 6f74 6520 7269 6e67 3a20 284d 6163  *note ring: (Mac
│ │ │ │ -00092880: 6175 6c61 7932 446f 6329 5269 6e67 2c2c  aulay2Doc)Ring,,
│ │ │ │ -00092890: 2074 6865 206e 6577 2063 6f65 6666 6963   the new coeffic
│ │ │ │ -000928a0: 6965 6e74 2072 696e 6720 2877 6869 6368  ient ring (which
│ │ │ │ -000928b0: 0a20 2020 2020 2020 206d 7573 7420 6265  .        must be
│ │ │ │ -000928c0: 2061 2066 6965 6c64 290a 2020 2a20 4f75   a field).  * Ou
│ │ │ │ -000928d0: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ -000928e0: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ -000928f0: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ -00092900: 2c2c 2061 2072 6174 696f 6e61 6c20 6d61  ,, a rational ma
│ │ │ │ -00092910: 7020 6465 6669 6e65 6420 6f76 6572 204b  p defined over K
│ │ │ │ -00092920: 2c0a 2020 2020 2020 2020 6f62 7461 696e  ,.        obtain
│ │ │ │ -00092930: 6564 2062 7920 636f 6572 6369 6e67 2074  ed by coercing t
│ │ │ │ -00092940: 6865 2063 6f65 6666 6963 6965 6e74 7320  he coefficients 
│ │ │ │ -00092950: 6f66 2074 6865 2066 6f72 6d73 2064 6566  of the forms def
│ │ │ │ -00092960: 696e 696e 6720 7068 6920 696e 746f 204b  ining phi into K
│ │ │ │ -00092970: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -00092980: 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 7420 6973  =========..It is
│ │ │ │ -00092990: 206e 6563 6573 7361 7279 2074 6861 7420   necessary that 
│ │ │ │ -000929a0: 616c 6c20 666f 726d 7320 696e 2074 6865  all forms in the
│ │ │ │ -000929b0: 206f 6c64 2063 6f65 6666 6963 6965 6e74   old coefficient
│ │ │ │ -000929c0: 2072 696e 6720 4620 6361 6e20 6265 0a61   ring F can be.a
│ │ │ │ -000929d0: 7574 6f6d 6174 6963 616c 6c79 2063 6f65  utomatically coe
│ │ │ │ -000929e0: 7263 6564 2069 6e74 6f20 7468 6520 6e65  rced into the ne
│ │ │ │ -000929f0: 7720 636f 6566 6669 6369 656e 7420 7269  w coefficient ri
│ │ │ │ -00092a00: 6e67 204b 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ng K...+--------
│ │ │ │ +00092790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000927a0: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +000927b0: 3d3d 3d3d 3d0a 0a20 202a 204f 7065 7261  =====..  * Opera
│ │ │ │ +000927c0: 746f 723a 202a 6e6f 7465 202a 2a3a 2028  tor: *note **: (
│ │ │ │ +000927d0: 4d61 6361 756c 6179 3244 6f63 295f 7374  Macaulay2Doc)_st
│ │ │ │ +000927e0: 5f73 742c 0a20 202a 2055 7361 6765 3a20  _st,.  * Usage: 
│ │ │ │ +000927f0: 0a20 2020 2020 2020 2070 6869 202a 2a20  .        phi ** 
│ │ │ │ +00092800: 4b0a 2020 2a20 496e 7075 7473 3a0a 2020  K.  * Inputs:.  
│ │ │ │ +00092810: 2020 2020 2a20 7068 692c 2061 202a 6e6f      * phi, a *no
│ │ │ │ +00092820: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ +00092830: 2052 6174 696f 6e61 6c4d 6170 2c2c 2064   RationalMap,, d
│ │ │ │ +00092840: 6566 696e 6564 206f 7665 7220 6120 636f  efined over a co
│ │ │ │ +00092850: 6566 6669 6369 656e 740a 2020 2020 2020  efficient.      
│ │ │ │ +00092860: 2020 7269 6e67 2046 0a20 2020 2020 202a    ring F.      *
│ │ │ │ +00092870: 204b 2c20 6120 2a6e 6f74 6520 7269 6e67   K, a *note ring
│ │ │ │ +00092880: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00092890: 5269 6e67 2c2c 2074 6865 206e 6577 2063  Ring,, the new c
│ │ │ │ +000928a0: 6f65 6666 6963 6965 6e74 2072 696e 6720  oefficient ring 
│ │ │ │ +000928b0: 2877 6869 6368 0a20 2020 2020 2020 206d  (which.        m
│ │ │ │ +000928c0: 7573 7420 6265 2061 2066 6965 6c64 290a  ust be a field).
│ │ │ │ +000928d0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +000928e0: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ +000928f0: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +00092900: 6e61 6c4d 6170 2c2c 2061 2072 6174 696f  nalMap,, a ratio
│ │ │ │ +00092910: 6e61 6c20 6d61 7020 6465 6669 6e65 6420  nal map defined 
│ │ │ │ +00092920: 6f76 6572 204b 2c0a 2020 2020 2020 2020  over K,.        
│ │ │ │ +00092930: 6f62 7461 696e 6564 2062 7920 636f 6572  obtained by coer
│ │ │ │ +00092940: 6369 6e67 2074 6865 2063 6f65 6666 6963  cing the coeffic
│ │ │ │ +00092950: 6965 6e74 7320 6f66 2074 6865 2066 6f72  ients of the for
│ │ │ │ +00092960: 6d73 2064 6566 696e 696e 6720 7068 6920  ms defining phi 
│ │ │ │ +00092970: 696e 746f 204b 0a0a 4465 7363 7269 7074  into K..Descript
│ │ │ │ +00092980: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +00092990: 0a49 7420 6973 206e 6563 6573 7361 7279  .It is necessary
│ │ │ │ +000929a0: 2074 6861 7420 616c 6c20 666f 726d 7320   that all forms 
│ │ │ │ +000929b0: 696e 2074 6865 206f 6c64 2063 6f65 6666  in the old coeff
│ │ │ │ +000929c0: 6963 6965 6e74 2072 696e 6720 4620 6361  icient ring F ca
│ │ │ │ +000929d0: 6e20 6265 0a61 7574 6f6d 6174 6963 616c  n be.automatical
│ │ │ │ +000929e0: 6c79 2063 6f65 7263 6564 2069 6e74 6f20  ly coerced into 
│ │ │ │ +000929f0: 7468 6520 6e65 7720 636f 6566 6669 6369  the new coeffici
│ │ │ │ +00092a00: 656e 7420 7269 6e67 204b 2e0a 0a2b 2d2d  ent ring K...+--
│ │ │ │  00092a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00092a50: 2d2d 2d2d 2d2b 0a7c 6931 203a 2051 515b  -----+.|i1 : QQ[
│ │ │ │ -00092a60: 7661 7273 2830 2e2e 3529 5d20 2020 2020  vars(0..5)]     
│ │ │ │ -00092a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00092a60: 203a 2051 515b 7661 7273 2830 2e2e 3529   : QQ[vars(0..5)
│ │ │ │ +00092a70: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  00092a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092aa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00092aa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00092ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092af0: 2020 2020 207c 0a7c 6f31 203d 2051 515b       |.|o1 = QQ[
│ │ │ │ -00092b00: 612e 2e66 5d20 2020 2020 2020 2020 2020  a..f]           
│ │ │ │ +00092af0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00092b00: 203d 2051 515b 612e 2e66 5d20 2020 2020   = QQ[a..f]     
│ │ │ │  00092b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092b40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00092b40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00092b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092b90: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c       |.|o1 : Pol
│ │ │ │ -00092ba0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ -00092bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092b90: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00092ba0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +00092bb0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  00092bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092be0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00092be0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00092bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00092c30: 2d2d 2d2d 2d2b 0a7c 6932 203a 2070 6869  -----+.|i2 : phi
│ │ │ │ -00092c40: 203d 2072 6174 696f 6e61 6c4d 6170 207b   = rationalMap {
│ │ │ │ -00092c50: 655e 322d 642a 662c 2063 2a65 2d62 2a66  e^2-d*f, c*e-b*f
│ │ │ │ -00092c60: 2c20 632a 642d 622a 652c 2063 5e32 2d61  , c*d-b*e, c^2-a
│ │ │ │ -00092c70: 2a66 2c20 622a 632d 612a 652c 2062 5e32  *f, b*c-a*e, b^2
│ │ │ │ -00092c80: 2d61 2a64 7d7c 0a7c 2020 2020 2020 2020  -a*d}|.|        
│ │ │ │ +00092c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00092c40: 203a 2070 6869 203d 2072 6174 696f 6e61   : phi = rationa
│ │ │ │ +00092c50: 6c4d 6170 207b 655e 322d 642a 662c 2063  lMap {e^2-d*f, c
│ │ │ │ +00092c60: 2a65 2d62 2a66 2c20 632a 642d 622a 652c  *e-b*f, c*d-b*e,
│ │ │ │ +00092c70: 2063 5e32 2d61 2a66 2c20 622a 632d 612a   c^2-a*f, b*c-a*
│ │ │ │ +00092c80: 652c 2062 5e32 2d61 2a64 7d7c 0a7c 2020  e, b^2-a*d}|.|  
│ │ │ │  00092c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092cd0: 2020 2020 207c 0a7c 6f32 203d 202d 2d20       |.|o2 = -- 
│ │ │ │ -00092ce0: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ -00092cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092cd0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00092ce0: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +00092cf0: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  00092d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092d20: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -00092d30: 7263 653a 2050 726f 6a28 5151 5b61 2c20  rce: Proj(QQ[a, 
│ │ │ │ -00092d40: 622c 2063 2c20 642c 2065 2c20 665d 2920  b, c, d, e, f]) 
│ │ │ │ -00092d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092d20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00092d30: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ +00092d40: 5151 5b61 2c20 622c 2063 2c20 642c 2065  QQ[a, b, c, d, e
│ │ │ │ +00092d50: 2c20 665d 2920 2020 2020 2020 2020 2020  , f])           
│ │ │ │  00092d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092d70: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -00092d80: 6765 743a 2050 726f 6a28 5151 5b61 2c20  get: Proj(QQ[a, 
│ │ │ │ -00092d90: 622c 2063 2c20 642c 2065 2c20 665d 2920  b, c, d, e, f]) 
│ │ │ │ -00092da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092d70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00092d80: 2020 2074 6172 6765 743a 2050 726f 6a28     target: Proj(
│ │ │ │ +00092d90: 5151 5b61 2c20 622c 2063 2c20 642c 2065  QQ[a, b, c, d, e
│ │ │ │ +00092da0: 2c20 665d 2920 2020 2020 2020 2020 2020  , f])           
│ │ │ │  00092db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092dc0: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -00092dd0: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ -00092de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092dc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00092dd0: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
│ │ │ │ +00092de0: 733a 207b 2020 2020 2020 2020 2020 2020  s: {            
│ │ │ │  00092df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00092e20: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00092e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092e10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00092e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092e30: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00092e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092e60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00092e70: 2020 2020 2020 2020 2020 2020 2020 6520                e 
│ │ │ │ -00092e80: 202d 2064 2a66 2c20 2020 2020 2020 2020   - d*f,         
│ │ │ │ +00092e60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00092e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092e80: 2020 2020 6520 202d 2064 2a66 2c20 2020      e  - d*f,   
│ │ │ │  00092e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092eb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00092eb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00092ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00092f10: 2020 2020 2020 2020 2020 2020 2020 632a                c*
│ │ │ │ -00092f20: 6520 2d20 622a 662c 2020 2020 2020 2020  e - b*f,        
│ │ │ │ +00092f00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00092f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092f20: 2020 2020 632a 6520 2d20 622a 662c 2020      c*e - b*f,  
│ │ │ │  00092f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092f50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00092f50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00092f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092fa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00092fb0: 2020 2020 2020 2020 2020 2020 2020 632a                c*
│ │ │ │ -00092fc0: 6420 2d20 622a 652c 2020 2020 2020 2020  d - b*e,        
│ │ │ │ +00092fa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00092fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00092fc0: 2020 2020 632a 6420 2d20 622a 652c 2020      c*d - b*e,  
│ │ │ │  00092fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092ff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00092ff0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093040: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093050: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00093060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093040: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093060: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00093070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000930a0: 2020 2020 2020 2020 2020 2020 2020 6320                c 
│ │ │ │ -000930b0: 202d 2061 2a66 2c20 2020 2020 2020 2020   - a*f,         
│ │ │ │ +00093090: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000930a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000930b0: 2020 2020 6320 202d 2061 2a66 2c20 2020      c  - a*f,   
│ │ │ │  000930c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000930d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000930e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000930e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000930f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093130: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093140: 2020 2020 2020 2020 2020 2020 2020 622a                b*
│ │ │ │ -00093150: 6320 2d20 612a 652c 2020 2020 2020 2020  c - a*e,        
│ │ │ │ +00093130: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093150: 2020 2020 622a 6320 2d20 612a 652c 2020      b*c - a*e,  
│ │ │ │  00093160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093180: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00093180: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000931a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000931b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000931c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000931d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000931e0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -000931f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000931d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000931e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000931f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00093200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093220: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093230: 2020 2020 2020 2020 2020 2020 2020 6220                b 
│ │ │ │ -00093240: 202d 2061 2a64 2020 2020 2020 2020 2020   - a*d          
│ │ │ │ +00093220: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093240: 2020 2020 6220 202d 2061 2a64 2020 2020      b  - a*d    
│ │ │ │  00093250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093280: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ -00093290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093270: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093290: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  000932a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000932b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000932c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000932c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000932d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000932e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000932f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093310: 2020 2020 207c 0a7c 6f32 203a 2052 6174       |.|o2 : Rat
│ │ │ │ -00093320: 696f 6e61 6c4d 6170 2028 7175 6164 7261  ionalMap (quadra
│ │ │ │ -00093330: 7469 6320 7261 7469 6f6e 616c 206d 6170  tic rational map
│ │ │ │ -00093340: 2066 726f 6d20 5050 5e35 2074 6f20 5050   from PP^5 to PP
│ │ │ │ -00093350: 5e35 2920 2020 2020 2020 2020 2020 2020  ^5)             
│ │ │ │ -00093360: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00093310: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00093320: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +00093330: 7175 6164 7261 7469 6320 7261 7469 6f6e  quadratic ration
│ │ │ │ +00093340: 616c 206d 6170 2066 726f 6d20 5050 5e35  al map from PP^5
│ │ │ │ +00093350: 2074 6f20 5050 5e35 2920 2020 2020 2020   to PP^5)       
│ │ │ │ +00093360: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00093370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000933a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000933b0: 2d2d 2d2d 2d2b 0a7c 6933 203a 204b 203d  -----+.|i3 : K =
│ │ │ │ -000933c0: 205a 5a2f 3635 3532 313b 2020 2020 2020   ZZ/65521;      
│ │ │ │ +000933b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +000933c0: 203a 204b 203d 205a 5a2f 3635 3532 313b   : K = ZZ/65521;
│ │ │ │  000933d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000933e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000933f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093400: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00093400: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00093410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00093450: 2d2d 2d2d 2d2b 0a7c 6934 203a 2070 6869  -----+.|i4 : phi
│ │ │ │ -00093460: 202a 2a20 4b20 2020 2020 2020 2020 2020   ** K           
│ │ │ │ +00093450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +00093460: 203a 2070 6869 202a 2a20 4b20 2020 2020   : phi ** K     
│ │ │ │  00093470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000934a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000934a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000934b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000934c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000934d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000934e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000934f0: 2020 2020 207c 0a7c 6f34 203d 202d 2d20       |.|o4 = -- 
│ │ │ │ -00093500: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ -00093510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000934f0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +00093500: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +00093510: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  00093520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093540: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -00093550: 7263 653a 2050 726f 6a28 4b5b 612c 2062  rce: Proj(K[a, b
│ │ │ │ -00093560: 2c20 632c 2064 2c20 652c 2066 5d29 2020  , c, d, e, f])  
│ │ │ │ -00093570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093540: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093550: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ +00093560: 4b5b 612c 2062 2c20 632c 2064 2c20 652c  K[a, b, c, d, e,
│ │ │ │ +00093570: 2066 5d29 2020 2020 2020 2020 2020 2020   f])            
│ │ │ │  00093580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093590: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -000935a0: 6765 743a 2050 726f 6a28 4b5b 612c 2062  get: Proj(K[a, b
│ │ │ │ -000935b0: 2c20 632c 2064 2c20 652c 2066 5d29 2020  , c, d, e, f])  
│ │ │ │ -000935c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000935a0: 2020 2074 6172 6765 743a 2050 726f 6a28     target: Proj(
│ │ │ │ +000935b0: 4b5b 612c 2062 2c20 632c 2064 2c20 652c  K[a, b, c, d, e,
│ │ │ │ +000935c0: 2066 5d29 2020 2020 2020 2020 2020 2020   f])            
│ │ │ │  000935d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000935e0: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -000935f0: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ -00093600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000935e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000935f0: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
│ │ │ │ +00093600: 733a 207b 2020 2020 2020 2020 2020 2020  s: {            
│ │ │ │  00093610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093630: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093640: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00093650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093630: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093650: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00093660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093690: 2020 2020 2020 2020 2020 2020 2020 6520                e 
│ │ │ │ -000936a0: 202d 2064 2a66 2c20 2020 2020 2020 2020   - d*f,         
│ │ │ │ +00093680: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000936a0: 2020 2020 6520 202d 2064 2a66 2c20 2020      e  - d*f,   
│ │ │ │  000936b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000936c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000936d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000936d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000936e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000936f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093720: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093730: 2020 2020 2020 2020 2020 2020 2020 632a                c*
│ │ │ │ -00093740: 6520 2d20 622a 662c 2020 2020 2020 2020  e - b*f,        
│ │ │ │ +00093720: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093740: 2020 2020 632a 6520 2d20 622a 662c 2020      c*e - b*f,  
│ │ │ │  00093750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093770: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00093770: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000937a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000937b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000937c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000937d0: 2020 2020 2020 2020 2020 2020 2020 632a                c*
│ │ │ │ -000937e0: 6420 2d20 622a 652c 2020 2020 2020 2020  d - b*e,        
│ │ │ │ +000937c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000937d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000937e0: 2020 2020 632a 6420 2d20 622a 652c 2020      c*d - b*e,  
│ │ │ │  000937f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00093810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093860: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093870: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00093880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093860: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093880: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00093890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000938a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000938b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000938c0: 2020 2020 2020 2020 2020 2020 2020 6320                c 
│ │ │ │ -000938d0: 202d 2061 2a66 2c20 2020 2020 2020 2020   - a*f,         
│ │ │ │ +000938b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000938c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000938d0: 2020 2020 6320 202d 2061 2a66 2c20 2020      c  - a*f,   
│ │ │ │  000938e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000938f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093900: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00093900: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093950: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093960: 2020 2020 2020 2020 2020 2020 2020 622a                b*
│ │ │ │ -00093970: 6320 2d20 612a 652c 2020 2020 2020 2020  c - a*e,        
│ │ │ │ +00093950: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093970: 2020 2020 622a 6320 2d20 612a 652c 2020      b*c - a*e,  
│ │ │ │  00093980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000939a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000939a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000939b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000939c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000939d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000939e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000939f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093a00: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00093a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000939f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093a10: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00093a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093a40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093a50: 2020 2020 2020 2020 2020 2020 2020 6220                b 
│ │ │ │ -00093a60: 202d 2061 2a64 2020 2020 2020 2020 2020   - a*d          
│ │ │ │ +00093a40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093a60: 2020 2020 6220 202d 2061 2a64 2020 2020      b  - a*d    
│ │ │ │  00093a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093a90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093aa0: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ -00093ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093a90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093ab0: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  00093ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093ae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00093ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093b30: 2020 2020 207c 0a7c 6f34 203a 2052 6174       |.|o4 : Rat
│ │ │ │ -00093b40: 696f 6e61 6c4d 6170 2028 7175 6164 7261  ionalMap (quadra
│ │ │ │ -00093b50: 7469 6320 7261 7469 6f6e 616c 206d 6170  tic rational map
│ │ │ │ -00093b60: 2066 726f 6d20 5050 5e35 2074 6f20 5050   from PP^5 to PP
│ │ │ │ -00093b70: 5e35 2920 2020 2020 2020 2020 2020 2020  ^5)             
│ │ │ │ -00093b80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00093b30: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +00093b40: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +00093b50: 7175 6164 7261 7469 6320 7261 7469 6f6e  quadratic ration
│ │ │ │ +00093b60: 616c 206d 6170 2066 726f 6d20 5050 5e35  al map from PP^5
│ │ │ │ +00093b70: 2074 6f20 5050 5e35 2920 2020 2020 2020   to PP^5)       
│ │ │ │ +00093b80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00093b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00093bd0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2070 6869  -----+.|i5 : phi
│ │ │ │ -00093be0: 202a 2a20 6672 6163 284b 5b74 5d29 2020   ** frac(K[t])  
│ │ │ │ -00093bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +00093be0: 203a 2070 6869 202a 2a20 6672 6163 284b   : phi ** frac(K
│ │ │ │ +00093bf0: 5b74 5d29 2020 2020 2020 2020 2020 2020  [t])            
│ │ │ │  00093c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093c20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00093c20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093c70: 2020 2020 207c 0a7c 6f35 203d 202d 2d20       |.|o5 = -- 
│ │ │ │ -00093c80: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ -00093c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093c70: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +00093c80: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +00093c90: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  00093ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093cc0: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -00093cd0: 7263 653a 2050 726f 6a28 6672 6163 284b  rce: Proj(frac(K
│ │ │ │ -00093ce0: 5b74 5d29 5b61 2c20 622c 2063 2c20 642c  [t])[a, b, c, d,
│ │ │ │ -00093cf0: 2065 2c20 665d 2920 2020 2020 2020 2020   e, f])         
│ │ │ │ +00093cc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093cd0: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ +00093ce0: 6672 6163 284b 5b74 5d29 5b61 2c20 622c  frac(K[t])[a, b,
│ │ │ │ +00093cf0: 2063 2c20 642c 2065 2c20 665d 2920 2020   c, d, e, f])   
│ │ │ │  00093d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093d10: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -00093d20: 6765 743a 2050 726f 6a28 6672 6163 284b  get: Proj(frac(K
│ │ │ │ -00093d30: 5b74 5d29 5b61 2c20 622c 2063 2c20 642c  [t])[a, b, c, d,
│ │ │ │ -00093d40: 2065 2c20 665d 2920 2020 2020 2020 2020   e, f])         
│ │ │ │ +00093d10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093d20: 2020 2074 6172 6765 743a 2050 726f 6a28     target: Proj(
│ │ │ │ +00093d30: 6672 6163 284b 5b74 5d29 5b61 2c20 622c  frac(K[t])[a, b,
│ │ │ │ +00093d40: 2063 2c20 642c 2065 2c20 665d 2920 2020   c, d, e, f])   
│ │ │ │  00093d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093d60: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -00093d70: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ -00093d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093d60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093d70: 2020 2064 6566 696e 696e 6720 666f 726d     defining form
│ │ │ │ +00093d80: 733a 207b 2020 2020 2020 2020 2020 2020  s: {            
│ │ │ │  00093d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093db0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093dc0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00093dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093db0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093dd0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00093de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093e00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093e10: 2020 2020 2020 2020 2020 2020 2020 6520                e 
│ │ │ │ -00093e20: 202d 2064 2a66 2c20 2020 2020 2020 2020   - d*f,         
│ │ │ │ +00093e00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093e20: 2020 2020 6520 202d 2064 2a66 2c20 2020      e  - d*f,   
│ │ │ │  00093e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093e50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00093e50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093ea0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093eb0: 2020 2020 2020 2020 2020 2020 2020 632a                c*
│ │ │ │ -00093ec0: 6520 2d20 622a 662c 2020 2020 2020 2020  e - b*f,        
│ │ │ │ +00093ea0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093ec0: 2020 2020 632a 6520 2d20 622a 662c 2020      c*e - b*f,  
│ │ │ │  00093ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093ef0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00093ef0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093f40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093f50: 2020 2020 2020 2020 2020 2020 2020 632a                c*
│ │ │ │ -00093f60: 6420 2d20 622a 652c 2020 2020 2020 2020  d - b*e,        
│ │ │ │ +00093f40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093f60: 2020 2020 632a 6420 2d20 622a 652c 2020      c*d - b*e,  
│ │ │ │  00093f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093f90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00093f90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00093fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00093ff0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00094000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093fe0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00093ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00094000: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00094010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094030: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00094040: 2020 2020 2020 2020 2020 2020 2020 6320                c 
│ │ │ │ -00094050: 202d 2061 2a66 2c20 2020 2020 2020 2020   - a*f,         
│ │ │ │ +00094030: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00094040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00094050: 2020 2020 6320 202d 2061 2a66 2c20 2020      c  - a*f,   
│ │ │ │  00094060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094080: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00094080: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00094090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000940a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000940b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000940c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000940d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000940e0: 2020 2020 2020 2020 2020 2020 2020 622a                b*
│ │ │ │ -000940f0: 6320 2d20 612a 652c 2020 2020 2020 2020  c - a*e,        
│ │ │ │ +000940d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000940e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000940f0: 2020 2020 622a 6320 2d20 612a 652c 2020      b*c - a*e,  
│ │ │ │  00094100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094120: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00094120: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00094130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094170: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00094180: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00094190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00094170: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00094180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00094190: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  000941a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000941b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000941c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000941d0: 2020 2020 2020 2020 2020 2020 2020 6220                b 
│ │ │ │ -000941e0: 202d 2061 2a64 2020 2020 2020 2020 2020   - a*d          
│ │ │ │ +000941c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000941d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000941e0: 2020 2020 6220 202d 2061 2a64 2020 2020      b  - a*d    
│ │ │ │  000941f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094210: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00094220: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ -00094230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00094210: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00094220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00094230: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  00094240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094260: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00094260: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00094270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000942a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000942b0: 2020 2020 207c 0a7c 6f35 203a 2052 6174       |.|o5 : Rat
│ │ │ │ -000942c0: 696f 6e61 6c4d 6170 2028 7175 6164 7261  ionalMap (quadra
│ │ │ │ -000942d0: 7469 6320 7261 7469 6f6e 616c 206d 6170  tic rational map
│ │ │ │ -000942e0: 2066 726f 6d20 5050 5e35 2074 6f20 5050   from PP^5 to PP
│ │ │ │ -000942f0: 5e35 2920 2020 2020 2020 2020 2020 2020  ^5)             
│ │ │ │ -00094300: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000942b0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +000942c0: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +000942d0: 7175 6164 7261 7469 6320 7261 7469 6f6e  quadratic ration
│ │ │ │ +000942e0: 616c 206d 6170 2066 726f 6d20 5050 5e35  al map from PP^5
│ │ │ │ +000942f0: 2074 6f20 5050 5e35 2920 2020 2020 2020   to PP^5)       
│ │ │ │ +00094300: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00094310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00094350: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ -00094360: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00094370: 6e6f 7465 2063 6f65 6666 6963 6965 6e74  note coefficient
│ │ │ │ -00094380: 5269 6e67 2852 6174 696f 6e61 6c4d 6170  Ring(RationalMap
│ │ │ │ -00094390: 293a 2063 6f65 6666 6963 6965 6e74 5269  ): coefficientRi
│ │ │ │ -000943a0: 6e67 5f6c 7052 6174 696f 6e61 6c4d 6170  ng_lpRationalMap
│ │ │ │ -000943b0: 5f72 702c 202d 2d0a 2020 2020 636f 6566  _rp, --.    coef
│ │ │ │ -000943c0: 6669 6369 656e 7420 7269 6e67 206f 6620  ficient ring of 
│ │ │ │ -000943d0: 6120 7261 7469 6f6e 616c 206d 6170 0a0a  a rational map..
│ │ │ │ -000943e0: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
│ │ │ │ -000943f0: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   method:.=======
│ │ │ │ +00094350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ +00094360: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +00094370: 0a20 202a 202a 6e6f 7465 2063 6f65 6666  .  * *note coeff
│ │ │ │ +00094380: 6963 6965 6e74 5269 6e67 2852 6174 696f  icientRing(Ratio
│ │ │ │ +00094390: 6e61 6c4d 6170 293a 2063 6f65 6666 6963  nalMap): coeffic
│ │ │ │ +000943a0: 6965 6e74 5269 6e67 5f6c 7052 6174 696f  ientRing_lpRatio
│ │ │ │ +000943b0: 6e61 6c4d 6170 5f72 702c 202d 2d0a 2020  nalMap_rp, --.  
│ │ │ │ +000943c0: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ +000943d0: 6e67 206f 6620 6120 7261 7469 6f6e 616c  ng of a rational
│ │ │ │ +000943e0: 206d 6170 0a0a 5761 7973 2074 6f20 7573   map..Ways to us
│ │ │ │ +000943f0: 6520 7468 6973 206d 6574 686f 643a 0a3d  e this method:.=
│ │ │ │  00094400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00094410: 3d0a 0a20 202a 202a 6e6f 7465 2052 6174  =..  * *note Rat
│ │ │ │ -00094420: 696f 6e61 6c4d 6170 202a 2a20 5269 6e67  ionalMap ** Ring
│ │ │ │ -00094430: 3a20 5261 7469 6f6e 616c 4d61 7020 5f73  : RationalMap _s
│ │ │ │ -00094440: 745f 7374 2052 696e 672c 202d 2d20 6368  t_st Ring, -- ch
│ │ │ │ -00094450: 616e 6765 2074 6865 0a20 2020 2063 6f65  ange the.    coe
│ │ │ │ -00094460: 6666 6963 6965 6e74 2072 696e 6720 6f66  fficient ring of
│ │ │ │ -00094470: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ -00094480: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -00094490: 696e 666f 2c20 4e6f 6465 3a20 5261 7469  info, Node: Rati
│ │ │ │ -000944a0: 6f6e 616c 4d61 7020 3d3d 2052 6174 696f  onalMap == Ratio
│ │ │ │ -000944b0: 6e61 6c4d 6170 2c20 4e65 7874 3a20 5261  nalMap, Next: Ra
│ │ │ │ -000944c0: 7469 6f6e 616c 4d61 7020 5e20 5a5a 2c20  tionalMap ^ ZZ, 
│ │ │ │ -000944d0: 5072 6576 3a20 5261 7469 6f6e 616c 4d61  Prev: RationalMa
│ │ │ │ -000944e0: 7020 5f73 745f 7374 2052 696e 672c 2055  p _st_st Ring, U
│ │ │ │ -000944f0: 703a 2054 6f70 0a0a 5261 7469 6f6e 616c  p: Top..Rational
│ │ │ │ -00094500: 4d61 7020 3d3d 2052 6174 696f 6e61 6c4d  Map == RationalM
│ │ │ │ -00094510: 6170 202d 2d20 6571 7561 6c69 7479 206f  ap -- equality o
│ │ │ │ -00094520: 6620 7261 7469 6f6e 616c 206d 6170 730a  f rational maps.
│ │ │ │ -00094530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00094410: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ +00094420: 7465 2052 6174 696f 6e61 6c4d 6170 202a  te RationalMap *
│ │ │ │ +00094430: 2a20 5269 6e67 3a20 5261 7469 6f6e 616c  * Ring: Rational
│ │ │ │ +00094440: 4d61 7020 5f73 745f 7374 2052 696e 672c  Map _st_st Ring,
│ │ │ │ +00094450: 202d 2d20 6368 616e 6765 2074 6865 0a20   -- change the. 
│ │ │ │ +00094460: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
│ │ │ │ +00094470: 696e 6720 6f66 2061 2072 6174 696f 6e61  ing of a rationa
│ │ │ │ +00094480: 6c20 6d61 700a 1f0a 4669 6c65 3a20 4372  l map...File: Cr
│ │ │ │ +00094490: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ +000944a0: 3a20 5261 7469 6f6e 616c 4d61 7020 3d3d  : RationalMap ==
│ │ │ │ +000944b0: 2052 6174 696f 6e61 6c4d 6170 2c20 4e65   RationalMap, Ne
│ │ │ │ +000944c0: 7874 3a20 5261 7469 6f6e 616c 4d61 7020  xt: RationalMap 
│ │ │ │ +000944d0: 5e20 5a5a 2c20 5072 6576 3a20 5261 7469  ^ ZZ, Prev: Rati
│ │ │ │ +000944e0: 6f6e 616c 4d61 7020 5f73 745f 7374 2052  onalMap _st_st R
│ │ │ │ +000944f0: 696e 672c 2055 703a 2054 6f70 0a0a 5261  ing, Up: Top..Ra
│ │ │ │ +00094500: 7469 6f6e 616c 4d61 7020 3d3d 2052 6174  tionalMap == Rat
│ │ │ │ +00094510: 696f 6e61 6c4d 6170 202d 2d20 6571 7561  ionalMap -- equa
│ │ │ │ +00094520: 6c69 7479 206f 6620 7261 7469 6f6e 616c  lity of rational
│ │ │ │ +00094530: 206d 6170 730a 2a2a 2a2a 2a2a 2a2a 2a2a   maps.**********
│ │ │ │  00094540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00094550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00094560: 2a2a 2a2a 2a2a 2a0a 0a53 796e 6f70 7369  *******..Synopsi
│ │ │ │ -00094570: 730a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  s.========..  * 
│ │ │ │ -00094580: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
│ │ │ │ -00094590: 3d3d 3a20 284d 6163 6175 6c61 7932 446f  ==: (Macaulay2Do
│ │ │ │ -000945a0: 6329 3d3d 2c0a 2020 2a20 5573 6167 653a  c)==,.  * Usage:
│ │ │ │ -000945b0: 200a 2020 2020 2020 2020 7068 6920 3d3d   .        phi ==
│ │ │ │ -000945c0: 2070 7369 0a20 202a 2049 6e70 7574 733a   psi.  * Inputs:
│ │ │ │ -000945d0: 0a20 2020 2020 202a 2070 6869 2c20 6120  .      * phi, a 
│ │ │ │ -000945e0: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ -000945f0: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ -00094600: 0a20 2020 2020 202a 2070 7369 2c20 6120  .      * psi, a 
│ │ │ │ -00094610: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ -00094620: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ -00094630: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00094640: 2020 2020 2a20 6120 2a6e 6f74 6520 426f      * a *note Bo
│ │ │ │ -00094650: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61  olean value: (Ma
│ │ │ │ -00094660: 6361 756c 6179 3244 6f63 2942 6f6f 6c65  caulay2Doc)Boole
│ │ │ │ -00094670: 616e 2c2c 2020 7768 6574 6865 7220 7068  an,,  whether ph
│ │ │ │ -00094680: 6920 616e 6420 7073 6920 6172 650a 2020  i and psi are.  
│ │ │ │ -00094690: 2020 2020 2020 7468 6520 7361 6d65 2072        the same r
│ │ │ │ -000946a0: 6174 696f 6e61 6c20 6d61 700a 0a44 6573  ational map..Des
│ │ │ │ -000946b0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -000946c0: 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ====..+---------
│ │ │ │ +00094560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53  *************..S
│ │ │ │ +00094570: 796e 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d  ynopsis.========
│ │ │ │ +00094580: 0a0a 2020 2a20 4f70 6572 6174 6f72 3a20  ..  * Operator: 
│ │ │ │ +00094590: 2a6e 6f74 6520 3d3d 3a20 284d 6163 6175  *note ==: (Macau
│ │ │ │ +000945a0: 6c61 7932 446f 6329 3d3d 2c0a 2020 2a20  lay2Doc)==,.  * 
│ │ │ │ +000945b0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +000945c0: 7068 6920 3d3d 2070 7369 0a20 202a 2049  phi == psi.  * I
│ │ │ │ +000945d0: 6e70 7574 733a 0a20 2020 2020 202a 2070  nputs:.      * p
│ │ │ │ +000945e0: 6869 2c20 6120 2a6e 6f74 6520 7261 7469  hi, a *note rati
│ │ │ │ +000945f0: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ +00094600: 616c 4d61 702c 0a20 2020 2020 202a 2070  alMap,.      * p
│ │ │ │ +00094610: 7369 2c20 6120 2a6e 6f74 6520 7261 7469  si, a *note rati
│ │ │ │ +00094620: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ +00094630: 616c 4d61 702c 0a20 202a 204f 7574 7075  alMap,.  * Outpu
│ │ │ │ +00094640: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ +00094650: 6f74 6520 426f 6f6c 6561 6e20 7661 6c75  ote Boolean valu
│ │ │ │ +00094660: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +00094670: 2942 6f6f 6c65 616e 2c2c 2020 7768 6574  )Boolean,,  whet
│ │ │ │ +00094680: 6865 7220 7068 6920 616e 6420 7073 6920  her phi and psi 
│ │ │ │ +00094690: 6172 650a 2020 2020 2020 2020 7468 6520  are.        the 
│ │ │ │ +000946a0: 7361 6d65 2072 6174 696f 6e61 6c20 6d61  same rational ma
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│ │ │ │ +000946c0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d  ==========..+---
│ │ │ │  000946d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000946e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000946f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00094710: 2d2d 2d2d 2b0a 7c69 3120 3a20 5151 5b78  ----+.|i1 : QQ[x
│ │ │ │ -00094720: 5f30 2e2e 785f 355d 2020 2020 2020 2020  _0..x_5]        
│ │ │ │ +00094710: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00094720: 3a20 5151 5b78 5f30 2e2e 785f 355d 2020  : QQ[x_0..x_5]  
│ │ │ │  00094730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094760: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00094760: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00094770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000947a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000947b0: 2020 2020 7c0a 7c6f 3120 3d20 5151 5b78      |.|o1 = QQ[x
│ │ │ │ -000947c0: 202e 2e78 205d 2020 2020 2020 2020 2020   ..x ]          
│ │ │ │ +000947b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
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│ │ │ │  000947d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000947e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000947f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094800: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  00094830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00094850: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00094860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000948a0: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79      |.|o1 : Poly
│ │ │ │ -000948b0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +000948a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +000948b0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │  000948c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000948d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000948e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000948f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000948f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
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│ │ │ │  00094920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00094940: 2d2d 2d2d 2b0a 7c69 3220 3a20 7068 6920  ----+.|i2 : phi 
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│ │ │ │  000949a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000949e0: 2020 2020 7c0a 7c6f 3220 3d20 2d2d 2072      |.|o2 = -- r
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│ │ │ │  00094a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00095220: 2020 2020 3020 3120 2020 2030 2033 2020      0 1    0 3  
│ │ │ │  00095230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095250: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095260: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ -00095270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095250: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095270: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │  00095280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000952a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000952a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000952b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000952c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000952d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000952e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000952f0: 2020 2020 7c0a 7c6f 3220 3a20 5261 7469      |.|o2 : Rati
│ │ │ │ -00095300: 6f6e 616c 4d61 7020 2863 7562 6963 2072  onalMap (cubic r
│ │ │ │ -00095310: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ -00095320: 2050 505e 3520 746f 2050 505e 3529 2020   PP^5 to PP^5)  
│ │ │ │ -00095330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095340: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +000952f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00095300: 3a20 5261 7469 6f6e 616c 4d61 7020 2863  : RationalMap (c
│ │ │ │ +00095310: 7562 6963 2072 6174 696f 6e61 6c20 6d61  ubic rational ma
│ │ │ │ +00095320: 7020 6672 6f6d 2050 505e 3520 746f 2050  p from PP^5 to P
│ │ │ │ +00095330: 505e 3529 2020 2020 2020 2020 2020 2020  P^5)            
│ │ │ │ +00095340: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  00095350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00095390: 2d2d 2d2d 7c0a 7c2a 785f 332d 785f 302a  ----|.|*x_3-x_0*
│ │ │ │ -000953a0: 785f 312a 785f 342c 785f 302a 785f 325e  x_1*x_4,x_0*x_2^
│ │ │ │ -000953b0: 322d 785f 305e 322a 785f 352c 785f 302a  2-x_0^2*x_5,x_0*
│ │ │ │ -000953c0: 785f 312a 785f 322d 785f 305e 322a 785f  x_1*x_2-x_0^2*x_
│ │ │ │ -000953d0: 342c 785f 302a 785f 315e 322d 785f 305e  4,x_0*x_1^2-x_0^
│ │ │ │ -000953e0: 322a 785f 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  2*x_|.|---------
│ │ │ │ +00095390: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 785f  ----------|.|*x_
│ │ │ │ +000953a0: 332d 785f 302a 785f 312a 785f 342c 785f  3-x_0*x_1*x_4,x_
│ │ │ │ +000953b0: 302a 785f 325e 322d 785f 305e 322a 785f  0*x_2^2-x_0^2*x_
│ │ │ │ +000953c0: 352c 785f 302a 785f 312a 785f 322d 785f  5,x_0*x_1*x_2-x_
│ │ │ │ +000953d0: 305e 322a 785f 342c 785f 302a 785f 315e  0^2*x_4,x_0*x_1^
│ │ │ │ +000953e0: 322d 785f 305e 322a 785f 7c0a 7c2d 2d2d  2-x_0^2*x_|.|---
│ │ │ │  000953f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00095430: 2d2d 2d2d 7c0a 7c33 7d20 2020 2020 2020  ----|.|3}       
│ │ │ │ +00095430: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c33 7d20  ----------|.|3} 
│ │ │ │  00095440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095480: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00095480: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00095490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000954a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000954b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000954c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000954d0: 2d2d 2d2d 2b0a 7c69 3320 3a20 7073 6920  ----+.|i3 : psi 
│ │ │ │ -000954e0: 3d20 7261 7469 6f6e 616c 4d61 7020 7b78  = rationalMap {x
│ │ │ │ -000954f0: 5f34 5e32 2d78 5f33 2a78 5f35 2c78 5f32  _4^2-x_3*x_5,x_2
│ │ │ │ -00095500: 2a78 5f34 2d78 5f31 2a78 5f35 2c78 5f32  *x_4-x_1*x_5,x_2
│ │ │ │ -00095510: 2a78 5f33 2d78 5f31 2a78 5f34 2c78 5f32  *x_3-x_1*x_4,x_2
│ │ │ │ -00095520: 5e32 2d78 7c0a 7c20 2020 2020 2020 2020  ^2-x|.|         
│ │ │ │ +000954d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ +000954e0: 3a20 7073 6920 3d20 7261 7469 6f6e 616c  : psi = rational
│ │ │ │ +000954f0: 4d61 7020 7b78 5f34 5e32 2d78 5f33 2a78  Map {x_4^2-x_3*x
│ │ │ │ +00095500: 5f35 2c78 5f32 2a78 5f34 2d78 5f31 2a78  _5,x_2*x_4-x_1*x
│ │ │ │ +00095510: 5f35 2c78 5f32 2a78 5f33 2d78 5f31 2a78  _5,x_2*x_3-x_1*x
│ │ │ │ +00095520: 5f34 2c78 5f32 5e32 2d78 7c0a 7c20 2020  _4,x_2^2-x|.|   
│ │ │ │  00095530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095570: 2020 2020 7c0a 7c6f 3320 3d20 2d2d 2072      |.|o3 = -- r
│ │ │ │ -00095580: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │ -00095590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095570: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +00095580: 3d20 2d2d 2072 6174 696f 6e61 6c20 6d61  = -- rational ma
│ │ │ │ +00095590: 7020 2d2d 2020 2020 2020 2020 2020 2020  p --            
│ │ │ │  000955a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000955b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000955c0: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ -000955d0: 6365 3a20 5072 6f6a 2851 515b 7820 2c20  ce: Proj(QQ[x , 
│ │ │ │ -000955e0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -000955f0: 7820 5d29 2020 2020 2020 2020 2020 2020  x ])            
│ │ │ │ +000955c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000955d0: 2020 736f 7572 6365 3a20 5072 6f6a 2851    source: Proj(Q
│ │ │ │ +000955e0: 515b 7820 2c20 7820 2c20 7820 2c20 7820  Q[x , x , x , x 
│ │ │ │ +000955f0: 2c20 7820 2c20 7820 5d29 2020 2020 2020  , x , x ])      
│ │ │ │  00095600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095610: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095620: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00095630: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -00095640: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +00095610: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095630: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
│ │ │ │ +00095640: 2020 2034 2020 2035 2020 2020 2020 2020     4   5        
│ │ │ │  00095650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095660: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ -00095670: 6574 3a20 5072 6f6a 2851 515b 7820 2c20  et: Proj(QQ[x , 
│ │ │ │ -00095680: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -00095690: 7820 5d29 2020 2020 2020 2020 2020 2020  x ])            
│ │ │ │ +00095660: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095670: 2020 7461 7267 6574 3a20 5072 6f6a 2851    target: Proj(Q
│ │ │ │ +00095680: 515b 7820 2c20 7820 2c20 7820 2c20 7820  Q[x , x , x , x 
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│ │ │ │  000956a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000956b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000956c0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000956d0: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -000956e0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +000956b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000956c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000956d0: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
│ │ │ │ +000956e0: 2020 2034 2020 2035 2020 2020 2020 2020     4   5        
│ │ │ │  000956f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095700: 2020 2020 7c0a 7c20 2020 2020 6465 6669      |.|     defi
│ │ │ │ -00095710: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │ -00095720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095700: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095710: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ +00095720: 3a20 7b20 2020 2020 2020 2020 2020 2020  : {             
│ │ │ │  00095730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095760: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00095770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095750: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095770: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00095780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000957a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000957b0: 2020 2020 2020 2020 2020 2020 2078 2020               x  
│ │ │ │ -000957c0: 2d20 7820 7820 2c20 2020 2020 2020 2020  - x x ,         
│ │ │ │ +000957a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000957b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000957c0: 2020 2078 2020 2d20 7820 7820 2c20 2020     x  - x x ,   
│ │ │ │  000957d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000957e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000957f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095800: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ -00095810: 2020 2033 2035 2020 2020 2020 2020 2020     3 5          
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│ │ │ │ +00095800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095810: 2020 2020 3420 2020 2033 2035 2020 2020      4    3 5    
│ │ │ │  00095820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095840: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00095840: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00095850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095890: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000958a0: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ -000958b0: 2020 2d20 7820 7820 2c20 2020 2020 2020    - x x ,       
│ │ │ │ +00095890: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000958a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000958b0: 2020 2078 2078 2020 2d20 7820 7820 2c20     x x  - x x , 
│ │ │ │  000958c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000958d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000958e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000958f0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00095900: 3420 2020 2031 2035 2020 2020 2020 2020  4    1 5        
│ │ │ │ +000958e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000958f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095900: 2020 2020 3220 3420 2020 2031 2035 2020      2 4    1 5  
│ │ │ │  00095910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095930: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00095930: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00095940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095980: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095990: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ -000959a0: 2020 2d20 7820 7820 2c20 2020 2020 2020    - x x ,       
│ │ │ │ +00095980: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000959a0: 2020 2078 2078 2020 2d20 7820 7820 2c20     x x  - x x , 
│ │ │ │  000959b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000959c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000959d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000959e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000959f0: 3320 2020 2031 2034 2020 2020 2020 2020  3    1 4        
│ │ │ │ +000959d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000959e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000959f0: 2020 2020 3220 3320 2020 2031 2034 2020      2 3    1 4  
│ │ │ │  00095a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095a20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00095a20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00095a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095a70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095a80: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00095a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095a70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095a90: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00095aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095ac0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095ad0: 2020 2020 2020 2020 2020 2020 2078 2020               x  
│ │ │ │ -00095ae0: 2d20 7820 7820 2c20 2020 2020 2020 2020  - x x ,         
│ │ │ │ +00095ac0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095ae0: 2020 2078 2020 2d20 7820 7820 2c20 2020     x  - x x ,   
│ │ │ │  00095af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095b10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095b20: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00095b30: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ +00095b10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095b30: 2020 2020 3220 2020 2030 2035 2020 2020      2    0 5    
│ │ │ │  00095b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095b60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00095b60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00095b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095bb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095bc0: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ -00095bd0: 2020 2d20 7820 7820 2c20 2020 2020 2020    - x x ,       
│ │ │ │ +00095bb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095bd0: 2020 2078 2078 2020 2d20 7820 7820 2c20     x x  - x x , 
│ │ │ │  00095be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095c00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095c10: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00095c20: 3220 2020 2030 2034 2020 2020 2020 2020  2    0 4        
│ │ │ │ +00095c00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095c20: 2020 2020 3120 3220 2020 2030 2034 2020      1 2    0 4  
│ │ │ │  00095c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095c50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00095c50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00095c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095ca0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095cb0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00095cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095ca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095cc0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00095cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095cf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095d00: 2020 2020 2020 2020 2020 2020 2078 2020               x  
│ │ │ │ -00095d10: 2d20 7820 7820 2020 2020 2020 2020 2020  - x x           
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│ │ │ │ +00095d10: 2020 2078 2020 2d20 7820 7820 2020 2020     x  - x x     
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│ │ │ │  00095d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095d40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095d50: 2020 2020 2020 2020 2020 2020 2020 3120                1 
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│ │ │ │  00095d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095d90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095da0: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
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│ │ │ │  00095e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095e30: 2020 2020 7c0a 7c6f 3320 3a20 5261 7469      |.|o3 : Rati
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│ │ │ │ +00095e80: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │  00095e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00095ed0: 2d2d 2d2d 7c0a 7c5f 302a 785f 352c 785f  ----|.|_0*x_5,x_
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│ │ │ │  00095f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095f20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00095f20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00095f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00095f80: 3d3d 2070 7369 2020 2020 2020 2020 2020  == psi          
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│ │ │ │ +00095f80: 3a20 7068 6920 3d3d 2070 7369 2020 2020  : phi == psi    
│ │ │ │  00095f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095fc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00095fc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00095fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096010: 2020 2020 7c0a 7c6f 3420 3d20 7472 7565      |.|o4 = true
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│ │ │ │ +00096010: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +00096020: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │  00096030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096060: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00096060: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00096070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00096080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00096090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000960a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000960b0: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
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│ │ │ │ -000960d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00096180: 696c 653a 2043 7265 6d6f 6e61 2e69 6e66  ile: Cremona.inf
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│ │ │ │ -00096210: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ -00096220: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379 6e6f  **********..Syno
│ │ │ │ -00096230: 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  psis.========.. 
│ │ │ │ -00096240: 202a 204f 7065 7261 746f 723a 202a 6e6f   * Operator: *no
│ │ │ │ -00096250: 7465 205e 3a20 284d 6163 6175 6c61 7932  te ^: (Macaulay2
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│ │ │ │ -00096270: 3a20 0a20 2020 2020 2020 2070 6869 5e6e  : .        phi^n
│ │ │ │ -00096280: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -00096290: 2020 202a 2070 6869 2c20 6120 2a6e 6f74     * phi, a *not
│ │ │ │ -000962a0: 6520 7261 7469 6f6e 616c 206d 6170 3a20  e rational map: 
│ │ │ │ -000962b0: 5261 7469 6f6e 616c 4d61 702c 0a20 2020  RationalMap,.   
│ │ │ │ -000962c0: 2020 202a 206e 2c20 616e 202a 6e6f 7465     * n, an *note
│ │ │ │ -000962d0: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ -000962e0: 6c61 7932 446f 6329 5a5a 2c0a 2020 2a20  lay2Doc)ZZ,.  * 
│ │ │ │ -000962f0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00096300: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ -00096310: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ -00096320: 6170 2c2c 2070 6869 5e6e 0a0a 4465 7363  ap,, phi^n..Desc
│ │ │ │ -00096330: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -00096340: 3d3d 3d0a 0a49 6620 7468 6520 6d61 7020  ===..If the map 
│ │ │ │ -00096350: 6973 2062 6972 6174 696f 6e61 6c2c 2074  is birational, t
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│ │ │ │ -000963e0: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ -000963f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00096410: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ -00096420: 5e20 5a5a 3a20 5261 7469 6f6e 616c 4d61  ^ ZZ: RationalMa
│ │ │ │ -00096430: 7020 5e20 5a5a 2c20 2d2d 2070 6f77 6572  p ^ ZZ, -- power
│ │ │ │ -00096440: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
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│ │ │ │ -000964a0: 4d61 7020 5e20 5a5a 2c20 5570 3a20 546f  Map ^ ZZ, Up: To
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│ │ │ │ -000964f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000960b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179  ----------+..Way
│ │ │ │ +000960c0: 7320 746f 2075 7365 2074 6869 7320 6d65  s to use this me
│ │ │ │ +000960d0: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  thod:.==========
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│ │ │ │ +000960f0: 2020 2a20 2a6e 6f74 6520 5261 7469 6f6e    * *note Ration
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│ │ │ │ +00096110: 6c4d 6170 3a20 5261 7469 6f6e 616c 4d61  lMap: RationalMa
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│ │ │ │ +00096130: 2c20 2d2d 2065 7175 616c 6974 790a 2020  , -- equality.  
│ │ │ │ +00096140: 2020 6f66 2072 6174 696f 6e61 6c20 6d61    of rational ma
│ │ │ │ +00096150: 7073 0a20 202a 2022 5261 7469 6f6e 616c  ps.  * "Rational
│ │ │ │ +00096160: 4d61 7020 3d3d 205a 5a22 0a20 202a 2022  Map == ZZ".  * "
│ │ │ │ +00096170: 5a5a 203d 3d20 5261 7469 6f6e 616c 4d61  ZZ == RationalMa
│ │ │ │ +00096180: 7022 0a1f 0a46 696c 653a 2043 7265 6d6f  p"...File: Cremo
│ │ │ │ +00096190: 6e61 2e69 6e66 6f2c 204e 6f64 653a 2052  na.info, Node: R
│ │ │ │ +000961a0: 6174 696f 6e61 6c4d 6170 205e 205a 5a2c  ationalMap ^ ZZ,
│ │ │ │ +000961b0: 204e 6578 743a 2052 6174 696f 6e61 6c4d   Next: RationalM
│ │ │ │ +000961c0: 6170 205e 5f73 745f 7374 2049 6465 616c  ap ^_st_st Ideal
│ │ │ │ +000961d0: 2c20 5072 6576 3a20 5261 7469 6f6e 616c  , Prev: Rational
│ │ │ │ +000961e0: 4d61 7020 3d3d 2052 6174 696f 6e61 6c4d  Map == RationalM
│ │ │ │ +000961f0: 6170 2c20 5570 3a20 546f 700a 0a52 6174  ap, Up: Top..Rat
│ │ │ │ +00096200: 696f 6e61 6c4d 6170 205e 205a 5a20 2d2d  ionalMap ^ ZZ --
│ │ │ │ +00096210: 2070 6f77 6572 0a2a 2a2a 2a2a 2a2a 2a2a   power.*********
│ │ │ │ +00096220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00096230: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ +00096240: 3d3d 3d0a 0a20 202a 204f 7065 7261 746f  ===..  * Operato
│ │ │ │ +00096250: 723a 202a 6e6f 7465 205e 3a20 284d 6163  r: *note ^: (Mac
│ │ │ │ +00096260: 6175 6c61 7932 446f 6329 5e2c 0a20 202a  aulay2Doc)^,.  *
│ │ │ │ +00096270: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +00096280: 2070 6869 5e6e 0a20 202a 2049 6e70 7574   phi^n.  * Input
│ │ │ │ +00096290: 733a 0a20 2020 2020 202a 2070 6869 2c20  s:.      * phi, 
│ │ │ │ +000962a0: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ +000962b0: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
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│ │ │ │ +000962d0: 202a 6e6f 7465 2069 6e74 6567 6572 3a20   *note integer: 
│ │ │ │ +000962e0: 284d 6163 6175 6c61 7932 446f 6329 5a5a  (Macaulay2Doc)ZZ
│ │ │ │ +000962f0: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ +00096300: 2020 2020 202a 2061 202a 6e6f 7465 2072       * a *note r
│ │ │ │ +00096310: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ +00096320: 696f 6e61 6c4d 6170 2c2c 2070 6869 5e6e  ionalMap,, phi^n
│ │ │ │ +00096330: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00096340: 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 7468  =========..If th
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│ │ │ │ +00096360: 6e61 6c2c 2074 6865 6e20 6e65 6761 7469  nal, then negati
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│ │ │ │ +00096380: 2075 7365 6420 616e 6420 7468 6520 696e   used and the in
│ │ │ │ +00096390: 7665 7273 6520 7769 6c6c 0a62 6520 636f  verse will.be co
│ │ │ │ +000963a0: 6d70 7574 6564 2075 7369 6e67 202a 6e6f  mputed using *no
│ │ │ │ +000963b0: 7465 2069 6e76 6572 7365 3a20 696e 7665  te inverse: inve
│ │ │ │ +000963c0: 7273 655f 6c70 5261 7469 6f6e 616c 4d61  rse_lpRationalMa
│ │ │ │ +000963d0: 705f 7270 2c28 7068 6929 2e0a 0a57 6179  p_rp,(phi)...Way
│ │ │ │ +000963e0: 7320 746f 2075 7365 2074 6869 7320 6d65  s to use this me
│ │ │ │ +000963f0: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  thod:.==========
│ │ │ │ +00096400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00096410: 2020 2a20 2a6e 6f74 6520 5261 7469 6f6e    * *note Ration
│ │ │ │ +00096420: 616c 4d61 7020 5e20 5a5a 3a20 5261 7469  alMap ^ ZZ: Rati
│ │ │ │ +00096430: 6f6e 616c 4d61 7020 5e20 5a5a 2c20 2d2d  onalMap ^ ZZ, --
│ │ │ │ +00096440: 2070 6f77 6572 0a1f 0a46 696c 653a 2043   power...File: C
│ │ │ │ +00096450: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ +00096460: 653a 2052 6174 696f 6e61 6c4d 6170 205e  e: RationalMap ^
│ │ │ │ +00096470: 5f73 745f 7374 2049 6465 616c 2c20 4e65  _st_st Ideal, Ne
│ │ │ │ +00096480: 7874 3a20 5261 7469 6f6e 616c 4d61 7020  xt: RationalMap 
│ │ │ │ +00096490: 5f75 735f 7374 2c20 5072 6576 3a20 5261  _us_st, Prev: Ra
│ │ │ │ +000964a0: 7469 6f6e 616c 4d61 7020 5e20 5a5a 2c20  tionalMap ^ ZZ, 
│ │ │ │ +000964b0: 5570 3a20 546f 700a 0a52 6174 696f 6e61  Up: Top..Rationa
│ │ │ │ +000964c0: 6c4d 6170 205e 2a2a 2049 6465 616c 202d  lMap ^** Ideal -
│ │ │ │ +000964d0: 2d20 696e 7665 7273 6520 696d 6167 6520  - inverse image 
│ │ │ │ +000964e0: 7669 6120 6120 7261 7469 6f6e 616c 206d  via a rational m
│ │ │ │ +000964f0: 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ap.*************
│ │ │ │  00096500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00096510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00096520: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ -00096530: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 204f  .========..  * O
│ │ │ │ -00096540: 7065 7261 746f 723a 202a 6e6f 7465 205e  perator: *note ^
│ │ │ │ -00096550: 2a2a 3a20 284d 6163 6175 6c61 7932 446f  **: (Macaulay2Do
│ │ │ │ -00096560: 6329 5e5f 7374 5f73 742c 0a20 202a 2055  c)^_st_st,.  * U
│ │ │ │ -00096570: 7361 6765 3a20 0a20 2020 2020 2020 2070  sage: .        p
│ │ │ │ -00096580: 6869 5e2a 2a20 490a 2020 2a20 496e 7075  hi^** I.  * Inpu
│ │ │ │ -00096590: 7473 3a0a 2020 2020 2020 2a20 7068 692c  ts:.      * phi,
│ │ │ │ -000965a0: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ -000965b0: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ -000965c0: 6170 2c0a 2020 2020 2020 2a20 492c 2061  ap,.      * I, a
│ │ │ │ -000965d0: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028  n *note ideal: (
│ │ │ │ -000965e0: 4d61 6361 756c 6179 3244 6f63 2949 6465  Macaulay2Doc)Ide
│ │ │ │ -000965f0: 616c 2c2c 2061 2068 6f6d 6f67 656e 656f  al,, a homogeneo
│ │ │ │ -00096600: 7573 2069 6465 616c 2069 6e20 7468 650a  us ideal in the.
│ │ │ │ -00096610: 2020 2020 2020 2020 636f 6f72 6469 6e61          coordina
│ │ │ │ -00096620: 7465 2072 696e 6720 6f66 2074 6865 2074  te ring of the t
│ │ │ │ -00096630: 6172 6765 7420 6f66 2070 6869 0a20 202a  arget of phi.  *
│ │ │ │ -00096640: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -00096650: 2a20 616e 202a 6e6f 7465 2069 6465 616c  * an *note ideal
│ │ │ │ -00096660: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00096670: 4964 6561 6c2c 2c20 7468 6520 6964 6561  Ideal,, the idea
│ │ │ │ -00096680: 6c20 6f66 2074 6865 2063 6c6f 7375 7265  l of the closure
│ │ │ │ -00096690: 206f 6620 7468 650a 2020 2020 2020 2020   of the.        
│ │ │ │ -000966a0: 696e 7665 7273 6520 696d 6167 6520 6f66  inverse image of
│ │ │ │ -000966b0: 2056 2849 2920 7669 6120 7068 690a 0a44   V(I) via phi..D
│ │ │ │ -000966c0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -000966d0: 3d3d 3d3d 3d3d 0a0a 496e 206d 6f73 7420  ======..In most 
│ │ │ │ -000966e0: 6361 7365 7320 7468 6973 2069 7320 6571  cases this is eq
│ │ │ │ -000966f0: 7569 7661 6c65 6e74 2074 6f20 7068 695e  uivalent to phi^
│ │ │ │ -00096700: 2a20 492c 2077 6869 6368 2069 7320 6661  * I, which is fa
│ │ │ │ -00096710: 7374 6572 2062 7574 206d 6179 206e 6f74  ster but may not
│ │ │ │ -00096720: 2074 616b 650a 696e 746f 2061 6363 6f75   take.into accou
│ │ │ │ -00096730: 6e74 206f 7468 6572 2072 6570 7265 7365  nt other represe
│ │ │ │ -00096740: 6e74 6174 696f 6e73 206f 6620 7468 6520  ntations of the 
│ │ │ │ -00096750: 6d61 702e 0a0a 496e 2074 6865 2065 7861  map...In the exa
│ │ │ │ -00096760: 6d70 6c65 2062 656c 6f77 2c20 7765 2061  mple below, we a
│ │ │ │ -00096770: 7070 6c79 2074 6865 206d 6574 686f 6420  pply the method 
│ │ │ │ -00096780: 746f 2063 6865 636b 2074 6865 2062 6972  to check the bir
│ │ │ │ -00096790: 6174 696f 6e61 6c69 7479 206f 6620 6120  ationality of a 
│ │ │ │ -000967a0: 6d61 700a 2864 6574 6572 6d69 6e69 7374  map.(determinist
│ │ │ │ -000967b0: 6963 616c 6c79 292e 0a0a 2b2d 2d2d 2d2d  ically)...+-----
│ │ │ │ -000967c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00096520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ +00096530: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ +00096540: 0a20 202a 204f 7065 7261 746f 723a 202a  .  * Operator: *
│ │ │ │ +00096550: 6e6f 7465 205e 2a2a 3a20 284d 6163 6175  note ^**: (Macau
│ │ │ │ +00096560: 6c61 7932 446f 6329 5e5f 7374 5f73 742c  lay2Doc)^_st_st,
│ │ │ │ +00096570: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00096580: 2020 2020 2070 6869 5e2a 2a20 490a 2020       phi^** I.  
│ │ │ │ +00096590: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ +000965a0: 2a20 7068 692c 2061 202a 6e6f 7465 2072  * phi, a *note r
│ │ │ │ +000965b0: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ +000965c0: 696f 6e61 6c4d 6170 2c0a 2020 2020 2020  ionalMap,.      
│ │ │ │ +000965d0: 2a20 492c 2061 6e20 2a6e 6f74 6520 6964  * I, an *note id
│ │ │ │ +000965e0: 6561 6c3a 2028 4d61 6361 756c 6179 3244  eal: (Macaulay2D
│ │ │ │ +000965f0: 6f63 2949 6465 616c 2c2c 2061 2068 6f6d  oc)Ideal,, a hom
│ │ │ │ +00096600: 6f67 656e 656f 7573 2069 6465 616c 2069  ogeneous ideal i
│ │ │ │ +00096610: 6e20 7468 650a 2020 2020 2020 2020 636f  n the.        co
│ │ │ │ +00096620: 6f72 6469 6e61 7465 2072 696e 6720 6f66  ordinate ring of
│ │ │ │ +00096630: 2074 6865 2074 6172 6765 7420 6f66 2070   the target of p
│ │ │ │ +00096640: 6869 0a20 202a 204f 7574 7075 7473 3a0a  hi.  * Outputs:.
│ │ │ │ +00096650: 2020 2020 2020 2a20 616e 202a 6e6f 7465        * an *note
│ │ │ │ +00096660: 2069 6465 616c 3a20 284d 6163 6175 6c61   ideal: (Macaula
│ │ │ │ +00096670: 7932 446f 6329 4964 6561 6c2c 2c20 7468  y2Doc)Ideal,, th
│ │ │ │ +00096680: 6520 6964 6561 6c20 6f66 2074 6865 2063  e ideal of the c
│ │ │ │ +00096690: 6c6f 7375 7265 206f 6620 7468 650a 2020  losure of the.  
│ │ │ │ +000966a0: 2020 2020 2020 696e 7665 7273 6520 696d        inverse im
│ │ │ │ +000966b0: 6167 6520 6f66 2056 2849 2920 7669 6120  age of V(I) via 
│ │ │ │ +000966c0: 7068 690a 0a44 6573 6372 6970 7469 6f6e  phi..Description
│ │ │ │ +000966d0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 496e  .===========..In
│ │ │ │ +000966e0: 206d 6f73 7420 6361 7365 7320 7468 6973   most cases this
│ │ │ │ +000966f0: 2069 7320 6571 7569 7661 6c65 6e74 2074   is equivalent t
│ │ │ │ +00096700: 6f20 7068 695e 2a20 492c 2077 6869 6368  o phi^* I, which
│ │ │ │ +00096710: 2069 7320 6661 7374 6572 2062 7574 206d   is faster but m
│ │ │ │ +00096720: 6179 206e 6f74 2074 616b 650a 696e 746f  ay not take.into
│ │ │ │ +00096730: 2061 6363 6f75 6e74 206f 7468 6572 2072   account other r
│ │ │ │ +00096740: 6570 7265 7365 6e74 6174 696f 6e73 206f  epresentations o
│ │ │ │ +00096750: 6620 7468 6520 6d61 702e 0a0a 496e 2074  f the map...In t
│ │ │ │ +00096760: 6865 2065 7861 6d70 6c65 2062 656c 6f77  he example below
│ │ │ │ +00096770: 2c20 7765 2061 7070 6c79 2074 6865 206d  , we apply the m
│ │ │ │ +00096780: 6574 686f 6420 746f 2063 6865 636b 2074  ethod to check t
│ │ │ │ +00096790: 6865 2062 6972 6174 696f 6e61 6c69 7479  he birationality
│ │ │ │ +000967a0: 206f 6620 6120 6d61 700a 2864 6574 6572   of a map.(deter
│ │ │ │ +000967b0: 6d69 6e69 7374 6963 616c 6c79 292e 0a0a  ministically)...
│ │ │ │ +000967c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000967d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000967e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000967f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00096800: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00096810: 7068 6920 3d20 7175 6164 726f 5175 6164  phi = quadroQuad
│ │ │ │ -00096820: 7269 6343 7265 6d6f 6e61 5472 616e 7366  ricCremonaTransf
│ │ │ │ -00096830: 6f72 6d61 7469 6f6e 2835 2c31 2920 2020  ormation(5,1)   
│ │ │ │ -00096840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096850: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00096810: 7c69 3120 3a20 7068 6920 3d20 7175 6164  |i1 : phi = quad
│ │ │ │ +00096820: 726f 5175 6164 7269 6343 7265 6d6f 6e61  roQuadricCremona
│ │ │ │ +00096830: 5472 616e 7366 6f72 6d61 7469 6f6e 2835  Transformation(5
│ │ │ │ +00096840: 2c31 2920 2020 2020 2020 2020 2020 2020  ,1)             
│ │ │ │ +00096850: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096860: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00096870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000968a0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -000968b0: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -000968c0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +000968a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000968b0: 7c6f 3120 3d20 2d2d 2072 6174 696f 6e61  |o1 = -- rationa
│ │ │ │ +000968c0: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │  000968d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000968e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000968f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096900: 736f 7572 6365 3a20 5072 6f6a 2851 515b  source: Proj(QQ[
│ │ │ │ -00096910: 782c 2079 2c20 7a2c 2074 2c20 752c 2076  x, y, z, t, u, v
│ │ │ │ -00096920: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +000968f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096900: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ +00096910: 6f6a 2851 515b 782c 2079 2c20 7a2c 2074  oj(QQ[x, y, z, t
│ │ │ │ +00096920: 2c20 752c 2076 5d29 2020 2020 2020 2020  , u, v])        
│ │ │ │  00096930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096940: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096950: 7461 7267 6574 3a20 5072 6f6a 2851 515b  target: Proj(QQ[
│ │ │ │ -00096960: 782c 2079 2c20 7a2c 2074 2c20 752c 2076  x, y, z, t, u, v
│ │ │ │ -00096970: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +00096940: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096950: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ +00096960: 6f6a 2851 515b 782c 2079 2c20 7a2c 2074  oj(QQ[x, y, z, t
│ │ │ │ +00096970: 2c20 752c 2076 5d29 2020 2020 2020 2020  , u, v])        
│ │ │ │  00096980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096990: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000969a0: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -000969b0: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +00096990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000969a0: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ +000969b0: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │  000969c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000969d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000969e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000969f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096a00: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000969e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000969f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096a00: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00096a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096a30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096a50: 2079 2a7a 202d 2076 202c 2020 2020 2020   y*z - v ,      
│ │ │ │ +00096a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096a40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096a50: 2020 2020 2020 2079 2a7a 202d 2076 202c         y*z - v ,
│ │ │ │  00096a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096a90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00096aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096af0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00096ad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096ae0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096af0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00096b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096b20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096b40: 2078 2a7a 202d 2075 202c 2020 2020 2020   x*z - u ,      
│ │ │ │ +00096b20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096b30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096b40: 2020 2020 2020 2078 2a7a 202d 2075 202c         x*z - u ,
│ │ │ │  00096b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096b70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096b70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096b80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00096b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096bc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096be0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00096bc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096bd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096be0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00096bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096c10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096c30: 2078 2a79 202d 2074 202c 2020 2020 2020   x*y - t ,      
│ │ │ │ +00096c10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096c20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096c30: 2020 2020 2020 2078 2a79 202d 2074 202c         x*y - t ,
│ │ │ │  00096c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096c60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096c60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096c70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00096c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096cb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096cd0: 202d 207a 2a74 202b 2075 2a76 2c20 2020   - z*t + u*v,   
│ │ │ │ -00096ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096cb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096cc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096cd0: 2020 2020 2020 202d 207a 2a74 202b 2075         - z*t + u
│ │ │ │ +00096ce0: 2a76 2c20 2020 2020 2020 2020 2020 2020  *v,             
│ │ │ │  00096cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096d00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096d00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096d10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00096d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096d70: 202d 2079 2a75 202b 2074 2a76 2c20 2020   - y*u + t*v,   
│ │ │ │ -00096d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096d50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096d60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096d70: 2020 2020 2020 202d 2079 2a75 202b 2074         - y*u + t
│ │ │ │ +00096d80: 2a76 2c20 2020 2020 2020 2020 2020 2020  *v,             
│ │ │ │  00096d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096da0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096da0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096db0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00096dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096df0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096e10: 2074 2a75 202d 2078 2a76 2020 2020 2020   t*u - x*v      
│ │ │ │ +00096df0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096e00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096e10: 2020 2020 2020 2074 2a75 202d 2078 2a76         t*u - x*v
│ │ │ │  00096e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096e60: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00096e40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096e50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00096e60: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │  00096e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096e90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096e90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096ea0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00096eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096ee0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -00096ef0: 5261 7469 6f6e 616c 4d61 7020 2843 7265  RationalMap (Cre
│ │ │ │ -00096f00: 6d6f 6e61 2074 7261 6e73 666f 726d 6174  mona transformat
│ │ │ │ -00096f10: 696f 6e20 6f66 2050 505e 3520 6f66 2074  ion of PP^5 of t
│ │ │ │ -00096f20: 7970 6520 2832 2c32 2929 2020 2020 2020  ype (2,2))      
│ │ │ │ -00096f30: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00096f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00096ee0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096ef0: 7c6f 3120 3a20 5261 7469 6f6e 616c 4d61  |o1 : RationalMa
│ │ │ │ +00096f00: 7020 2843 7265 6d6f 6e61 2074 7261 6e73  p (Cremona trans
│ │ │ │ +00096f10: 666f 726d 6174 696f 6e20 6f66 2050 505e  formation of PP^
│ │ │ │ +00096f20: 3520 6f66 2074 7970 6520 2832 2c32 2929  5 of type (2,2))
│ │ │ │ +00096f30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096f40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00096f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00096f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00096f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00096f80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -00096f90: 4b20 3a3d 2066 7261 6328 5151 5b76 6172  K := frac(QQ[var
│ │ │ │ -00096fa0: 7328 302e 2e35 295d 293b 2070 6869 203d  s(0..5)]); phi =
│ │ │ │ -00096fb0: 2070 6869 202a 2a20 4b20 2020 2020 2020   phi ** K       
│ │ │ │ +00096f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00096f90: 7c69 3220 3a20 4b20 3a3d 2066 7261 6328  |i2 : K := frac(
│ │ │ │ +00096fa0: 5151 5b76 6172 7328 302e 2e35 295d 293b  QQ[vars(0..5)]);
│ │ │ │ +00096fb0: 2070 6869 203d 2070 6869 202a 2a20 4b20   phi = phi ** K 
│ │ │ │  00096fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096fd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00096fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096fd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00096fe0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00096ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097020: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00097030: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -00097040: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +00097020: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097030: 7c6f 3320 3d20 2d2d 2072 6174 696f 6e61  |o3 = -- rationa
│ │ │ │ +00097040: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │  00097050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097070: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097080: 736f 7572 6365 3a20 5072 6f6a 2866 7261  source: Proj(fra
│ │ │ │ -00097090: 6328 5151 5b61 2e2e 665d 295b 782c 2079  c(QQ[a..f])[x, y
│ │ │ │ -000970a0: 2c20 7a2c 2074 2c20 752c 2076 5d29 2020  , z, t, u, v])  
│ │ │ │ -000970b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000970c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000970d0: 7461 7267 6574 3a20 5072 6f6a 2866 7261  target: Proj(fra
│ │ │ │ -000970e0: 6328 5151 5b61 2e2e 665d 295b 782c 2079  c(QQ[a..f])[x, y
│ │ │ │ -000970f0: 2c20 7a2c 2074 2c20 752c 2076 5d29 2020  , z, t, u, v])  
│ │ │ │ -00097100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097110: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097120: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -00097130: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +00097070: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097080: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ +00097090: 6f6a 2866 7261 6328 5151 5b61 2e2e 665d  oj(frac(QQ[a..f]
│ │ │ │ +000970a0: 295b 782c 2079 2c20 7a2c 2074 2c20 752c  )[x, y, z, t, u,
│ │ │ │ +000970b0: 2076 5d29 2020 2020 2020 2020 2020 2020   v])            
│ │ │ │ +000970c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000970d0: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ +000970e0: 6f6a 2866 7261 6328 5151 5b61 2e2e 665d  oj(frac(QQ[a..f]
│ │ │ │ +000970f0: 295b 782c 2079 2c20 7a2c 2074 2c20 752c  )[x, y, z, t, u,
│ │ │ │ +00097100: 2076 5d29 2020 2020 2020 2020 2020 2020   v])            
│ │ │ │ +00097110: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097120: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ +00097130: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │  00097140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097160: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097180: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00097160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097170: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097180: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00097190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000971a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000971b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000971c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000971d0: 2079 2a7a 202d 2076 202c 2020 2020 2020   y*z - v ,      
│ │ │ │ +000971b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000971c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000971d0: 2020 2020 2020 2079 2a7a 202d 2076 202c         y*z - v ,
│ │ │ │  000971e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000971f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097200: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097210: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097250: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097270: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00097250: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097260: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097270: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00097280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000972a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000972b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000972c0: 2078 2a7a 202d 2075 202c 2020 2020 2020   x*z - u ,      
│ │ │ │ +000972a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000972b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000972c0: 2020 2020 2020 2078 2a7a 202d 2075 202c         x*z - u ,
│ │ │ │  000972d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000972e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000972f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000972f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097300: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097340: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097360: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00097340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097350: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097360: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00097370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097390: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000973a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000973b0: 2078 2a79 202d 2074 202c 2020 2020 2020   x*y - t ,      
│ │ │ │ +00097390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000973a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000973b0: 2020 2020 2020 2078 2a79 202d 2074 202c         x*y - t ,
│ │ │ │  000973c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000973d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000973e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000973f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000973e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000973f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097450: 202d 207a 2a74 202b 2075 2a76 2c20 2020   - z*t + u*v,   
│ │ │ │ -00097460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097440: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097450: 2020 2020 2020 202d 207a 2a74 202b 2075         - z*t + u
│ │ │ │ +00097460: 2a76 2c20 2020 2020 2020 2020 2020 2020  *v,             
│ │ │ │  00097470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097480: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097490: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000974a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000974b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000974c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000974d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000974e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000974f0: 202d 2079 2a75 202b 2074 2a76 2c20 2020   - y*u + t*v,   
│ │ │ │ -00097500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000974d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000974e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000974f0: 2020 2020 2020 202d 2079 2a75 202b 2074         - y*u + t
│ │ │ │ +00097500: 2a76 2c20 2020 2020 2020 2020 2020 2020  *v,             
│ │ │ │  00097510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097530: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097590: 2074 2a75 202d 2078 2a76 2020 2020 2020   t*u - x*v      
│ │ │ │ +00097570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097580: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097590: 2020 2020 2020 2074 2a75 202d 2078 2a76         t*u - x*v
│ │ │ │  000975a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000975b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000975c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000975d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000975e0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +000975c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000975d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000975e0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │  000975f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097610: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097620: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097660: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -00097670: 5261 7469 6f6e 616c 4d61 7020 2871 7561  RationalMap (qua
│ │ │ │ -00097680: 6472 6174 6963 2072 6174 696f 6e61 6c20  dratic rational 
│ │ │ │ -00097690: 6d61 7020 6672 6f6d 2050 505e 3520 746f  map from PP^5 to
│ │ │ │ -000976a0: 2050 505e 3529 2020 2020 2020 2020 2020   PP^5)          
│ │ │ │ -000976b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -000976c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00097660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097670: 7c6f 3320 3a20 5261 7469 6f6e 616c 4d61  |o3 : RationalMa
│ │ │ │ +00097680: 7020 2871 7561 6472 6174 6963 2072 6174  p (quadratic rat
│ │ │ │ +00097690: 696f 6e61 6c20 6d61 7020 6672 6f6d 2050  ional map from P
│ │ │ │ +000976a0: 505e 3520 746f 2050 505e 3529 2020 2020  P^5 to PP^5)    
│ │ │ │ +000976b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000976c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000976d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000976e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000976f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00097700: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00097710: 7020 3d20 7472 696d 206d 696e 6f72 7328  p = trim minors(
│ │ │ │ -00097720: 322c 2876 6172 7320 4b29 7c7c 2876 6172  2,(vars K)||(var
│ │ │ │ -00097730: 7320 736f 7572 6365 2070 6869 2929 2020  s source phi))  
│ │ │ │ -00097740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097750: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00097710: 7c69 3420 3a20 7020 3d20 7472 696d 206d  |i4 : p = trim m
│ │ │ │ +00097720: 696e 6f72 7328 322c 2876 6172 7320 4b29  inors(2,(vars K)
│ │ │ │ +00097730: 7c7c 2876 6172 7320 736f 7572 6365 2070  ||(vars source p
│ │ │ │ +00097740: 6869 2929 2020 2020 2020 2020 2020 2020  hi))            
│ │ │ │ +00097750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097760: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000977a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000977b0: 2020 2020 2020 2020 2020 202d 6520 2020             -e   
│ │ │ │ -000977c0: 2020 2020 202d 6420 2020 2020 2020 202d       -d        -
│ │ │ │ -000977d0: 6320 2020 2020 2020 202d 6220 2020 2020  c        -b     
│ │ │ │ -000977e0: 2020 202d 6120 2020 2020 2020 2020 2020     -a           
│ │ │ │ -000977f0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -00097800: 6964 6561 6c20 2875 202b 202d 2d2a 762c  ideal (u + --*v,
│ │ │ │ -00097810: 2074 202b 202d 2d2a 762c 207a 202b 202d   t + --*v, z + -
│ │ │ │ -00097820: 2d2a 762c 2079 202b 202d 2d2a 762c 2078  -*v, y + --*v, x
│ │ │ │ -00097830: 202b 202d 2d2a 7629 2020 2020 2020 2020   + --*v)        
│ │ │ │ -00097840: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097850: 2020 2020 2020 2020 2020 2020 6620 2020              f   
│ │ │ │ -00097860: 2020 2020 2020 6620 2020 2020 2020 2020        f         
│ │ │ │ -00097870: 6620 2020 2020 2020 2020 6620 2020 2020  f         f     
│ │ │ │ -00097880: 2020 2020 6620 2020 2020 2020 2020 2020      f           
│ │ │ │ -00097890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000978a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000977a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000977b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000977c0: 202d 6520 2020 2020 2020 202d 6420 2020   -e        -d   
│ │ │ │ +000977d0: 2020 2020 202d 6320 2020 2020 2020 202d       -c        -
│ │ │ │ +000977e0: 6220 2020 2020 2020 202d 6120 2020 2020  b        -a     
│ │ │ │ +000977f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097800: 7c6f 3420 3d20 6964 6561 6c20 2875 202b  |o4 = ideal (u +
│ │ │ │ +00097810: 202d 2d2a 762c 2074 202b 202d 2d2a 762c   --*v, t + --*v,
│ │ │ │ +00097820: 207a 202b 202d 2d2a 762c 2079 202b 202d   z + --*v, y + -
│ │ │ │ +00097830: 2d2a 762c 2078 202b 202d 2d2a 7629 2020  -*v, x + --*v)  
│ │ │ │ +00097840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097850: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097860: 2020 6620 2020 2020 2020 2020 6620 2020    f         f   
│ │ │ │ +00097870: 2020 2020 2020 6620 2020 2020 2020 2020        f         
│ │ │ │ +00097880: 6620 2020 2020 2020 2020 6620 2020 2020  f         f     
│ │ │ │ +00097890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000978a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000978b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000978c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000978d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000978e0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ -000978f0: 4964 6561 6c20 6f66 2066 7261 6328 5151  Ideal of frac(QQ
│ │ │ │ -00097900: 5b61 2e2e 665d 295b 782c 2079 2c20 7a2c  [a..f])[x, y, z,
│ │ │ │ -00097910: 2074 2c20 752c 2076 5d20 2020 2020 2020   t, u, v]       
│ │ │ │ +000978e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000978f0: 7c6f 3420 3a20 4964 6561 6c20 6f66 2066  |o4 : Ideal of f
│ │ │ │ +00097900: 7261 6328 5151 5b61 2e2e 665d 295b 782c  rac(QQ[a..f])[x,
│ │ │ │ +00097910: 2079 2c20 7a2c 2074 2c20 752c 2076 5d20   y, z, t, u, v] 
│ │ │ │  00097920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097930: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00097940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00097930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097940: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00097950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00097980: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -00097990: 7120 3d20 7068 6920 7020 2020 2020 2020  q = phi p       
│ │ │ │ +00097980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00097990: 7c69 3520 3a20 7120 3d20 7068 6920 7020  |i5 : q = phi p 
│ │ │ │  000979a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000979b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000979c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000979d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000979e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000979d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000979e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000979f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097a20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097a20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097a30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097a60: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00097a70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097a80: 2020 2020 2020 2020 2020 2062 2a65 202d             b*e -
│ │ │ │ -00097a90: 2064 2a66 2020 2020 2020 2020 632a 6420   d*f        c*d 
│ │ │ │ -00097aa0: 2d20 652a 6620 2020 2020 2020 202d 2061  - e*f        - a
│ │ │ │ -00097ab0: 2a62 202b 2064 2020 2020 2020 2020 2020  *b + d          
│ │ │ │ -00097ac0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -00097ad0: 6964 6561 6c20 2875 202b 202d 2d2d 2d2d  ideal (u + -----
│ │ │ │ -00097ae0: 2d2d 2d2d 2a76 2c20 7420 2b20 2d2d 2d2d  ----*v, t + ----
│ │ │ │ -00097af0: 2d2d 2d2d 2d2a 762c 207a 202b 202d 2d2d  -----*v, z + ---
│ │ │ │ -00097b00: 2d2d 2d2d 2d2d 2d2a 762c 2079 202b 2020  -------*v, y +  
│ │ │ │ -00097b10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097b20: 2020 2020 2020 2020 2020 2064 2a65 202d             d*e -
│ │ │ │ -00097b30: 2061 2a66 2020 2020 2020 2020 642a 6520   a*f        d*e 
│ │ │ │ -00097b40: 2d20 612a 6620 2020 2020 2020 2020 642a  - a*f         d*
│ │ │ │ -00097b50: 6520 2d20 612a 6620 2020 2020 2020 2020  e - a*f         
│ │ │ │ -00097b60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00097a60: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00097a70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097a80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097a90: 2062 2a65 202d 2064 2a66 2020 2020 2020   b*e - d*f      
│ │ │ │ +00097aa0: 2020 632a 6420 2d20 652a 6620 2020 2020    c*d - e*f     
│ │ │ │ +00097ab0: 2020 202d 2061 2a62 202b 2064 2020 2020     - a*b + d    
│ │ │ │ +00097ac0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097ad0: 7c6f 3520 3d20 6964 6561 6c20 2875 202b  |o5 = ideal (u +
│ │ │ │ +00097ae0: 202d 2d2d 2d2d 2d2d 2d2d 2a76 2c20 7420   ---------*v, t 
│ │ │ │ +00097af0: 2b20 2d2d 2d2d 2d2d 2d2d 2d2a 762c 207a  + ---------*v, z
│ │ │ │ +00097b00: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2a 762c   + ----------*v,
│ │ │ │ +00097b10: 2079 202b 2020 2020 2020 2020 2020 7c0a   y +          |.
│ │ │ │ +00097b20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097b30: 2064 2a65 202d 2061 2a66 2020 2020 2020   d*e - a*f      
│ │ │ │ +00097b40: 2020 642a 6520 2d20 612a 6620 2020 2020    d*e - a*f     
│ │ │ │ +00097b50: 2020 2020 642a 6520 2d20 612a 6620 2020      d*e - a*f   
│ │ │ │ +00097b60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097b70: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │  00097b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00097bb0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00097bc0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00097bd0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00097be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00097bc0: 7c20 2020 2020 2020 2020 2020 2020 2032  |              2
│ │ │ │ +00097bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097be0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00097bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097c10: 2d20 612a 6320 2b20 6520 2020 2020 2020  - a*c + e       
│ │ │ │ -00097c20: 2020 2d20 622a 6320 2b20 6620 2020 2020    - b*c + f     
│ │ │ │ -00097c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097c00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097c10: 7c20 2020 2020 2d20 612a 6320 2b20 6520  |     - a*c + e 
│ │ │ │ +00097c20: 2020 2020 2020 2020 2d20 622a 6320 2b20          - b*c + 
│ │ │ │ +00097c30: 6620 2020 2020 2020 2020 2020 2020 2020  f               
│ │ │ │  00097c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097c50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2a76 2c20 7820  ----------*v, x 
│ │ │ │ -00097c70: 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2a76 2920  + ----------*v) 
│ │ │ │ -00097c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097c50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097c60: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +00097c70: 2a76 2c20 7820 2b20 2d2d 2d2d 2d2d 2d2d  *v, x + --------
│ │ │ │ +00097c80: 2d2d 2a76 2920 2020 2020 2020 2020 2020  --*v)           
│ │ │ │  00097c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097ca0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097cb0: 2064 2a65 202d 2061 2a66 2020 2020 2020   d*e - a*f      
│ │ │ │ -00097cc0: 2020 2064 2a65 202d 2061 2a66 2020 2020     d*e - a*f    
│ │ │ │ -00097cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097ca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097cb0: 7c20 2020 2020 2064 2a65 202d 2061 2a66  |      d*e - a*f
│ │ │ │ +00097cc0: 2020 2020 2020 2020 2064 2a65 202d 2061           d*e - a
│ │ │ │ +00097cd0: 2a66 2020 2020 2020 2020 2020 2020 2020  *f              
│ │ │ │  00097ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097cf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097cf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097d00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097d40: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ -00097d50: 4964 6561 6c20 6f66 2066 7261 6328 5151  Ideal of frac(QQ
│ │ │ │ -00097d60: 5b61 2e2e 665d 295b 782c 2079 2c20 7a2c  [a..f])[x, y, z,
│ │ │ │ -00097d70: 2074 2c20 752c 2076 5d20 2020 2020 2020   t, u, v]       
│ │ │ │ +00097d40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097d50: 7c6f 3520 3a20 4964 6561 6c20 6f66 2066  |o5 : Ideal of f
│ │ │ │ +00097d60: 7261 6328 5151 5b61 2e2e 665d 295b 782c  rac(QQ[a..f])[x,
│ │ │ │ +00097d70: 2079 2c20 7a2c 2074 2c20 752c 2076 5d20   y, z, t, u, v] 
│ │ │ │  00097d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097d90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00097da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00097d90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097da0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00097db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00097de0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00097df0: 7469 6d65 2070 6869 5e2a 2a20 7120 2020  time phi^** q   
│ │ │ │ -00097e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00097df0: 7c69 3620 3a20 7469 6d65 2070 6869 5e2a  |i6 : time phi^*
│ │ │ │ +00097e00: 2a20 7120 2020 2020 2020 2020 2020 2020  * q             
│ │ │ │  00097e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097e30: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00097e40: 7365 6420 302e 3135 3232 3638 7320 2863  sed 0.152268s (c
│ │ │ │ -00097e50: 7075 293b 2030 2e31 3530 3538 3873 2028  pu); 0.150588s (
│ │ │ │ -00097e60: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -00097e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097e80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097e30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097e40: 7c20 2d2d 2075 7365 6420 302e 3238 3634  | -- used 0.2864
│ │ │ │ +00097e50: 3833 7320 2863 7075 293b 2030 2e32 3035  83s (cpu); 0.205
│ │ │ │ +00097e60: 3032 3673 2028 7468 7265 6164 293b 2030  026s (thread); 0
│ │ │ │ +00097e70: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00097e80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097e90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097ed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097ee0: 2020 2020 2020 2020 2020 202d 6520 2020             -e   
│ │ │ │ -00097ef0: 2020 2020 202d 6420 2020 2020 2020 202d       -d        -
│ │ │ │ -00097f00: 6320 2020 2020 2020 202d 6220 2020 2020  c        -b     
│ │ │ │ -00097f10: 2020 202d 6120 2020 2020 2020 2020 2020     -a           
│ │ │ │ -00097f20: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -00097f30: 6964 6561 6c20 2875 202b 202d 2d2a 762c  ideal (u + --*v,
│ │ │ │ -00097f40: 2074 202b 202d 2d2a 762c 207a 202b 202d   t + --*v, z + -
│ │ │ │ -00097f50: 2d2a 762c 2079 202b 202d 2d2a 762c 2078  -*v, y + --*v, x
│ │ │ │ -00097f60: 202b 202d 2d2a 7629 2020 2020 2020 2020   + --*v)        
│ │ │ │ -00097f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097f80: 2020 2020 2020 2020 2020 2020 6620 2020              f   
│ │ │ │ -00097f90: 2020 2020 2020 6620 2020 2020 2020 2020        f         
│ │ │ │ -00097fa0: 6620 2020 2020 2020 2020 6620 2020 2020  f         f     
│ │ │ │ -00097fb0: 2020 2020 6620 2020 2020 2020 2020 2020      f           
│ │ │ │ -00097fc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00097fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00097ed0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097ee0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097ef0: 202d 6520 2020 2020 2020 202d 6420 2020   -e        -d   
│ │ │ │ +00097f00: 2020 2020 202d 6320 2020 2020 2020 202d       -c        -
│ │ │ │ +00097f10: 6220 2020 2020 2020 202d 6120 2020 2020  b        -a     
│ │ │ │ +00097f20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097f30: 7c6f 3620 3d20 6964 6561 6c20 2875 202b  |o6 = ideal (u +
│ │ │ │ +00097f40: 202d 2d2a 762c 2074 202b 202d 2d2a 762c   --*v, t + --*v,
│ │ │ │ +00097f50: 207a 202b 202d 2d2a 762c 2079 202b 202d   z + --*v, y + -
│ │ │ │ +00097f60: 2d2a 762c 2078 202b 202d 2d2a 7629 2020  -*v, x + --*v)  
│ │ │ │ +00097f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097f80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00097f90: 2020 6620 2020 2020 2020 2020 6620 2020    f         f   
│ │ │ │ +00097fa0: 2020 2020 2020 6620 2020 2020 2020 2020        f         
│ │ │ │ +00097fb0: 6620 2020 2020 2020 2020 6620 2020 2020  f         f     
│ │ │ │ +00097fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00097fd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00097fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098010: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -00098020: 4964 6561 6c20 6f66 2066 7261 6328 5151  Ideal of frac(QQ
│ │ │ │ -00098030: 5b61 2e2e 665d 295b 782c 2079 2c20 7a2c  [a..f])[x, y, z,
│ │ │ │ -00098040: 2074 2c20 752c 2076 5d20 2020 2020 2020   t, u, v]       
│ │ │ │ +00098010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00098020: 7c6f 3620 3a20 4964 6561 6c20 6f66 2066  |o6 : Ideal of f
│ │ │ │ +00098030: 7261 6328 5151 5b61 2e2e 665d 295b 782c  rac(QQ[a..f])[x,
│ │ │ │ +00098040: 2079 2c20 7a2c 2074 2c20 752c 2076 5d20   y, z, t, u, v] 
│ │ │ │  00098050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098060: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00098070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00098060: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00098070: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00098080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000980a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000980b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ -000980c0: 6f6f 203d 3d20 7020 2020 2020 2020 2020  oo == p         
│ │ │ │ +000980b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000980c0: 7c69 3720 3a20 6f6f 203d 3d20 7020 2020  |i7 : oo == p   
│ │ │ │  000980d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000980e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000980f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098100: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00098110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00098110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00098120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098150: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ -00098160: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00098150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00098160: 7c6f 3720 3d20 7472 7565 2020 2020 2020  |o7 = true      
│ │ │ │  00098170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000981a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -000981b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000981a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000981b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000981c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000981d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000981e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000981f0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -00098200: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -00098210: 2a20 2a6e 6f74 6520 5261 7469 6f6e 616c  * *note Rational
│ │ │ │ -00098220: 4d61 7020 5f2a 3a20 5261 7469 6f6e 616c  Map _*: Rational
│ │ │ │ -00098230: 4d61 7020 5f75 735f 7374 2c20 2d2d 2064  Map _us_st, -- d
│ │ │ │ -00098240: 6972 6563 7420 696d 6167 6520 7669 6120  irect image via 
│ │ │ │ -00098250: 6120 7261 7469 6f6e 616c 0a20 2020 206d  a rational.    m
│ │ │ │ -00098260: 6170 0a20 202a 202a 6e6f 7465 2052 6174  ap.  * *note Rat
│ │ │ │ -00098270: 696f 6e61 6c4d 6170 202a 2a20 5269 6e67  ionalMap ** Ring
│ │ │ │ -00098280: 3a20 5261 7469 6f6e 616c 4d61 7020 5f73  : RationalMap _s
│ │ │ │ -00098290: 745f 7374 2052 696e 672c 202d 2d20 6368  t_st Ring, -- ch
│ │ │ │ -000982a0: 616e 6765 2074 6865 0a20 2020 2063 6f65  ange the.    coe
│ │ │ │ -000982b0: 6666 6963 6965 6e74 2072 696e 6720 6f66  fficient ring of
│ │ │ │ -000982c0: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ -000982d0: 2020 2a20 2a6e 6f74 6520 7461 7267 6574    * *note target
│ │ │ │ -000982e0: 2852 6174 696f 6e61 6c4d 6170 293a 2074  (RationalMap): t
│ │ │ │ -000982f0: 6172 6765 745f 6c70 5261 7469 6f6e 616c  arget_lpRational
│ │ │ │ -00098300: 4d61 705f 7270 2c20 2d2d 2063 6f6f 7264  Map_rp, -- coord
│ │ │ │ -00098310: 696e 6174 6520 7269 6e67 206f 660a 2020  inate ring of.  
│ │ │ │ -00098320: 2020 7468 6520 7461 7267 6574 2066 6f72    the target for
│ │ │ │ -00098330: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ -00098340: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ -00098350: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ +000981f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00098200: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00098210: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 5261  ==..  * *note Ra
│ │ │ │ +00098220: 7469 6f6e 616c 4d61 7020 5f2a 3a20 5261  tionalMap _*: Ra
│ │ │ │ +00098230: 7469 6f6e 616c 4d61 7020 5f75 735f 7374  tionalMap _us_st
│ │ │ │ +00098240: 2c20 2d2d 2064 6972 6563 7420 696d 6167  , -- direct imag
│ │ │ │ +00098250: 6520 7669 6120 6120 7261 7469 6f6e 616c  e via a rational
│ │ │ │ +00098260: 0a20 2020 206d 6170 0a20 202a 202a 6e6f  .    map.  * *no
│ │ │ │ +00098270: 7465 2052 6174 696f 6e61 6c4d 6170 202a  te RationalMap *
│ │ │ │ +00098280: 2a20 5269 6e67 3a20 5261 7469 6f6e 616c  * Ring: Rational
│ │ │ │ +00098290: 4d61 7020 5f73 745f 7374 2052 696e 672c  Map _st_st Ring,
│ │ │ │ +000982a0: 202d 2d20 6368 616e 6765 2074 6865 0a20   -- change the. 
│ │ │ │ +000982b0: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
│ │ │ │ +000982c0: 696e 6720 6f66 2061 2072 6174 696f 6e61  ing of a rationa
│ │ │ │ +000982d0: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ +000982e0: 7461 7267 6574 2852 6174 696f 6e61 6c4d  target(RationalM
│ │ │ │ +000982f0: 6170 293a 2074 6172 6765 745f 6c70 5261  ap): target_lpRa
│ │ │ │ +00098300: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ +00098310: 2063 6f6f 7264 696e 6174 6520 7269 6e67   coordinate ring
│ │ │ │ +00098320: 206f 660a 2020 2020 7468 6520 7461 7267   of.    the targ
│ │ │ │ +00098330: 6574 2066 6f72 2061 2072 6174 696f 6e61  et for a rationa
│ │ │ │ +00098340: 6c20 6d61 700a 0a57 6179 7320 746f 2075  l map..Ways to u
│ │ │ │ +00098350: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │  00098360: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00098370: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 5261  ==..  * *note Ra
│ │ │ │ -00098380: 7469 6f6e 616c 4d61 7020 5e2a 2a20 4964  tionalMap ^** Id
│ │ │ │ -00098390: 6561 6c3a 2052 6174 696f 6e61 6c4d 6170  eal: RationalMap
│ │ │ │ -000983a0: 205e 5f73 745f 7374 2049 6465 616c 2c20   ^_st_st Ideal, 
│ │ │ │ -000983b0: 2d2d 2069 6e76 6572 7365 2069 6d61 6765  -- inverse image
│ │ │ │ -000983c0: 0a20 2020 2076 6961 2061 2072 6174 696f  .    via a ratio
│ │ │ │ -000983d0: 6e61 6c20 6d61 700a 1f0a 4669 6c65 3a20  nal map...File: 
│ │ │ │ -000983e0: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ -000983f0: 6465 3a20 5261 7469 6f6e 616c 4d61 7020  de: RationalMap 
│ │ │ │ -00098400: 5f75 735f 7374 2c20 4e65 7874 3a20 5261  _us_st, Next: Ra
│ │ │ │ -00098410: 7469 6f6e 616c 4d61 7020 7c20 4964 6561  tionalMap | Idea
│ │ │ │ -00098420: 6c2c 2050 7265 763a 2052 6174 696f 6e61  l, Prev: Rationa
│ │ │ │ -00098430: 6c4d 6170 205e 5f73 745f 7374 2049 6465  lMap ^_st_st Ide
│ │ │ │ -00098440: 616c 2c20 5570 3a20 546f 700a 0a52 6174  al, Up: Top..Rat
│ │ │ │ -00098450: 696f 6e61 6c4d 6170 205f 2a20 2d2d 2064  ionalMap _* -- d
│ │ │ │ -00098460: 6972 6563 7420 696d 6167 6520 7669 6120  irect image via 
│ │ │ │ -00098470: 6120 7261 7469 6f6e 616c 206d 6170 0a2a  a rational map.*
│ │ │ │ -00098480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00098370: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00098380: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ +00098390: 5e2a 2a20 4964 6561 6c3a 2052 6174 696f  ^** Ideal: Ratio
│ │ │ │ +000983a0: 6e61 6c4d 6170 205e 5f73 745f 7374 2049  nalMap ^_st_st I
│ │ │ │ +000983b0: 6465 616c 2c20 2d2d 2069 6e76 6572 7365  deal, -- inverse
│ │ │ │ +000983c0: 2069 6d61 6765 0a20 2020 2076 6961 2061   image.    via a
│ │ │ │ +000983d0: 2072 6174 696f 6e61 6c20 6d61 700a 1f0a   rational map...
│ │ │ │ +000983e0: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ +000983f0: 666f 2c20 4e6f 6465 3a20 5261 7469 6f6e  fo, Node: Ration
│ │ │ │ +00098400: 616c 4d61 7020 5f75 735f 7374 2c20 4e65  alMap _us_st, Ne
│ │ │ │ +00098410: 7874 3a20 5261 7469 6f6e 616c 4d61 7020  xt: RationalMap 
│ │ │ │ +00098420: 7c20 4964 6561 6c2c 2050 7265 763a 2052  | Ideal, Prev: R
│ │ │ │ +00098430: 6174 696f 6e61 6c4d 6170 205e 5f73 745f  ationalMap ^_st_
│ │ │ │ +00098440: 7374 2049 6465 616c 2c20 5570 3a20 546f  st Ideal, Up: To
│ │ │ │ +00098450: 700a 0a52 6174 696f 6e61 6c4d 6170 205f  p..RationalMap _
│ │ │ │ +00098460: 2a20 2d2d 2064 6972 6563 7420 696d 6167  * -- direct imag
│ │ │ │ +00098470: 6520 7669 6120 6120 7261 7469 6f6e 616c  e via a rational
│ │ │ │ +00098480: 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a   map.***********
│ │ │ │  00098490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000984a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000984b0: 0a0a 5379 6e6f 7073 6973 0a3d 3d3d 3d3d  ..Synopsis.=====
│ │ │ │ -000984c0: 3d3d 3d0a 0a20 202a 204f 7065 7261 746f  ===..  * Operato
│ │ │ │ -000984d0: 723a 202a 6e6f 7465 205f 2a3a 2028 4d61  r: *note _*: (Ma
│ │ │ │ -000984e0: 6361 756c 6179 3244 6f63 295f 7573 5f73  caulay2Doc)_us_s
│ │ │ │ -000984f0: 742c 0a20 202a 2055 7361 6765 3a20 0a20  t,.  * Usage: . 
│ │ │ │ -00098500: 2020 2020 2020 2070 6869 5f2a 2049 0a20         phi_* I. 
│ │ │ │ -00098510: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -00098520: 202a 2070 6869 2c20 6120 2a6e 6f74 6520   * phi, a *note 
│ │ │ │ -00098530: 7261 7469 6f6e 616c 206d 6170 3a20 5261  rational map: Ra
│ │ │ │ -00098540: 7469 6f6e 616c 4d61 702c 2c20 4920 6120  tionalMap,, I a 
│ │ │ │ -00098550: 686f 6d6f 6765 6e65 6f75 7320 6964 6561  homogeneous idea
│ │ │ │ -00098560: 6c20 696e 2074 6865 0a20 2020 2020 2020  l in the.       
│ │ │ │ -00098570: 2063 6f6f 7264 696e 6174 6520 7269 6e67   coordinate ring
│ │ │ │ -00098580: 206f 6620 7468 6520 736f 7572 6365 206f   of the source o
│ │ │ │ -00098590: 6620 7068 690a 2020 2a20 4f75 7470 7574  f phi.  * Output
│ │ │ │ -000985a0: 733a 0a20 2020 2020 202a 2061 6e20 2a6e  s:.      * an *n
│ │ │ │ -000985b0: 6f74 6520 6964 6561 6c3a 2028 4d61 6361  ote ideal: (Maca
│ │ │ │ -000985c0: 756c 6179 3244 6f63 2949 6465 616c 2c2c  ulay2Doc)Ideal,,
│ │ │ │ -000985d0: 2074 6865 2069 6465 616c 206f 6620 7468   the ideal of th
│ │ │ │ -000985e0: 6520 636c 6f73 7572 6520 6f66 2074 6865  e closure of the
│ │ │ │ -000985f0: 0a20 2020 2020 2020 2064 6972 6563 7420  .        direct 
│ │ │ │ -00098600: 696d 6167 6520 6f66 2056 2849 2920 7669  image of V(I) vi
│ │ │ │ -00098610: 6120 7068 690a 0a44 6573 6372 6970 7469  a phi..Descripti
│ │ │ │ -00098620: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00098630: 496e 206d 6f73 7420 6361 7365 7320 7468  In most cases th
│ │ │ │ -00098640: 6973 2069 7320 6571 7569 7661 6c65 6e74  is is equivalent
│ │ │ │ -00098650: 2074 6f20 7068 6920 492c 2077 6869 6368   to phi I, which
│ │ │ │ -00098660: 2069 7320 6661 7374 6572 2062 7574 206d   is faster but m
│ │ │ │ -00098670: 6179 206e 6f74 2074 616b 650a 696e 746f  ay not take.into
│ │ │ │ -00098680: 2061 6363 6f75 6e74 206f 7468 6572 2072   account other r
│ │ │ │ -00098690: 6570 7265 7365 6e74 6174 696f 6e73 206f  epresentations o
│ │ │ │ -000986a0: 6620 7468 6520 6d61 702e 0a0a 5365 6520  f the map...See 
│ │ │ │ -000986b0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -000986c0: 202a 202a 6e6f 7465 2052 6174 696f 6e61   * *note Rationa
│ │ │ │ -000986d0: 6c4d 6170 205e 2a2a 2049 6465 616c 3a20  lMap ^** Ideal: 
│ │ │ │ -000986e0: 5261 7469 6f6e 616c 4d61 7020 5e5f 7374  RationalMap ^_st
│ │ │ │ -000986f0: 5f73 7420 4964 6561 6c2c 202d 2d20 696e  _st Ideal, -- in
│ │ │ │ -00098700: 7665 7273 6520 696d 6167 650a 2020 2020  verse image.    
│ │ │ │ -00098710: 7669 6120 6120 7261 7469 6f6e 616c 206d  via a rational m
│ │ │ │ -00098720: 6170 0a20 202a 202a 6e6f 7465 2073 6f75  ap.  * *note sou
│ │ │ │ -00098730: 7263 6528 5261 7469 6f6e 616c 4d61 7029  rce(RationalMap)
│ │ │ │ -00098740: 3a20 736f 7572 6365 5f6c 7052 6174 696f  : source_lpRatio
│ │ │ │ -00098750: 6e61 6c4d 6170 5f72 702c 202d 2d20 636f  nalMap_rp, -- co
│ │ │ │ -00098760: 6f72 6469 6e61 7465 2072 696e 6720 6f66  ordinate ring of
│ │ │ │ -00098770: 0a20 2020 2074 6865 2073 6f75 7263 6520  .    the source 
│ │ │ │ -00098780: 666f 7220 6120 7261 7469 6f6e 616c 206d  for a rational m
│ │ │ │ -00098790: 6170 0a0a 5761 7973 2074 6f20 7573 6520  ap..Ways to use 
│ │ │ │ -000987a0: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ -000987b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000987c0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -000987d0: 2052 6174 696f 6e61 6c4d 6170 205f 2a3a   RationalMap _*:
│ │ │ │ -000987e0: 2052 6174 696f 6e61 6c4d 6170 205f 7573   RationalMap _us
│ │ │ │ -000987f0: 5f73 742c 202d 2d20 6469 7265 6374 2069  _st, -- direct i
│ │ │ │ -00098800: 6d61 6765 2076 6961 2061 2072 6174 696f  mage via a ratio
│ │ │ │ -00098810: 6e61 6c0a 2020 2020 6d61 700a 1f0a 4669  nal.    map...Fi
│ │ │ │ -00098820: 6c65 3a20 4372 656d 6f6e 612e 696e 666f  le: Cremona.info
│ │ │ │ -00098830: 2c20 4e6f 6465 3a20 5261 7469 6f6e 616c  , Node: Rational
│ │ │ │ -00098840: 4d61 7020 7c20 4964 6561 6c2c 204e 6578  Map | Ideal, Nex
│ │ │ │ -00098850: 743a 2052 6174 696f 6e61 6c4d 6170 207c  t: RationalMap |
│ │ │ │ -00098860: 7c20 4964 6561 6c2c 2050 7265 763a 2052  | Ideal, Prev: R
│ │ │ │ -00098870: 6174 696f 6e61 6c4d 6170 205f 7573 5f73  ationalMap _us_s
│ │ │ │ -00098880: 742c 2055 703a 2054 6f70 0a0a 5261 7469  t, Up: Top..Rati
│ │ │ │ -00098890: 6f6e 616c 4d61 7020 7c20 4964 6561 6c20  onalMap | Ideal 
│ │ │ │ -000988a0: 2d2d 2072 6573 7472 6963 7469 6f6e 206f  -- restriction o
│ │ │ │ -000988b0: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ -000988c0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +000984b0: 2a2a 2a2a 2a2a 0a0a 5379 6e6f 7073 6973  ******..Synopsis
│ │ │ │ +000984c0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 204f  .========..  * O
│ │ │ │ +000984d0: 7065 7261 746f 723a 202a 6e6f 7465 205f  perator: *note _
│ │ │ │ +000984e0: 2a3a 2028 4d61 6361 756c 6179 3244 6f63  *: (Macaulay2Doc
│ │ │ │ +000984f0: 295f 7573 5f73 742c 0a20 202a 2055 7361  )_us_st,.  * Usa
│ │ │ │ +00098500: 6765 3a20 0a20 2020 2020 2020 2070 6869  ge: .        phi
│ │ │ │ +00098510: 5f2a 2049 0a20 202a 2049 6e70 7574 733a  _* I.  * Inputs:
│ │ │ │ +00098520: 0a20 2020 2020 202a 2070 6869 2c20 6120  .      * phi, a 
│ │ │ │ +00098530: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +00098540: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ +00098550: 2c20 4920 6120 686f 6d6f 6765 6e65 6f75  , I a homogeneou
│ │ │ │ +00098560: 7320 6964 6561 6c20 696e 2074 6865 0a20  s ideal in the. 
│ │ │ │ +00098570: 2020 2020 2020 2063 6f6f 7264 696e 6174         coordinat
│ │ │ │ +00098580: 6520 7269 6e67 206f 6620 7468 6520 736f  e ring of the so
│ │ │ │ +00098590: 7572 6365 206f 6620 7068 690a 2020 2a20  urce of phi.  * 
│ │ │ │ +000985a0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +000985b0: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a   an *note ideal:
│ │ │ │ +000985c0: 2028 4d61 6361 756c 6179 3244 6f63 2949   (Macaulay2Doc)I
│ │ │ │ +000985d0: 6465 616c 2c2c 2074 6865 2069 6465 616c  deal,, the ideal
│ │ │ │ +000985e0: 206f 6620 7468 6520 636c 6f73 7572 6520   of the closure 
│ │ │ │ +000985f0: 6f66 2074 6865 0a20 2020 2020 2020 2064  of the.        d
│ │ │ │ +00098600: 6972 6563 7420 696d 6167 6520 6f66 2056  irect image of V
│ │ │ │ +00098610: 2849 2920 7669 6120 7068 690a 0a44 6573  (I) via phi..Des
│ │ │ │ +00098620: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00098630: 3d3d 3d3d 0a0a 496e 206d 6f73 7420 6361  ====..In most ca
│ │ │ │ +00098640: 7365 7320 7468 6973 2069 7320 6571 7569  ses this is equi
│ │ │ │ +00098650: 7661 6c65 6e74 2074 6f20 7068 6920 492c  valent to phi I,
│ │ │ │ +00098660: 2077 6869 6368 2069 7320 6661 7374 6572   which is faster
│ │ │ │ +00098670: 2062 7574 206d 6179 206e 6f74 2074 616b   but may not tak
│ │ │ │ +00098680: 650a 696e 746f 2061 6363 6f75 6e74 206f  e.into account o
│ │ │ │ +00098690: 7468 6572 2072 6570 7265 7365 6e74 6174  ther representat
│ │ │ │ +000986a0: 696f 6e73 206f 6620 7468 6520 6d61 702e  ions of the map.
│ │ │ │ +000986b0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +000986c0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2052  ===..  * *note R
│ │ │ │ +000986d0: 6174 696f 6e61 6c4d 6170 205e 2a2a 2049  ationalMap ^** I
│ │ │ │ +000986e0: 6465 616c 3a20 5261 7469 6f6e 616c 4d61  deal: RationalMa
│ │ │ │ +000986f0: 7020 5e5f 7374 5f73 7420 4964 6561 6c2c  p ^_st_st Ideal,
│ │ │ │ +00098700: 202d 2d20 696e 7665 7273 6520 696d 6167   -- inverse imag
│ │ │ │ +00098710: 650a 2020 2020 7669 6120 6120 7261 7469  e.    via a rati
│ │ │ │ +00098720: 6f6e 616c 206d 6170 0a20 202a 202a 6e6f  onal map.  * *no
│ │ │ │ +00098730: 7465 2073 6f75 7263 6528 5261 7469 6f6e  te source(Ration
│ │ │ │ +00098740: 616c 4d61 7029 3a20 736f 7572 6365 5f6c  alMap): source_l
│ │ │ │ +00098750: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +00098760: 202d 2d20 636f 6f72 6469 6e61 7465 2072   -- coordinate r
│ │ │ │ +00098770: 696e 6720 6f66 0a20 2020 2074 6865 2073  ing of.    the s
│ │ │ │ +00098780: 6f75 7263 6520 666f 7220 6120 7261 7469  ource for a rati
│ │ │ │ +00098790: 6f6e 616c 206d 6170 0a0a 5761 7973 2074  onal map..Ways t
│ │ │ │ +000987a0: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ +000987b0: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ +000987c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +000987d0: 202a 6e6f 7465 2052 6174 696f 6e61 6c4d   *note RationalM
│ │ │ │ +000987e0: 6170 205f 2a3a 2052 6174 696f 6e61 6c4d  ap _*: RationalM
│ │ │ │ +000987f0: 6170 205f 7573 5f73 742c 202d 2d20 6469  ap _us_st, -- di
│ │ │ │ +00098800: 7265 6374 2069 6d61 6765 2076 6961 2061  rect image via a
│ │ │ │ +00098810: 2072 6174 696f 6e61 6c0a 2020 2020 6d61   rational.    ma
│ │ │ │ +00098820: 700a 1f0a 4669 6c65 3a20 4372 656d 6f6e  p...File: Cremon
│ │ │ │ +00098830: 612e 696e 666f 2c20 4e6f 6465 3a20 5261  a.info, Node: Ra
│ │ │ │ +00098840: 7469 6f6e 616c 4d61 7020 7c20 4964 6561  tionalMap | Idea
│ │ │ │ +00098850: 6c2c 204e 6578 743a 2052 6174 696f 6e61  l, Next: Rationa
│ │ │ │ +00098860: 6c4d 6170 207c 7c20 4964 6561 6c2c 2050  lMap || Ideal, P
│ │ │ │ +00098870: 7265 763a 2052 6174 696f 6e61 6c4d 6170  rev: RationalMap
│ │ │ │ +00098880: 205f 7573 5f73 742c 2055 703a 2054 6f70   _us_st, Up: Top
│ │ │ │ +00098890: 0a0a 5261 7469 6f6e 616c 4d61 7020 7c20  ..RationalMap | 
│ │ │ │ +000988a0: 4964 6561 6c20 2d2d 2072 6573 7472 6963  Ideal -- restric
│ │ │ │ +000988b0: 7469 6f6e 206f 6620 6120 7261 7469 6f6e  tion of a ration
│ │ │ │ +000988c0: 616c 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a  al map.*********
│ │ │ │  000988d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000988e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000988f0: 2a2a 2a2a 2a0a 0a53 796e 6f70 7369 730a  *****..Synopsis.
│ │ │ │ -00098900: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 4f70  ========..  * Op
│ │ │ │ -00098910: 6572 6174 6f72 3a20 2a6e 6f74 6520 7c3a  erator: *note |:
│ │ │ │ -00098920: 2028 4d61 6361 756c 6179 3244 6f63 297c   (Macaulay2Doc)|
│ │ │ │ -00098930: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ -00098940: 2020 2020 2020 5068 6920 7c20 490a 2020        Phi | I.  
│ │ │ │ -00098950: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00098960: 2a20 5068 692c 2061 202a 6e6f 7465 2072  * Phi, a *note r
│ │ │ │ -00098970: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ -00098980: 696f 6e61 6c4d 6170 2c2c 2024 5c70 6869  ionalMap,, $\phi
│ │ │ │ -00098990: 3a58 205c 6461 7368 7269 6768 7461 7272  :X \dashrightarr
│ │ │ │ -000989a0: 6f77 2059 240a 2020 2020 2020 2a20 492c  ow Y$.      * I,
│ │ │ │ -000989b0: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a   an *note ideal:
│ │ │ │ -000989c0: 2028 4d61 6361 756c 6179 3244 6f63 2949   (Macaulay2Doc)I
│ │ │ │ -000989d0: 6465 616c 2c2c 2061 2068 6f6d 6f67 656e  deal,, a homogen
│ │ │ │ -000989e0: 656f 7573 2069 6465 616c 206f 6620 610a  eous ideal of a.
│ │ │ │ -000989f0: 2020 2020 2020 2020 7375 6276 6172 6965          subvarie
│ │ │ │ -00098a00: 7479 2024 5a5c 7375 6273 6574 2058 240a  ty $Z\subset X$.
│ │ │ │ -00098a10: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00098a20: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ -00098a30: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00098a40: 6e61 6c4d 6170 2c2c 2074 6865 2072 6573  nalMap,, the res
│ │ │ │ -00098a50: 7472 6963 7469 6f6e 206f 6620 245c 7068  triction of $\ph
│ │ │ │ -00098a60: 6924 2074 6f20 245a 242c 0a20 2020 2020  i$ to $Z$,.     
│ │ │ │ -00098a70: 2020 2024 5c70 6869 7c5f 7b5a 7d3a 205a     $\phi|_{Z}: Z
│ │ │ │ -00098a80: 205c 6461 7368 7269 6768 7461 7272 6f77   \dashrightarrow
│ │ │ │ -00098a90: 2059 240a 0a44 6573 6372 6970 7469 6f6e   Y$..Description
│ │ │ │ -00098aa0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ -00098ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000988f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a53 796e  ***********..Syn
│ │ │ │ +00098900: 6f70 7369 730a 3d3d 3d3d 3d3d 3d3d 0a0a  opsis.========..
│ │ │ │ +00098910: 2020 2a20 4f70 6572 6174 6f72 3a20 2a6e    * Operator: *n
│ │ │ │ +00098920: 6f74 6520 7c3a 2028 4d61 6361 756c 6179  ote |: (Macaulay
│ │ │ │ +00098930: 3244 6f63 297c 2c0a 2020 2a20 5573 6167  2Doc)|,.  * Usag
│ │ │ │ +00098940: 653a 200a 2020 2020 2020 2020 5068 6920  e: .        Phi 
│ │ │ │ +00098950: 7c20 490a 2020 2a20 496e 7075 7473 3a0a  | I.  * Inputs:.
│ │ │ │ +00098960: 2020 2020 2020 2a20 5068 692c 2061 202a        * Phi, a *
│ │ │ │ +00098970: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ +00098980: 703a 2052 6174 696f 6e61 6c4d 6170 2c2c  p: RationalMap,,
│ │ │ │ +00098990: 2024 5c70 6869 3a58 205c 6461 7368 7269   $\phi:X \dashri
│ │ │ │ +000989a0: 6768 7461 7272 6f77 2059 240a 2020 2020  ghtarrow Y$.    
│ │ │ │ +000989b0: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520    * I, an *note 
│ │ │ │ +000989c0: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ +000989d0: 3244 6f63 2949 6465 616c 2c2c 2061 2068  2Doc)Ideal,, a h
│ │ │ │ +000989e0: 6f6d 6f67 656e 656f 7573 2069 6465 616c  omogeneous ideal
│ │ │ │ +000989f0: 206f 6620 610a 2020 2020 2020 2020 7375   of a.        su
│ │ │ │ +00098a00: 6276 6172 6965 7479 2024 5a5c 7375 6273  bvariety $Z\subs
│ │ │ │ +00098a10: 6574 2058 240a 2020 2a20 4f75 7470 7574  et X$.  * Output
│ │ │ │ +00098a20: 733a 0a20 2020 2020 202a 2061 202a 6e6f  s:.      * a *no
│ │ │ │ +00098a30: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ +00098a40: 2052 6174 696f 6e61 6c4d 6170 2c2c 2074   RationalMap,, t
│ │ │ │ +00098a50: 6865 2072 6573 7472 6963 7469 6f6e 206f  he restriction o
│ │ │ │ +00098a60: 6620 245c 7068 6924 2074 6f20 245a 242c  f $\phi$ to $Z$,
│ │ │ │ +00098a70: 0a20 2020 2020 2020 2024 5c70 6869 7c5f  .        $\phi|_
│ │ │ │ +00098a80: 7b5a 7d3a 205a 205c 6461 7368 7269 6768  {Z}: Z \dashrigh
│ │ │ │ +00098a90: 7461 7272 6f77 2059 240a 0a44 6573 6372  tarrow Y$..Descr
│ │ │ │ +00098aa0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00098ab0: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ==..+-----------
│ │ │ │  00098ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00098af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00098b00: 3120 3a20 5035 203d 205a 5a2f 3139 3031  1 : P5 = ZZ/1901
│ │ │ │ -00098b10: 3831 5b78 5f30 2e2e 785f 355d 2020 2020  81[x_0..x_5]    
│ │ │ │ -00098b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00098b00: 2d2d 2b0a 7c69 3120 3a20 5035 203d 205a  --+.|i1 : P5 = Z
│ │ │ │ +00098b10: 5a2f 3139 3031 3831 5b78 5f30 2e2e 785f  Z/190181[x_0..x_
│ │ │ │ +00098b20: 355d 2020 2020 2020 2020 2020 2020 2020  5]              
│ │ │ │  00098b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098b50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00098b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098b90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00098ba0: 3120 3d20 5035 2020 2020 2020 2020 2020  1 = P5          
│ │ │ │ +00098b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098ba0: 2020 7c0a 7c6f 3120 3d20 5035 2020 2020    |.|o1 = P5    
│ │ │ │  00098bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098be0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098bf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00098c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098c30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00098c40: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269  1 : PolynomialRi
│ │ │ │ -00098c50: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00098c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098c40: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f    |.|o1 : Polyno
│ │ │ │ +00098c50: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │  00098c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098c80: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00098c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00098c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098c90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00098ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00098cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00098ce0: 3220 3a20 5068 6920 3d20 7261 7469 6f6e  2 : Phi = ration
│ │ │ │ -00098cf0: 616c 4d61 7020 7b78 5f34 5e32 2d78 5f33  alMap {x_4^2-x_3
│ │ │ │ -00098d00: 2a78 5f35 2c78 5f32 2a78 5f34 2d78 5f31  *x_5,x_2*x_4-x_1
│ │ │ │ -00098d10: 2a78 5f35 2c78 5f32 2a78 5f33 2d78 5f31  *x_5,x_2*x_3-x_1
│ │ │ │ -00098d20: 2a78 5f34 2c78 5f32 5e32 2d78 7c0a 7c20  *x_4,x_2^2-x|.| 
│ │ │ │ -00098d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00098ce0: 2d2d 2b0a 7c69 3220 3a20 5068 6920 3d20  --+.|i2 : Phi = 
│ │ │ │ +00098cf0: 7261 7469 6f6e 616c 4d61 7020 7b78 5f34  rationalMap {x_4
│ │ │ │ +00098d00: 5e32 2d78 5f33 2a78 5f35 2c78 5f32 2a78  ^2-x_3*x_5,x_2*x
│ │ │ │ +00098d10: 5f34 2d78 5f31 2a78 5f35 2c78 5f32 2a78  _4-x_1*x_5,x_2*x
│ │ │ │ +00098d20: 5f33 2d78 5f31 2a78 5f34 2c78 5f32 5e32  _3-x_1*x_4,x_2^2
│ │ │ │ +00098d30: 2d78 7c0a 7c20 2020 2020 2020 2020 2020  -x|.|           
│ │ │ │  00098d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098d70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00098d80: 3220 3d20 2d2d 2072 6174 696f 6e61 6c20  2 = -- rational 
│ │ │ │ -00098d90: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +00098d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098d80: 2020 7c0a 7c6f 3220 3d20 2d2d 2072 6174    |.|o2 = -- rat
│ │ │ │ +00098d90: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │  00098da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098de0: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │ +00098dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098dd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00098de0: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │  00098df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098e10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098e20: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -00098e30: 282d 2d2d 2d2d 2d5b 7820 2c20 7820 2c20  (------[x , x , 
│ │ │ │ -00098e40: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ -00098e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098e80: 2031 3930 3138 3120 2030 2020 2031 2020   190181  0   1  
│ │ │ │ -00098e90: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ -00098ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098ed0: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │ +00098e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098e20: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +00098e30: 3a20 5072 6f6a 282d 2d2d 2d2d 2d5b 7820  : Proj(------[x 
│ │ │ │ +00098e40: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +00098e50: 2c20 7820 5d29 2020 2020 2020 2020 2020  , x ])          
│ │ │ │ +00098e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098e70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00098e80: 2020 2020 2020 2031 3930 3138 3120 2030         190181  0
│ │ │ │ +00098e90: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ +00098ea0: 2020 2035 2020 2020 2020 2020 2020 2020     5            
│ │ │ │ +00098eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098ec0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00098ed0: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │  00098ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098f10: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -00098f20: 282d 2d2d 2d2d 2d5b 7820 2c20 7820 2c20  (------[x , x , 
│ │ │ │ -00098f30: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ -00098f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098f70: 2031 3930 3138 3120 2030 2020 2031 2020   190181  0   1  
│ │ │ │ -00098f80: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ -00098f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098fb0: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -00098fc0: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +00098f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098f10: 2020 7c0a 7c20 2020 2020 7461 7267 6574    |.|     target
│ │ │ │ +00098f20: 3a20 5072 6f6a 282d 2d2d 2d2d 2d5b 7820  : Proj(------[x 
│ │ │ │ +00098f30: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +00098f40: 2c20 7820 5d29 2020 2020 2020 2020 2020  , x ])          
│ │ │ │ +00098f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098f60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00098f70: 2020 2020 2020 2031 3930 3138 3120 2030         190181  0
│ │ │ │ +00098f80: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ +00098f90: 2020 2035 2020 2020 2020 2020 2020 2020     5            
│ │ │ │ +00098fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00098fb0: 2020 7c0a 7c20 2020 2020 6465 6669 6e69    |.|     defini
│ │ │ │ +00098fc0: 6e67 2066 6f72 6d73 3a20 7b20 2020 2020  ng forms: {     
│ │ │ │  00098fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099010: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00098ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099000: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099010: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00099020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099040: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099060: 2020 2020 2078 2020 2d20 7820 7820 2c20       x  - x x , 
│ │ │ │ -00099070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099050: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099060: 2020 2020 2020 2020 2020 2078 2020 2d20             x  - 
│ │ │ │ +00099070: 7820 7820 2c20 2020 2020 2020 2020 2020  x x ,           
│ │ │ │  00099080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000990a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000990b0: 2020 2020 2020 3420 2020 2033 2035 2020        4    3 5  
│ │ │ │ -000990c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000990a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000990b0: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +000990c0: 2033 2035 2020 2020 2020 2020 2020 2020   3 5            
│ │ │ │  000990d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000990e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000990f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000990e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000990f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00099100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099130: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099150: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -00099160: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00099130: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00099150: 2020 2020 2020 2020 2020 2078 2078 2020             x x  
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│ │ │ │  00099170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099180: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000991a0: 2020 2020 2020 3220 3420 2020 2031 2035        2 4    1 5
│ │ │ │ -000991b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099190: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000991a0: 2020 2020 2020 2020 2020 2020 3220 3420              2 4 
│ │ │ │ +000991b0: 2020 2031 2035 2020 2020 2020 2020 2020     1 5          
│ │ │ │  000991c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000991d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000991e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000991d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000991e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000991f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099240: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -00099250: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00099220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099230: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099240: 2020 2020 2020 2020 2020 2078 2078 2020             x x  
│ │ │ │ +00099250: 2d20 7820 7820 2c20 2020 2020 2020 2020  - x x ,         
│ │ │ │  00099260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099270: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099290: 2020 2020 2020 3220 3320 2020 2031 2034        2 3    1 4
│ │ │ │ -000992a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099280: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099290: 2020 2020 2020 2020 2020 2020 3220 3320              2 3 
│ │ │ │ +000992a0: 2020 2031 2034 2020 2020 2020 2020 2020     1 4          
│ │ │ │  000992b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000992c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000992d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000992c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000992d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000992e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000992f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099310: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099330: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00099310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099320: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099330: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00099340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099360: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099380: 2020 2020 2078 2020 2d20 7820 7820 2c20       x  - x x , 
│ │ │ │ -00099390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099370: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099380: 2020 2020 2020 2020 2020 2078 2020 2d20             x  - 
│ │ │ │ +00099390: 7820 7820 2c20 2020 2020 2020 2020 2020  x x ,           
│ │ │ │  000993a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000993b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000993c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000993d0: 2020 2020 2020 3220 2020 2030 2035 2020        2    0 5  
│ │ │ │ -000993e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000993b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000993c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000993d0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000993e0: 2030 2035 2020 2020 2020 2020 2020 2020   0 5            
│ │ │ │  000993f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099400: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099410: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00099420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099450: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099470: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -00099480: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00099450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099470: 2020 2020 2020 2020 2020 2078 2078 2020             x x  
│ │ │ │ +00099480: 2d20 7820 7820 2c20 2020 2020 2020 2020  - x x ,         
│ │ │ │  00099490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000994a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000994b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000994c0: 2020 2020 2020 3120 3220 2020 2030 2034        1 2    0 4
│ │ │ │ -000994d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000994a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000994b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000994c0: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
│ │ │ │ +000994d0: 2020 2030 2034 2020 2020 2020 2020 2020     0 4          
│ │ │ │  000994e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000994f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000994f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099500: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00099510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099560: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00099540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099550: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099560: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00099570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099590: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000995a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000995b0: 2020 2020 2078 2020 2d20 7820 7820 2020       x  - x x   
│ │ │ │ -000995c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000995a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000995b0: 2020 2020 2020 2020 2020 2078 2020 2d20             x  - 
│ │ │ │ +000995c0: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │  000995d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000995e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000995f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099600: 2020 2020 2020 3120 2020 2030 2033 2020        1    0 3  
│ │ │ │ -00099610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000995e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000995f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099600: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00099610: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
│ │ │ │  00099620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099630: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099650: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +00099630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099640: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099650: 2020 2020 2020 2020 2020 7d20 2020 2020            }     
│ │ │ │  00099660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099690: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000996a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000996b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000996c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000996d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000996e0: 3220 3a20 5261 7469 6f6e 616c 4d61 7020  2 : RationalMap 
│ │ │ │ -000996f0: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ -00099700: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ -00099710: 3520 746f 2050 505e 3529 2020 2020 2020  5 to PP^5)      
│ │ │ │ -00099720: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -00099730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000996d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000996e0: 2020 7c0a 7c6f 3220 3a20 5261 7469 6f6e    |.|o2 : Ration
│ │ │ │ +000996f0: 616c 4d61 7020 2871 7561 6472 6174 6963  alMap (quadratic
│ │ │ │ +00099700: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ +00099710: 6f6d 2050 505e 3520 746f 2050 505e 3529  om PP^5 to PP^5)
│ │ │ │ +00099720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099730: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  00099740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f  ------------|.|_
│ │ │ │ -00099780: 302a 785f 352c 785f 312a 785f 322d 785f  0*x_5,x_1*x_2-x_
│ │ │ │ -00099790: 302a 785f 342c 785f 315e 322d 785f 302a  0*x_4,x_1^2-x_0*
│ │ │ │ -000997a0: 785f 337d 2020 2020 2020 2020 2020 2020  x_3}            
│ │ │ │ +00099770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00099780: 2d2d 7c0a 7c5f 302a 785f 352c 785f 312a  --|.|_0*x_5,x_1*
│ │ │ │ +00099790: 785f 322d 785f 302a 785f 342c 785f 315e  x_2-x_0*x_4,x_1^
│ │ │ │ +000997a0: 322d 785f 302a 785f 337d 2020 2020 2020  2-x_0*x_3}      
│ │ │ │  000997b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000997c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -000997d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000997c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000997d0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000997e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000997f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00099820: 3320 3a20 4920 3d20 6964 6561 6c28 7261  3 : I = ideal(ra
│ │ │ │ -00099830: 6e64 6f6d 2832 2c50 3529 2c72 616e 646f  ndom(2,P5),rando
│ │ │ │ -00099840: 6d28 332c 5035 2929 3b20 2020 2020 2020  m(3,P5));       
│ │ │ │ +00099810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00099820: 2d2d 2b0a 7c69 3320 3a20 4920 3d20 6964  --+.|i3 : I = id
│ │ │ │ +00099830: 6561 6c28 7261 6e64 6f6d 2832 2c50 3529  eal(random(2,P5)
│ │ │ │ +00099840: 2c72 616e 646f 6d28 332c 5035 2929 3b20  ,random(3,P5)); 
│ │ │ │  00099850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099870: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00099880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000998a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000998b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000998c0: 3320 3a20 4964 6561 6c20 6f66 2050 3520  3 : Ideal of P5 
│ │ │ │ -000998d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000998b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000998c0: 2020 7c0a 7c6f 3320 3a20 4964 6561 6c20    |.|o3 : Ideal 
│ │ │ │ +000998d0: 6f66 2050 3520 2020 2020 2020 2020 2020  of P5           
│ │ │ │  000998e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000998f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00099910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00099900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099910: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00099920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00099960: 3420 3a20 5068 6927 203d 2050 6869 7c49  4 : Phi' = Phi|I
│ │ │ │ -00099970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00099960: 2d2d 2b0a 7c69 3420 3a20 5068 6927 203d  --+.|i4 : Phi' =
│ │ │ │ +00099970: 2050 6869 7c49 2020 2020 2020 2020 2020   Phi|I          
│ │ │ │  00099980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000999a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000999b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000999a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000999b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000999c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000999d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000999e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000999f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00099a00: 3420 3d20 2d2d 2072 6174 696f 6e61 6c20  4 = -- rational 
│ │ │ │ -00099a10: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +000999f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099a00: 2020 7c0a 7c6f 3420 3d20 2d2d 2072 6174    |.|o4 = -- rat
│ │ │ │ +00099a10: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │  00099a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099a50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00099a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099a70: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +00099a70: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │  00099a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099a90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099aa0: 2020 2020 736f 7572 6365 3a20 7375 6276      source: subv
│ │ │ │ -00099ab0: 6172 6965 7479 206f 6620 5072 6f6a 282d  ariety of Proj(-
│ │ │ │ -00099ac0: 2d2d 2d2d 2d5b 7820 2c20 7820 2c20 7820  -----[x , x , x 
│ │ │ │ -00099ad0: 2c20 7820 2c20 7820 2c20 7820 5d29 2064  , x , x , x ]) d
│ │ │ │ -00099ae0: 6566 696e 6564 2062 7920 2020 7c0a 7c20  efined by   |.| 
│ │ │ │ -00099af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099b00: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00099b10: 3930 3138 3120 2030 2020 2031 2020 2032  90181  0   1   2
│ │ │ │ -00099b20: 2020 2033 2020 2034 2020 2035 2020 2020     3   4   5    
│ │ │ │ -00099b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099b40: 2020 2020 2020 2020 2020 2020 7b20 2020              {   
│ │ │ │ -00099b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099aa0: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +00099ab0: 3a20 7375 6276 6172 6965 7479 206f 6620  : subvariety of 
│ │ │ │ +00099ac0: 5072 6f6a 282d 2d2d 2d2d 2d5b 7820 2c20  Proj(------[x , 
│ │ │ │ +00099ad0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ +00099ae0: 7820 5d29 2064 6566 696e 6564 2062 7920  x ]) defined by 
│ │ │ │ +00099af0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099b10: 2020 2020 2031 3930 3138 3120 2030 2020       190181  0  
│ │ │ │ +00099b20: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ +00099b30: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +00099b40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099b50: 2020 7b20 2020 2020 2020 2020 2020 2020    {             
│ │ │ │  00099b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00099b90: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00099ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099bb0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00099b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099b90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00099ba0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00099bb0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  00099bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099bd0: 2020 2020 2020 3220 2020 2020 7c0a 7c20        2     |.| 
│ │ │ │ -00099be0: 2020 2020 2020 2020 2020 2020 2078 2020               x  
│ │ │ │ -00099bf0: 2b20 3136 3536 3678 2078 2020 2d20 3437  + 16566x x  - 47
│ │ │ │ -00099c00: 3436 3878 2020 2d20 3730 3135 3878 2078  468x  - 70158x x
│ │ │ │ -00099c10: 2020 2b20 3930 3632 3678 2078 2020 2d20    + 90626x x  - 
│ │ │ │ -00099c20: 3234 3231 3278 2020 2d20 2020 7c0a 7c20  24212x  -   |.| 
│ │ │ │ -00099c30: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00099c40: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ -00099c50: 2020 2020 3120 2020 2020 2020 2020 3020      1         0 
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│ │ │ │ -00099c70: 2020 2020 2020 3220 2020 2020 7c0a 7c20        2     |.| 
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│ │ │ │  0009a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0009a620: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009a630: 3420 3a20 5261 7469 6f6e 616c 4d61 7020  4 : RationalMap 
│ │ │ │ -0009a640: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ -0009a650: 6e61 6c20 6d61 7020 6672 6f6d 2074 6872  nal map from thr
│ │ │ │ -0009a660: 6565 666f 6c64 2069 6e20 5050 5e35 2074  eefold in PP^5 t
│ │ │ │ -0009a670: 6f20 5050 5e35 2920 2020 2020 7c0a 7c2d  o PP^5)     |.|-
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│ │ │ │ +0009a630: 2020 7c0a 7c6f 3420 3a20 5261 7469 6f6e    |.|o4 : Ration
│ │ │ │ +0009a640: 616c 4d61 7020 2871 7561 6472 6174 6963  alMap (quadratic
│ │ │ │ +0009a650: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ +0009a660: 6f6d 2074 6872 6565 666f 6c64 2069 6e20  om threefold in 
│ │ │ │ +0009a670: 5050 5e35 2074 6f20 5050 5e35 2920 2020  PP^5 to PP^5)   
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│ │ │ │ -0009a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009a6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0009a6f0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0009a6f0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  0009a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a710: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ -0009a720: 3831 3438 7820 7820 202d 2033 3137 3739  8148x x  - 31779
│ │ │ │ -0009a730: 7820 7820 202b 2031 3936 3035 7820 7820  x x  + 19605x x 
│ │ │ │ -0009a740: 202d 2037 3639 7820 202d 2037 3738 3634   - 769x  - 77864
│ │ │ │ -0009a750: 7820 7820 202b 2031 3036 3630 7820 7820  x x  + 10660x x 
│ │ │ │ -0009a760: 202b 2020 2020 2020 2020 2020 7c0a 7c20   +          |.| 
│ │ │ │ -0009a770: 2020 2020 2030 2033 2020 2020 2020 2020       0 3        
│ │ │ │ -0009a780: 2031 2033 2020 2020 2020 2020 2032 2033   1 3         2 3
│ │ │ │ -0009a790: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -0009a7a0: 2030 2034 2020 2020 2020 2020 2031 2034   0 4         1 4
│ │ │ │ -0009a7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009a720: 2020 7c0a 7c33 3831 3438 7820 7820 202d    |.|38148x x  -
│ │ │ │ +0009a730: 2033 3137 3739 7820 7820 202b 2031 3936   31779x x  + 196
│ │ │ │ +0009a740: 3035 7820 7820 202d 2037 3639 7820 202d  05x x  - 769x  -
│ │ │ │ +0009a750: 2037 3738 3634 7820 7820 202b 2031 3036   77864x x  + 106
│ │ │ │ +0009a760: 3630 7820 7820 202b 2020 2020 2020 2020  60x x  +        
│ │ │ │ +0009a770: 2020 7c0a 7c20 2020 2020 2030 2033 2020    |.|      0 3  
│ │ │ │ +0009a780: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ +0009a790: 2020 2032 2033 2020 2020 2020 2033 2020     2 3       3  
│ │ │ │ +0009a7a0: 2020 2020 2020 2030 2034 2020 2020 2020         0 4      
│ │ │ │ +0009a7b0: 2020 2031 2034 2020 2020 2020 2020 2020     1 4          
│ │ │ │ +0009a7c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0009a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009a810: 3220 2020 2020 2020 2033 2020 2020 2020  2        3      
│ │ │ │ +0009a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0009a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0009a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a850: 2020 2020 2020 2020 2020 2020 7c0a 7c78              |.|x
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│ │ │ │ -0009a8a0: 202d 2020 2020 2020 2020 2020 7c0a 7c20   -          |.| 
│ │ │ │ -0009a8b0: 3220 2020 2020 2020 2032 2020 2020 2020  2        2      
│ │ │ │ -0009a8c0: 2020 2030 2031 2033 2020 2020 2020 2020     0 1 3        
│ │ │ │ -0009a8d0: 2031 2033 2020 2020 2020 2020 2030 2032   1 3         0 2
│ │ │ │ -0009a8e0: 2033 2020 2020 2020 2020 2031 2032 2033   3         1 2 3
│ │ │ │ -0009a8f0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -0009a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0009a870: 202b 2031 3231 3234 7820 7820 7820 202d   + 12124x x x  -
│ │ │ │ +0009a880: 2033 3035 3438 7820 7820 202d 2035 3737   30548x x  - 577
│ │ │ │ +0009a890: 3433 7820 7820 7820 202d 2031 3235 3930  43x x x  - 12590
│ │ │ │ +0009a8a0: 7820 7820 7820 202d 2020 2020 2020 2020  x x x  -        
│ │ │ │ +0009a8b0: 2020 7c0a 7c20 3220 2020 2020 2020 2032    |.| 2        2
│ │ │ │ +0009a8c0: 2020 2020 2020 2020 2030 2031 2033 2020           0 1 3  
│ │ │ │ +0009a8d0: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ +0009a8e0: 2020 2030 2032 2033 2020 2020 2020 2020     0 2 3        
│ │ │ │ +0009a8f0: 2031 2032 2033 2020 2020 2020 2020 2020   1 2 3          
│ │ │ │ +0009a900: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  0009a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0009a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0009a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0009a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a990: 2020 2020 2020 2020 2020 2020 7c0a 7c37              |.|7
│ │ │ │ -0009a9a0: 3431 3078 2078 2020 2d20 3636 3536 3578  410x x  - 66565x
│ │ │ │ -0009a9b0: 2078 2020 2d20 3439 3139 3278 2020 2d20   x  - 49192x  - 
│ │ │ │ -0009a9c0: 3731 3332 3178 2078 2020 2b20 3138 3433  71321x x  + 1843
│ │ │ │ -0009a9d0: 3378 2078 2020 2d20 3436 3436 3078 2078  3x x  - 46460x x
│ │ │ │ -0009a9e0: 2020 2b20 2020 2020 2020 2020 7c0a 7c20    +         |.| 
│ │ │ │ -0009a9f0: 2020 2020 3220 3420 2020 2020 2020 2020      2 4         
│ │ │ │ -0009aa00: 3320 3420 2020 2020 2020 2020 3420 2020  3 4         4   
│ │ │ │ -0009aa10: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ -0009aa20: 2020 3120 3520 2020 2020 2020 2020 3220    1 5         2 
│ │ │ │ -0009aa30: 3520 2020 2020 2020 2020 2020 7c0a 7c20  5           |.| 
│ │ │ │ -0009aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0009a9b0: 3636 3536 3578 2078 2020 2d20 3439 3139  66565x x  - 4919
│ │ │ │ +0009a9c0: 3278 2020 2d20 3731 3332 3178 2078 2020  2x  - 71321x x  
│ │ │ │ +0009a9d0: 2b20 3138 3433 3378 2078 2020 2d20 3436  + 18433x x  - 46
│ │ │ │ +0009a9e0: 3436 3078 2078 2020 2b20 2020 2020 2020  460x x  +       
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│ │ │ │ +0009aa00: 2020 2020 2020 3320 3420 2020 2020 2020        3 4       
│ │ │ │ +0009aa10: 2020 3420 2020 2020 2020 2020 3020 3520    4         0 5 
│ │ │ │ +0009aa20: 2020 2020 2020 2020 3120 3520 2020 2020          1 5     
│ │ │ │ +0009aa30: 2020 2020 3220 3520 2020 2020 2020 2020      2 5         
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│ │ │ │  0009aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009aa80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009aa90: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0009aaa0: 2020 2032 2020 2020 2020 2020 2020 2032     2           2
│ │ │ │ -0009aab0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0009aac0: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ -0009aad0: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ -0009aae0: 3933 3537 7820 7820 202d 2032 3535 3035  9357x x  - 25505
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│ │ │ │ -0009ab10: 3930 3838 7820 202b 2032 3636 3135 7820  9088x  + 26615x 
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│ │ │ │ -0009ab30: 2020 2020 2032 2033 2020 2020 2020 2020       2 3        
│ │ │ │ -0009ab40: 2030 2033 2020 2020 2020 2020 2031 2033   0 3         1 3
│ │ │ │ -0009ab50: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -0009ab60: 2020 2020 2033 2020 2020 2020 2020 2030       3         0
│ │ │ │ -0009ab70: 2031 2034 2020 2020 2020 2020 7c0a 7c2d   1 4        |.|-
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│ │ │ │ +0009ab10: 7820 202d 2037 3930 3838 7820 202b 2032  x  - 79088x  + 2
│ │ │ │ +0009ab20: 3636 3135 7820 7820 7820 2020 2020 2020  6615x x x       
│ │ │ │ +0009ab30: 2020 7c0a 7c20 2020 2020 2032 2033 2020    |.|      2 3  
│ │ │ │ +0009ab40: 2020 2020 2020 2030 2033 2020 2020 2020         0 3      
│ │ │ │ +0009ab50: 2020 2031 2033 2020 2020 2020 2020 2032     1 3         2
│ │ │ │ +0009ab60: 2033 2020 2020 2020 2020 2033 2020 2020   3         3    
│ │ │ │ +0009ab70: 2020 2020 2030 2031 2034 2020 2020 2020       0 1 4      
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│ │ │ │  0009ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0009abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0009abf0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0009ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ac10: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ -0009ac20: 3639 3878 2078 2020 2d20 3736 3839 3078  698x x  - 76890x
│ │ │ │ -0009ac30: 2078 2020 2d20 3535 3537 3478 202c 2020   x  - 55574x ,  
│ │ │ │ -0009ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009ac20: 2020 7c0a 7c34 3639 3878 2078 2020 2d20    |.|4698x x  - 
│ │ │ │ +0009ac30: 3736 3839 3078 2078 2020 2d20 3535 3537  76890x x  - 5557
│ │ │ │ +0009ac40: 3478 202c 2020 2020 2020 2020 2020 2020  4x ,            
│ │ │ │  0009ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ac60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009ac70: 2020 2020 3320 3520 2020 2020 2020 2020      3 5         
│ │ │ │ -0009ac80: 3420 3520 2020 2020 2020 2020 3520 2020  4 5         5   
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│ │ │ │  0009aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009acb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  0009acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ad00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  0009ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0009ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ad50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009ad60: 2d20 3130 3033 3678 2078 2020 2b20 3339  - 10036x x  + 39
│ │ │ │ -0009ad70: 3435 3778 2078 2078 2020 2b20 3837 3978  457x x x  + 879x
│ │ │ │ -0009ad80: 2078 2078 2020 2b20 3738 3536 3078 2078   x x  + 78560x x
│ │ │ │ -0009ad90: 2020 2b20 3833 3939 3778 2078 2078 2020    + 83997x x x  
│ │ │ │ -0009ada0: 2d20 2020 2020 2020 2020 2020 7c0a 7c20  -           |.| 
│ │ │ │ -0009adb0: 2020 2020 2020 2020 3120 3420 2020 2020          1 4     
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│ │ │ │ -0009add0: 3120 3220 3420 2020 2020 2020 2020 3220  1 2 4         2 
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│ │ │ │ +0009adc0: 3420 2020 2020 2020 2020 3020 3220 3420  4         0 2 4 
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│ │ │ │  0009ae30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0009ae80: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0009ae90: 2020 2020 2020 2032 2020 2020 7c0a 7c34         2    |.|4
│ │ │ │ -0009aea0: 3235 3532 7820 7820 7820 202b 2036 3937  2552x x x  + 697
│ │ │ │ -0009aeb0: 3532 7820 7820 7820 202d 2037 3837 3133  52x x x  - 78713
│ │ │ │ -0009aec0: 7820 7820 202b 2035 3236 3938 7820 7820  x x  + 52698x x 
│ │ │ │ -0009aed0: 202d 2034 3635 3937 7820 7820 202d 2034   - 46597x x  - 4
│ │ │ │ -0009aee0: 3236 3636 7820 7820 202b 2020 7c0a 7c20  2666x x  +  |.| 
│ │ │ │ -0009aef0: 2020 2020 2031 2033 2034 2020 2020 2020       1 3 4      
│ │ │ │ -0009af00: 2020 2032 2033 2034 2020 2020 2020 2020     2 3 4        
│ │ │ │ -0009af10: 2033 2034 2020 2020 2020 2020 2030 2034   3 4         0 4
│ │ │ │ -0009af20: 2020 2020 2020 2020 2031 2034 2020 2020           1 4    
│ │ │ │ -0009af30: 2020 2020 2032 2034 2020 2020 7c0a 7c2d       2 4    |.|-
│ │ │ │ -0009af40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0009aea0: 2020 7c0a 7c34 3235 3532 7820 7820 7820    |.|42552x x x 
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│ │ │ │ +0009aed0: 3938 7820 7820 202d 2034 3635 3937 7820  98x x  - 46597x 
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│ │ │ │ +0009aef0: 2020 7c0a 7c20 2020 2020 2031 2033 2034    |.|      1 3 4
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│ │ │ │ +0009af10: 2020 2020 2020 2033 2034 2020 2020 2020         3 4      
│ │ │ │ +0009af20: 2020 2030 2034 2020 2020 2020 2020 2031     0 4         1
│ │ │ │ +0009af30: 2034 2020 2020 2020 2020 2032 2034 2020   4         2 4  
│ │ │ │ +0009af40: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
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│ │ │ │  0009af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0009af90: 2020 2020 2020 2032 2020 2020 2020 2020         2        
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│ │ │ │ +0009af90: 2d2d 7c0a 7c20 2020 2020 2020 2032 2020  --|.|        2  
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│ │ │ │  0009afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009afd0: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ -0009afe0: 3431 3430 7820 7820 202d 2032 3932 3036  4140x x  - 29206
│ │ │ │ -0009aff0: 7820 202d 2038 3834 3035 7820 7820 7820  x  - 88405x x x 
│ │ │ │ -0009b000: 202d 2031 3537 3730 7820 7820 202b 2031   - 15770x x  + 1
│ │ │ │ -0009b010: 3830 3539 7820 7820 7820 202d 2031 3433  8059x x x  - 143
│ │ │ │ -0009b020: 3632 7820 7820 7820 202b 2020 7c0a 7c20  62x x x  +  |.| 
│ │ │ │ -0009b030: 2020 2020 2033 2034 2020 2020 2020 2020       3 4        
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│ │ │ │ -0009b050: 2020 2020 2020 2020 2031 2035 2020 2020           1 5    
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│ │ │ │ -0009b490: 3520 3a20 6465 7363 7269 6265 2050 6869  5 : describe Phi
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│ │ │ │ +0009b480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0009b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0009b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b520: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
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│ │ │ │ -0009b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009b580: 2020 2020 736f 7572 6365 2076 6172 6965      source varie
│ │ │ │ -0009b590: 7479 3a20 5050 5e35 2020 2020 2020 2020  ty: PP^5        
│ │ │ │ +0009b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b530: 2020 7c0a 7c6f 3520 3d20 7261 7469 6f6e    |.|o5 = ration
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│ │ │ │ +0009b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b580: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +0009b590: 2076 6172 6965 7479 3a20 5050 5e35 2020   variety: PP^5  
│ │ │ │  0009b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b5c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009b5d0: 2020 2020 7461 7267 6574 2076 6172 6965      target varie
│ │ │ │ -0009b5e0: 7479 3a20 5050 5e35 2020 2020 2020 2020  ty: PP^5        
│ │ │ │ +0009b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b5d0: 2020 7c0a 7c20 2020 2020 7461 7267 6574    |.|     target
│ │ │ │ +0009b5e0: 2076 6172 6965 7479 3a20 5050 5e35 2020   variety: PP^5  
│ │ │ │  0009b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b610: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ -0009b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b620: 2020 7c0a 7c20 2020 2020 636f 6566 6669    |.|     coeffi
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│ │ │ │ +0009b640: 3930 3138 3120 2020 2020 2020 2020 2020  90181           
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│ │ │ │ -0009b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0009b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0009b6c0: 3620 3a20 6465 7363 7269 6265 2050 6869  6 : describe Phi
│ │ │ │ -0009b6d0: 2720 2020 2020 2020 2020 2020 2020 2020  '               
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│ │ │ │ +0009b6c0: 2d2d 2b0a 7c69 3620 3a20 6465 7363 7269  --+.|i6 : descri
│ │ │ │ +0009b6d0: 6265 2050 6869 2720 2020 2020 2020 2020  be Phi'         
│ │ │ │  0009b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b700: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ +0009b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0009b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b750: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009b760: 3620 3d20 7261 7469 6f6e 616c 206d 6170  6 = rational map
│ │ │ │ -0009b770: 2064 6566 696e 6564 2062 7920 666f 726d   defined by form
│ │ │ │ -0009b780: 7320 6f66 2064 6567 7265 6520 3220 2020  s of degree 2   
│ │ │ │ -0009b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009b7b0: 2020 2020 736f 7572 6365 2076 6172 6965      source varie
│ │ │ │ -0009b7c0: 7479 3a20 636f 6d70 6c65 7465 2069 6e74  ty: complete int
│ │ │ │ -0009b7d0: 6572 7365 6374 696f 6e20 6f66 2074 7970  ersection of typ
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│ │ │ │ -0009b7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009b800: 2020 2020 7461 7267 6574 2076 6172 6965      target varie
│ │ │ │ -0009b810: 7479 3a20 5050 5e35 2020 2020 2020 2020  ty: PP^5        
│ │ │ │ +0009b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b760: 2020 7c0a 7c6f 3620 3d20 7261 7469 6f6e    |.|o6 = ration
│ │ │ │ +0009b770: 616c 206d 6170 2064 6566 696e 6564 2062  al map defined b
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│ │ │ │ +0009b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b7b0: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +0009b7c0: 2076 6172 6965 7479 3a20 636f 6d70 6c65   variety: comple
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│ │ │ │ +0009b7e0: 6f66 2074 7970 6520 2832 2c33 2920 696e  of type (2,3) in
│ │ │ │ +0009b7f0: 2050 505e 3520 2020 2020 2020 2020 2020   PP^5           
│ │ │ │ +0009b800: 2020 7c0a 7c20 2020 2020 7461 7267 6574    |.|     target
│ │ │ │ +0009b810: 2076 6172 6965 7479 3a20 5050 5e35 2020   variety: PP^5  
│ │ │ │  0009b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b840: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009b850: 2020 2020 636f 6566 6669 6369 656e 7420      coefficient 
│ │ │ │ -0009b860: 7269 6e67 3a20 5a5a 2f31 3930 3138 3120  ring: ZZ/190181 
│ │ │ │ -0009b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b850: 2020 7c0a 7c20 2020 2020 636f 6566 6669    |.|     coeffi
│ │ │ │ +0009b860: 6369 656e 7420 7269 6e67 3a20 5a5a 2f31  cient ring: ZZ/1
│ │ │ │ +0009b870: 3930 3138 3120 2020 2020 2020 2020 2020  90181           
│ │ │ │  0009b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b890: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0009b8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0009b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b8a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0009b8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009b8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ -0009b8f0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -0009b900: 0a0a 2020 2a20 2a6e 6f74 6520 5261 7469  ..  * *note Rati
│ │ │ │ -0009b910: 6f6e 616c 4d61 7020 7c7c 2049 6465 616c  onalMap || Ideal
│ │ │ │ -0009b920: 3a20 5261 7469 6f6e 616c 4d61 7020 7c7c  : RationalMap ||
│ │ │ │ -0009b930: 2049 6465 616c 2c20 2d2d 2072 6573 7472   Ideal, -- restr
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│ │ │ │ -0009b950: 7261 7469 6f6e 616c 206d 6170 0a0a 5761  rational map..Wa
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│ │ │ │ -0009b980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0009b990: 0a20 202a 202a 6e6f 7465 2052 6174 696f  .  * *note Ratio
│ │ │ │ -0009b9a0: 6e61 6c4d 6170 207c 2049 6465 616c 3a20  nalMap | Ideal: 
│ │ │ │ -0009b9b0: 5261 7469 6f6e 616c 4d61 7020 7c20 4964  RationalMap | Id
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│ │ │ │ -0009b9f0: 6174 696f 6e61 6c4d 6170 207c 2052 696e  ationalMap | Rin
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│ │ │ │ -0009ba10: 4d61 7020 7c20 5269 6e67 456c 656d 656e  Map | RingElemen
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│ │ │ │ -0009ba70: 6d5a 5a5f 636d 5a5a 5f72 702c 2050 7265  mZZ_cmZZ_rp, Pre
│ │ │ │ -0009ba80: 763a 2052 6174 696f 6e61 6c4d 6170 207c  v: RationalMap |
│ │ │ │ -0009ba90: 2049 6465 616c 2c20 5570 3a20 546f 700a   Ideal, Up: Top.
│ │ │ │ -0009baa0: 0a52 6174 696f 6e61 6c4d 6170 207c 7c20  .RationalMap || 
│ │ │ │ -0009bab0: 4964 6561 6c20 2d2d 2072 6573 7472 6963  Ideal -- restric
│ │ │ │ -0009bac0: 7469 6f6e 206f 6620 6120 7261 7469 6f6e  tion of a ration
│ │ │ │ -0009bad0: 616c 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a  al map.*********
│ │ │ │ +0009b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0009b910: 6520 5261 7469 6f6e 616c 4d61 7020 7c7c  e RationalMap ||
│ │ │ │ +0009b920: 2049 6465 616c 3a20 5261 7469 6f6e 616c   Ideal: Rational
│ │ │ │ +0009b930: 4d61 7020 7c7c 2049 6465 616c 2c20 2d2d  Map || Ideal, --
│ │ │ │ +0009b940: 2072 6573 7472 6963 7469 6f6e 206f 6620   restriction of 
│ │ │ │ +0009b950: 610a 2020 2020 7261 7469 6f6e 616c 206d  a.    rational m
│ │ │ │ +0009b960: 6170 0a0a 5761 7973 2074 6f20 7573 6520  ap..Ways to use 
│ │ │ │ +0009b970: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ +0009b980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0009b990: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +0009b9a0: 2052 6174 696f 6e61 6c4d 6170 207c 2049   RationalMap | I
│ │ │ │ +0009b9b0: 6465 616c 3a20 5261 7469 6f6e 616c 4d61  deal: RationalMa
│ │ │ │ +0009b9c0: 7020 7c20 4964 6561 6c2c 202d 2d20 7265  p | Ideal, -- re
│ │ │ │ +0009b9d0: 7374 7269 6374 696f 6e20 6f66 2061 0a20  striction of a. 
│ │ │ │ +0009b9e0: 2020 2072 6174 696f 6e61 6c20 6d61 700a     rational map.
│ │ │ │ +0009b9f0: 2020 2a20 2252 6174 696f 6e61 6c4d 6170    * "RationalMap
│ │ │ │ +0009ba00: 207c 2052 696e 6722 0a20 202a 2022 5261   | Ring".  * "Ra
│ │ │ │ +0009ba10: 7469 6f6e 616c 4d61 7020 7c20 5269 6e67  tionalMap | Ring
│ │ │ │ +0009ba20: 456c 656d 656e 7422 0a1f 0a46 696c 653a  Element"...File:
│ │ │ │ +0009ba30: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
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│ │ │ │ +0009ba50: 207c 7c20 4964 6561 6c2c 204e 6578 743a   || Ideal, Next:
│ │ │ │ +0009ba60: 2072 6174 696f 6e61 6c4d 6170 5f6c 7049   rationalMap_lpI
│ │ │ │ +0009ba70: 6465 616c 5f63 6d5a 5a5f 636d 5a5a 5f72  deal_cmZZ_cmZZ_r
│ │ │ │ +0009ba80: 702c 2050 7265 763a 2052 6174 696f 6e61  p, Prev: Rationa
│ │ │ │ +0009ba90: 6c4d 6170 207c 2049 6465 616c 2c20 5570  lMap | Ideal, Up
│ │ │ │ +0009baa0: 3a20 546f 700a 0a52 6174 696f 6e61 6c4d  : Top..RationalM
│ │ │ │ +0009bab0: 6170 207c 7c20 4964 6561 6c20 2d2d 2072  ap || Ideal -- r
│ │ │ │ +0009bac0: 6573 7472 6963 7469 6f6e 206f 6620 6120  estriction of a 
│ │ │ │ +0009bad0: 7261 7469 6f6e 616c 206d 6170 0a2a 2a2a  rational map.***
│ │ │ │  0009bae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0009baf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0009bb00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 5379  ************..Sy
│ │ │ │ -0009bb10: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ -0009bb20: 0a20 202a 204f 7065 7261 746f 723a 202a  .  * Operator: *
│ │ │ │ -0009bb30: 6e6f 7465 207c 7c3a 2028 4d61 6361 756c  note ||: (Macaul
│ │ │ │ -0009bb40: 6179 3244 6f63 297c 7c2c 0a20 202a 2055  ay2Doc)||,.  * U
│ │ │ │ -0009bb50: 7361 6765 3a20 0a20 2020 2020 2020 2050  sage: .        P
│ │ │ │ -0009bb60: 6869 207c 7c20 4a0a 2020 2a20 496e 7075  hi || J.  * Inpu
│ │ │ │ -0009bb70: 7473 3a0a 2020 2020 2020 2a20 5068 692c  ts:.      * Phi,
│ │ │ │ -0009bb80: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ -0009bb90: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ -0009bba0: 6170 2c2c 2024 5c70 6869 3a58 205c 6461  ap,, $\phi:X \da
│ │ │ │ -0009bbb0: 7368 7269 6768 7461 7272 6f77 2059 240a  shrightarrow Y$.
│ │ │ │ -0009bbc0: 2020 2020 2020 2a20 4a2c 2061 6e20 2a6e        * J, an *n
│ │ │ │ -0009bbd0: 6f74 6520 6964 6561 6c3a 2028 4d61 6361  ote ideal: (Maca
│ │ │ │ -0009bbe0: 756c 6179 3244 6f63 2949 6465 616c 2c2c  ulay2Doc)Ideal,,
│ │ │ │ -0009bbf0: 2061 2068 6f6d 6f67 656e 656f 7573 2069   a homogeneous i
│ │ │ │ -0009bc00: 6465 616c 206f 6620 610a 2020 2020 2020  deal of a.      
│ │ │ │ -0009bc10: 2020 7375 6276 6172 6965 7479 2024 5a5c    subvariety $Z\
│ │ │ │ -0009bc20: 7375 6273 6574 2059 240a 2020 2a20 4f75  subset Y$.  * Ou
│ │ │ │ -0009bc30: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ -0009bc40: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ -0009bc50: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ -0009bc60: 2c2c 2074 6865 2072 6573 7472 6963 7469  ,, the restricti
│ │ │ │ -0009bc70: 6f6e 206f 6620 245c 7068 6924 2074 6f0a  on of $\phi$ to.
│ │ │ │ -0009bc80: 2020 2020 2020 2020 247b 5c70 6869 7d5e          ${\phi}^
│ │ │ │ -0009bc90: 7b28 2d31 297d 205a 242c 2024 7b7b 5c70  {(-1)} Z$, ${{\p
│ │ │ │ -0009bca0: 6869 7d7c 7d5f 7b7b 5c70 6869 7d5e 7b28  hi}|}_{{\phi}^{(
│ │ │ │ -0009bcb0: 2d31 297d 205a 7d3a 207b 5c70 6869 7d5e  -1)} Z}: {\phi}^
│ │ │ │ -0009bcc0: 7b28 2d31 297d 205a 0a20 2020 2020 2020  {(-1)} Z.       
│ │ │ │ -0009bcd0: 205c 6461 7368 7269 6768 7461 7272 6f77   \dashrightarrow
│ │ │ │ -0009bce0: 205a 240a 0a44 6573 6372 6970 7469 6f6e   Z$..Description
│ │ │ │ -0009bcf0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ -0009bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0009bb00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0009bb10: 2a2a 0a0a 5379 6e6f 7073 6973 0a3d 3d3d  **..Synopsis.===
│ │ │ │ +0009bb20: 3d3d 3d3d 3d0a 0a20 202a 204f 7065 7261  =====..  * Opera
│ │ │ │ +0009bb30: 746f 723a 202a 6e6f 7465 207c 7c3a 2028  tor: *note ||: (
│ │ │ │ +0009bb40: 4d61 6361 756c 6179 3244 6f63 297c 7c2c  Macaulay2Doc)||,
│ │ │ │ +0009bb50: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +0009bb60: 2020 2020 2050 6869 207c 7c20 4a0a 2020       Phi || J.  
│ │ │ │ +0009bb70: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ +0009bb80: 2a20 5068 692c 2061 202a 6e6f 7465 2072  * Phi, a *note r
│ │ │ │ +0009bb90: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ +0009bba0: 696f 6e61 6c4d 6170 2c2c 2024 5c70 6869  ionalMap,, $\phi
│ │ │ │ +0009bbb0: 3a58 205c 6461 7368 7269 6768 7461 7272  :X \dashrightarr
│ │ │ │ +0009bbc0: 6f77 2059 240a 2020 2020 2020 2a20 4a2c  ow Y$.      * J,
│ │ │ │ +0009bbd0: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a   an *note ideal:
│ │ │ │ +0009bbe0: 2028 4d61 6361 756c 6179 3244 6f63 2949   (Macaulay2Doc)I
│ │ │ │ +0009bbf0: 6465 616c 2c2c 2061 2068 6f6d 6f67 656e  deal,, a homogen
│ │ │ │ +0009bc00: 656f 7573 2069 6465 616c 206f 6620 610a  eous ideal of a.
│ │ │ │ +0009bc10: 2020 2020 2020 2020 7375 6276 6172 6965          subvarie
│ │ │ │ +0009bc20: 7479 2024 5a5c 7375 6273 6574 2059 240a  ty $Z\subset Y$.
│ │ │ │ +0009bc30: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +0009bc40: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ +0009bc50: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +0009bc60: 6e61 6c4d 6170 2c2c 2074 6865 2072 6573  nalMap,, the res
│ │ │ │ +0009bc70: 7472 6963 7469 6f6e 206f 6620 245c 7068  triction of $\ph
│ │ │ │ +0009bc80: 6924 2074 6f0a 2020 2020 2020 2020 247b  i$ to.        ${
│ │ │ │ +0009bc90: 5c70 6869 7d5e 7b28 2d31 297d 205a 242c  \phi}^{(-1)} Z$,
│ │ │ │ +0009bca0: 2024 7b7b 5c70 6869 7d7c 7d5f 7b7b 5c70   ${{\phi}|}_{{\p
│ │ │ │ +0009bcb0: 6869 7d5e 7b28 2d31 297d 205a 7d3a 207b  hi}^{(-1)} Z}: {
│ │ │ │ +0009bcc0: 5c70 6869 7d5e 7b28 2d31 297d 205a 0a20  \phi}^{(-1)} Z. 
│ │ │ │ +0009bcd0: 2020 2020 2020 205c 6461 7368 7269 6768         \dashrigh
│ │ │ │ +0009bce0: 7461 7272 6f77 205a 240a 0a44 6573 6372  tarrow Z$..Descr
│ │ │ │ +0009bcf0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0009bd00: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ==..+-----------
│ │ │ │  0009bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0009bd50: 3120 3a20 5035 203d 205a 5a2f 3139 3031  1 : P5 = ZZ/1901
│ │ │ │ -0009bd60: 3831 5b78 5f30 2e2e 785f 355d 2020 2020  81[x_0..x_5]    
│ │ │ │ -0009bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0009bd50: 2d2d 2b0a 7c69 3120 3a20 5035 203d 205a  --+.|i1 : P5 = Z
│ │ │ │ +0009bd60: 5a2f 3139 3031 3831 5b78 5f30 2e2e 785f  Z/190181[x_0..x_
│ │ │ │ +0009bd70: 355d 2020 2020 2020 2020 2020 2020 2020  5]              
│ │ │ │  0009bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bd90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bda0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0009bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bde0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009bdf0: 3120 3d20 5035 2020 2020 2020 2020 2020  1 = P5          
│ │ │ │ +0009bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bdf0: 2020 7c0a 7c6f 3120 3d20 5035 2020 2020    |.|o1 = P5    
│ │ │ │  0009be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009be30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009be40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0009be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009be80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009be90: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269  1 : PolynomialRi
│ │ │ │ -0009bea0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +0009be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009be90: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f    |.|o1 : Polyno
│ │ │ │ +0009bea0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │  0009beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bed0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0009bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0009bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bee0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0009bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009bf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0009bf30: 3220 3a20 5068 6920 3d20 7261 7469 6f6e  2 : Phi = ration
│ │ │ │ -0009bf40: 616c 4d61 7020 7b78 5f34 5e32 2d78 5f33  alMap {x_4^2-x_3
│ │ │ │ -0009bf50: 2a78 5f35 2c78 5f32 2a78 5f34 2d78 5f31  *x_5,x_2*x_4-x_1
│ │ │ │ -0009bf60: 2a78 5f35 2c78 5f32 2a78 5f33 2d78 5f31  *x_5,x_2*x_3-x_1
│ │ │ │ -0009bf70: 2a78 5f34 2c78 5f32 5e32 2d78 7c0a 7c20  *x_4,x_2^2-x|.| 
│ │ │ │ -0009bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0009bf30: 2d2d 2b0a 7c69 3220 3a20 5068 6920 3d20  --+.|i2 : Phi = 
│ │ │ │ +0009bf40: 7261 7469 6f6e 616c 4d61 7020 7b78 5f34  rationalMap {x_4
│ │ │ │ +0009bf50: 5e32 2d78 5f33 2a78 5f35 2c78 5f32 2a78  ^2-x_3*x_5,x_2*x
│ │ │ │ +0009bf60: 5f34 2d78 5f31 2a78 5f35 2c78 5f32 2a78  _4-x_1*x_5,x_2*x
│ │ │ │ +0009bf70: 5f33 2d78 5f31 2a78 5f34 2c78 5f32 5e32  _3-x_1*x_4,x_2^2
│ │ │ │ +0009bf80: 2d78 7c0a 7c20 2020 2020 2020 2020 2020  -x|.|           
│ │ │ │  0009bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bfc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009bfd0: 3220 3d20 2d2d 2072 6174 696f 6e61 6c20  2 = -- rational 
│ │ │ │ -0009bfe0: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +0009bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bfd0: 2020 7c0a 7c6f 3220 3d20 2d2d 2072 6174    |.|o2 = -- rat
│ │ │ │ +0009bfe0: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │  0009bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c010: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c030: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │ +0009c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009c020: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0009c030: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │  0009c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c060: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009c070: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -0009c080: 282d 2d2d 2d2d 2d5b 7820 2c20 7820 2c20  (------[x , x , 
│ │ │ │ -0009c090: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ -0009c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c0b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c0d0: 2031 3930 3138 3120 2030 2020 2031 2020   190181  0   1  
│ │ │ │ -0009c0e0: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ -0009c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c100: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0009c080: 3a20 5072 6f6a 282d 2d2d 2d2d 2d5b 7820  : Proj(------[x 
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│ │ │ │  0009c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009c160: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
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│ │ │ │ -0009c1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0009c1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ +0009c200: 2020 7c0a 7c20 2020 2020 6465 6669 6e69    |.|     defini
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TRUNCATED DUE TO SIZE LIMIT: 10485760 bytes